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arXiv:1403.5955v1 [math.AP] 24 Mar 2014EXISTENCE RESULTS FOR SOME DAMPED SECOND-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS TOKADIAGANA InMemoryofProf. YahyaOuldHamidoune Abstract. In this paper we make a subtle use of operator theory techniq ues and the well- knownSchauderfixed-pointprincipletoestablish theexist enceofpseudo-almostautomor- phic solutions to some second-order damped integro-di fferential equations with pseudo- almost automorphic coe fficients. In order to illustrate our main results, we will stud y the existence of pseudo-almost automorphic solutions to a s tructurally damped plate-like boundary value problem. 1. Introduction Integro-differentialequationsplayanimportantrolewhenitcomestomo delingvarious natural phenomena, see, e.g., [ 9,10,15,27,28,29,34,43,49,52,53,57,60,61,62, 71]. In recent years, noteworthy progress has been made in stud ying the existence of periodic,almostperiodic,almostautomorphic,pseudo-al mostperiodic,andpseudo-almost automorphicsolutionsto first-orderintegro-di fferentialequations,see, e.g., [ 2,17,22,23, 24,25,35,36,37,44,45,60,61,62]. The most popular method used to deal with the existence of solutions to those first-order integro-di fferentialequations consists of the so- calledmethodofresolvents,see, e.g.,[ 1,14,15,36,37,44,45]. Fixα∈(0,1). LetHbe an infinite dimensional separable Hilbert space over the fi eld ofcomplexnumbersequippedwith the innerproductandnormg ivenrespectivelyby /an}bracketle{t·,·/an}bracketri}ht and/bardbl·/bardbl. The purpose of this paper consists of making use of a new appr oach to study theexistenceofpseudo-almostautomorphicsolutionstoth eclassofdampedsecond-order Volterraintegro-differentialequationsgivenby d2ϕ dt2+Bdϕ dt+Aϕ=/integraldisplayt −∞C(t−s)ϕ(s)ds+f(t,ϕ), (1.1) whereA:D(A)⊂H/mapsto→His an unbounded self-adjoint linear operator whose spectru m consistsofisolatedeigenvaluesgivenby 0<λ1<λ2<...<λ n→∞ asn→∞with eacheigenvaluehavinga finite multiplicity γjequalsto themultiplicityof thecorrespondingeigenspace, B:D(B)⊂H/mapsto→His a positiveself-adjointlinearoperator suchthatthereexisttwoconstants γ1,γ2>0andsuchthat γ1Aα≤B≤γ2Aα, thatis, γ1/an}bracketle{tAαϕ,ϕ/an}bracketri}ht≤/an}bracketle{tBϕ,ϕ/an}bracketri}ht≤γ2/an}bracketle{tAαϕ,ϕ/an}bracketri}ht for allϕ∈D(B1 2)=D(Aα 2), the mappings C(t) :D(A)⊂H/mapsto→Hconsist of (possibly unbounded)linear operators for each t∈R, and the function f:R×H/mapsto→His pseudo- almostautomorphicin thefirst variableuniformlyinthesec ondone. 2000Mathematics Subject Classification. 12H20; 45J05;43A60; 35L71; 35L10;37L05. Key words and phrases. second-order integro-di fferential equation; pseudo-almost automorphic; Schauder fixed point theorem; hyperbolic semigroup; structurally da mped plate-like boundary value problem. 12 TOKADIAGANA EquationsoftypeEq. ( 1.1)ariseveryofteninthestudyofnaturalphenomenainwhich acertainmemorye ffectistakenintoconsideration,see,e.g.,[ 3,6,46,50,51]. In[3,6]for instance,equationsoftypeEq. ( 1.1)appearedinthestudyofa viscoelasticwaveequation withmemory. The existence, uniqueness, and asymptotic behavior of solu tions to Eq. ( 1.1) have widely been studied, see, e.g., [ 3,6,7,8,41,42,46,50,51,54,55,56]. However, to the best of our knowledge, the existence of pseudo-almost au tomorphic solutions to Eq. (1.1) is an untreated original problem with important applicati ons, which constitutes the mainmotivationofthispaper. In this paper,we are interestedin the special case B=2γAαwhereγ >0 is a constant, thatis, d2ϕ dt2+2γAαdϕ dt+Aϕ=/integraldisplayt −∞C(t−s)ϕ(s)ds+f(t,ϕ),t∈R. (1.2) It should be mentioned that various versions of Eq. ( 1.2) have been investigated in the literature, see, e.g., Chen and Triggiani [ 11,12], Huang [ 30,31,32,33], and Xiao and Liang[65,66,67,68,69,70]. Considerthepolynomial Qγ nassociatedwiththeleft handsideofEq. ( 1.2),thatis, Qγ n(ρ) :=ρ2+2γλα nρ+λn (1.3) anddenoteits rootsby ρn 1:=dn+ienandρn 2:=rn+isnforalln≥1. Intherestofthepaper,wesupposethattheroots ρn 1andρn 2satisfy:ρn 1/nequalρn 2foralln≥1 andthat thefollowingcrucialassumptionholds: thereexis tsδ0>0suchthat sup n≥1/bracketleftig max(dn,rn)/bracketrightig ≤−δ0<0. (1.4) In order to investigate the existence of pseudo-almost auto morphic solutions to Eq. (1.2), our strategy consists of rewriting it as a first-order inte gro-differential equation in the product spaceE1 2:=D(A1 2)×Hand then study the existence of pseudo-almost auto- morphic solutions to the obtained first-order integro-di fferentialequation with the help of SchauderfixedpointprincipleandthengobacktoEq. ( 1.2). Recall thatthe innerproductof E1 2isdefinedasfollows: /parenleftigg/parenleftiggϕ1 ϕ2/parenrightigg ,ψ1 ψ2/parenrightigg E1 2:=/an}bracketle{tA1 2ϕ1,A1 2ψ1/an}bracketri}ht+/an}bracketle{tϕ2,ψ2/an}bracketri}ht forallϕ1,ψ1∈D(A1 2) andϕ2,ψ2∈H. Itscorrespondingnormwill bedenoted /bardbl·/bardblE1 2. Letting Φ:=ϕ ϕ′∈E1 2, thenEq. ( 1.2)canberewrittenin thefollowingform (1.5)dΦ dt=AΦ+/integraldisplayt −∞C(t−s)Φ(s)ds+F(t,Φ(t)),t∈R,EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 3 whereA,Caretheoperatormatricesdefinedby A=0I −A−Aα,C=C 0, withdomain D(A)=D(A)×[D(A1 2)∩D(Aα)]=D(C)(D(A)=D(A)×D(A1 2)if0<α≤1 2 andD(A)=D(A)×D(Aα) if1 2≤α <1), and the function F:R×E1 2/mapsto→E:=H×His givenby F(t,Φ)=0 f(t,ϕ). In order to investigate Eq. ( 1.5), we study the first-order di fferential equation in the spaceE1 2givenby, dϕ dt=(A+B)ϕ+F(t,ϕ),t∈R, (1.6) whereB:C(R,D(A))/mapsto→E1 2isthelinearoperatordefinedby Bϕ:=/integraldisplayt −∞C(t−s)ϕ(s)ds, ϕ∈C(R,D(A)) (1.7) withC(R,D(A))beingthecollectionofall continuousfunctionsfrom RintoD(A). In order to study the existence of solutionsto Eq. ( 1.6), we will make extensive use of hyperbolicsemigrouptoolsandfractionalpowersofoperat ors,andthatthelinearoperator Bsatisfiessomeadditionalassumptions. Ourexistenceresul twillthenbeobtainedthrough theuseofthewell-knownSchauderfixed-pointtheorem. Obvi ously,onceweestablishthe sought existence results for Eq. ( 1.6), then we can easily go back to Eq. ( 1.2) notably throughEq. ( 1.7). The concept of pseudo almost automorphy is a powerful notion introduced in the lit- erature by Liang et al.[39,40,63,64]. This concept has recently generated several de- velopmentsandextensions,whichhavebeensummarizedin a n ewbookbyDiagana[ 17]. The existence of almost periodic and asymptotically almost periodic solutions to integro- differential equations of the form Eq. ( 1.5) in a general context has recently been estab- lishedin[ 36,37]. Similarly,in[ 45],theexistenceofpseudo-almostautomorphicsolutions to Eq. (1.5)was studied. Themain methodused in the above-mentionedpa persare resol- vents operators. However, to the best of our knowledge, the e xistence of pseudo-almost automorphic solutions to Eq. ( 1.2) is an important untreated topic with some interesting applications. Amongotherthings, we will make extensiveus e of theSchauderfixedpoint toderivesomesufficientconditionsfortheexistenceofpseudo-almostautomo rphic(mild) solutionsto ( 1.6)andthentoEq. ( 1.2). 2. Preliminaries Some of the basic results discussed in this section are mainl ytaken fromthe following recentpapersbyDiagana[ 18,21]. Inthispaper,Hdenoteaninfinitedimensionalseparable Hilbertspaceoverthefieldofcomplexnumbersequippedwith theinnerproductandnorm givenrespectivelyby /an}bracketle{t·,·/an}bracketri}htand/bardbl·/bardbl. IfAisalinearoperatoruponaBanachspace( X,/bardbl·/bardbl), then the notations D(A),ρ(A),σ(A),N(A), andR(A) stand respectively for the domain, resolvent, spectrum, kernel, and the range of A. Similarly, if A:D:=D(A)⊂X/mapsto→Xis4 TOKADIAGANA a closed linear operator on a Banach space, one denotes its gr aph norm by/bardbl·/bardblDdefined by/bardblx/bardblD:=/bardblx/bardbl+/bardblAx/bardblfor allx∈D. From the closedness of A, one can easily see that (D,/bardbl·/bardblD) is a Banach space. Moreover, one sets R(λ,L) :=(λI−L)−1for allλ∈ρ(A). We setQ=I−Pfor a projection P. IfY,Zare Banach spaces, then the space B(Y,Z) denotes the collection of all bounded linear operators from YintoZequipped with its natural uniform operator topology /bardbl·/bardblB(Y,Z). We also set B(Y)=B(Y,Y). IfK⊂Xis a subset,we let coKdenotetheclosed convexhullof K. Additionally,Twill denotethe set definedby,T:={(t,s)∈R×R:t≥s}.If(X,/bardbl·/bardblX)and(Y,/bardbl·/bardblY)areBanachspaces,their productX×Y:={(x,y) :x∈X,y∈Y}is also a Banach when it is equipped with the normgivenby /bardbl(x,y)/bardblX×Y=/radicalig /bardblx/bardbl2 X+/bardbly/bardbl2 Yforall (x,y)∈X×Y. In this paper if β≥0, then we setEβ:=D(Aβ)×H, andE:=H×Hand equip them with their corresponding topologies /bardbl·/bardblEβand/bardbl·/bardblE. Recall that D(Aβ) will be equipped withthenormdefinedby, /bardblϕ/bardblβ:=/bardblAβϕ/bardblforallϕ∈D(Aβ). In the sequel, A:D(A)⊂H/mapsto→Hstands for a self-adjoint (possibly unbounded) linear operator on the Hilbert space Hwhose spectrum consists of isolated eigenvalues 0< λ1< λ2< ... < λ n→∞with each eigenvalue having a finite multiplicity γjequals to the multiplicity of the correspondingeigenspace. Let {ek j}be a (complete) orthonormal sequence of eigenvectors associated with the eigenvalues {λj}j≥1. Clearly, for each u∈ D(A),whereif u∈D(A) :=/braceleftig u∈H:∞/summationdisplay j=1λ2 j/bardblEju/bardbl2<∞/bracerightig ,thenAu=∞/summationdisplay j=1λjγj/summationdisplay k=1/an}bracketle{tu,ek j/an}bracketri}htek j=∞/summationdisplay j=1λjEju withEju=/summationtextγj k=1/an}bracketle{tu,ek j/an}bracketri}htek j.Note that{Ej}j≥1is a sequence of orthogonalprojectionson H. Moreover,each u∈Hcan written as follows: u=/summationtext∞ j=1Eju.It should also be mentioned thattheoperator−Aistheinfinitesimalgeneratorofananalyticsemigroup {S(t)}t≥0,which isexplicitlyexpressedintermsofthoseorthogonalprojec tionsEjby,forall u∈H, S(t)u=∞/summationdisplay j=1e−λjtEju whichin particularisexponentiallystable as /bardblS(t)/bardbl≤e−λ1t forallt≥0. 3. SectorialLinearOperators Thebasicresultsdiscussedinthissectionaremainlytaken fromDiagana[ 17,20]. Definition 3.1. A linearoperator B:D(B)⊂X/mapsto→X(notnecessarilydenselydefined)on a BanachspaceXissaid to be sectorialif the followinghold: thereexist con stantsω∈R, θ∈/parenleftbiggπ 2,π/parenrightbigg ,andM>0 suchthatρ(B)⊃Sθ,ω, Sθ,ω:=/braceleftig λ∈C:λ/nequalω,|arg(λ−ω)|<θ/bracerightig ,and (3.1) /bardblR(λ,B)/bardbl≤M |λ−ω|, λ∈Sθ,ω. (3.2)EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 5 Example 3.2. Letp≥1 and letΩ⊂Rdbe open bounded subset with C2boundary∂Ω. LetX:=Lp(Ω)betheLebesguespaceequippedwith thenorm, /bardbl·/bardblpdefinedby, /bardblϕ/bardblp=/parenleftig/integraldisplay Ω|ϕ(x)|pdx/parenrightig1/p. Definethe operator Aasfollows: D(B)=W2,p(Ω)∩W1,p 0(Ω),B(ϕ)=∆ϕ,∀ϕ∈D(B), where∆=d/summationdisplay k=1∂2 ∂x2 kistheLaplaceoperator. Itcanbecheckedthattheoperator Bissectorial onLp(Ω). It is well-known [ 47] that ifB:D(B)⊂X/mapsto→Xis a sectorial linear operator, then it generatesananalyticsemigroup( T(t))t≥0,whichmaps(0 ,∞)intoB(X)andsuchthatthere existM0,M1>0 with /bardblT(t)/bardbl≤M0eωt,t>0, (3.3) /bardblt(A−ω)T(t)/bardbl≤M1eωt,t>0. (3.4) In this paper, we suppose that the semigroup ( T(t))t≥0is hyperbolic,that is, there exist a projection Pand constants M,δ >0 such that T(t) commutes with P,N(P) is invariant withrespectto T(t),T(t) :R(Q)/mapsto→R(Q)isinvertible,andthefollowinghold (3.5) /bardblT(t)Px/bardbl≤Me−δt/bardblx/bardblfort≥0, (3.6) /bardblT(t)Qx/bardbl≤Meδt/bardblx/bardblfort≤0, whereQ:=I−Pand,fort≤0,T(t) :=(T(−t))−1. Recall that the analytic semigroup( T(t))t≥0associated with Bis hyperbolicif and only ifσ(B)∩iR=∅,see detailsin [ 26, Proposition1.15,pp.305]. Definition 3.3. Letα∈(0,1). A Banach space ( Xα,/bardbl·/bardblα) is said to be an intermediate space between D(B) andX, or a space of class Jα, ifD(B)⊂Xα⊂Xand there is a constantc>0suchthat (3.7) /bardblx/bardblα≤c/bardblx/bardbl1−α/bardblx/bardblα B,x∈D(B). Concreteexamplesof XαincludeD((−Bα))forα∈(0,1),thedomainsofthefractional powersof B, the real interpolationspaces DB(α,∞),α∈(0,1),defined as the space of all x∈Xsuchthat, [x]α=sup 0<t≤1/bardblt1−αBT(t)x/bardbl<∞ withthenorm /bardblx/bardblα=/bardblx/bardbl+[x]α, the abstract H¨ older spaces DB(α) :=D(B)/bardbl./bardblαas well as the complex interpolation spaces [X,D(B)]α. Fora hyperbolicanalyticsemigroup( T(t))t≥0, onecaneasilycheckthat similarestima- tions as both Eq. ( 3.5) and Eq. ( 3.6) still hold with the α-norms/bardbl·/bardblα. In fact, as the part ofAinR(Q)isbounded,it followsfromEq. ( 3.6)that /bardblBT(t)Qx/bardbl≤C′eδt/bardblx/bardblfort≤0. Hence,fromEq. ( 3.7)thereexistsaconstant c(α)>0suchthat (3.8) /bardblT(t)Qx/bardblα≤c(α)eδt/bardblx/bardblfort≤0.6 TOKADIAGANA Inadditiontothe above,thefollowingholds /bardblT(t)Px/bardblα≤/bardblT(1)/bardblB(X,Xα)/bardblT(t−1)Px/bardbl,t≥1, andhencefromEq. ( 3.5),oneobtains /bardblT(t)Px/bardblα≤M′e−δt/bardblx/bardbl,t≥1, whereM′dependson α. Fort∈(0,1],byEq. ( 3.4)andEq. ( 3.7), /bardblT(t)Px/bardblα≤M′′t−α/bardblx/bardbl. Hence,thereexistconstants M(α)>0andγ>0suchthat (3.9) /bardblT(t)Px/bardblα≤M(α)t−αe−γt/bardblx/bardblfort>0. Remark3.4.Note that if the analytic semigroup T(t) is exponential stable, that is, there exists constants N,δ >0 such that/bardblT(t)/bardbl≤Ne−δtfort≥0, then the projection P=I (Q=I−P(t)=0). In that case, Eq. ( 3.9) still holds and can be rewritten as follows: for allx∈X, (3.10) /bardblT(t)x/bardblα≤M(α)e−γ 2tt−α/bardblx/bardbl. Formoreoninterpolationspacesandrelatedissues,werefe rthereadertothefollowing excellentbooksAmann[ 5] andLunardi[ 47]. 3.1.Pseudo-AlmostAutomorphic Functions. LetBC(R,X)stand for the Banachspace ofallboundedcontinuousfunctions ϕ:R/mapsto→X,whichweequipwiththesup-normdefined by/bardblϕ/bardbl∞:=supt∈R/bardblϕ(t)/bardblforallϕ∈BC(R,X). Ifβ≥0,wewillalsobeusingthefollowing notions, /bardblΦ/bardblEβ,∞:=sup t∈R/bardblΦ(t)/bardblEβ forΦ∈BC(R,Eβ), and /bardblϕ/bardblβ,∞:=sup t∈R/bardblϕ(t)/bardblβ forϕ∈BC(R,D(Aβ)). Definition 3.5. [17] A function f∈C(R,X) is said to be almost automorphicif for every sequenceofrealnumbers( s′ n)n∈N,thereexistsasubsequence( sn)n∈Nsuchthat g(t) :=lim n→∞f(t+sn) iswelldefinedforeach t∈R, and lim n→∞g(t−sn)=f(t) foreacht∈R. If the convergenceabove is uniformin t∈R, thenfis almost periodic in the classical Bochner’s sense. Denote by AA(X) the collection of all almost automorphic functions R/mapsto→X. Notethat AA(X)equippedwiththesup-normturnsouttobea Banachspace. Amongotherthings,almostautomorphicfunctionssatisfy t hefollowingproperties. Theorem3.6. [17]If f,f1,f2∈AA(X),then (i)f1+f2∈AA(X), (ii)λf∈AA(X)foranyscalar λ, (iii)fα∈AA(X)where fα:R→Xisdefinedby f α(·)=f(·+α), (iv)the rangeRf:=/braceleftbigf(t) :t∈R/bracerightbigis relatively compact in X, thus f is bounded in norm,EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 7 (v)if fn→f uniformlyonRwhereeach f n∈AA(X),then f∈AA(X)too. Definition3.7. LetYbeanotherBanachspace. Ajointlycontinuousfunction F:R×Y/mapsto→ Xis said to be almost automorphic in t∈Rift/mapsto→F(t,x) is almost automorphic for all x∈K(K⊂Ybeinganyboundedsubset). Equivalently,foreverysequenc eofrealnumbers (s′ n)n∈N,thereexistsasubsequence( sn)n∈Nsuchthat G(t,x) :=lim n→∞F(t+sn,x) iswelldefinedin t∈Randforeach x∈K, and lim n→∞G(t−sn,x)=F(t,x) forallt∈Randx∈K. Thecollectionofsuchfunctionswill bedenotedby AA(R×X). Formoreonalmostautomorphicfunctionsandtheirgenerali zations,wereferthereader totherecentbookbyDiagana[ 17]. Define (see Diagana [ 17,19]) the space PAP0(R,X) as the collection of all functions ϕ∈BC(R,X)satisfying, lim r→∞1 2r/integraldisplayr −r/bardblϕ(s)/bardblds=0. Similarly, PAP0(R×X) will denotethe collectionof all boundedcontinuousfunct ions F:R×Y/mapsto→Xsuchthat lim T→∞1 2r/integraldisplayr −r/bardblF(s,x)/bardblds=0 uniformlyin x∈K,whereK⊂Yis anyboundedsubset. Definition 3.8. (Lianget al.[39] andXiao et al. [63]) A function f∈BC(R,X)is called pseudo almost automorphic if it can be expressed as f=g+φ,whereg∈AA(X) and φ∈PAP0(X). Thecollectionofsuchfunctionswill bedenotedby PAA(X). The functions gandφappearing in Definition 3.8are respectively called the almost automorphic andtheergodicperturbation componentsof f. Definition 3.9. LetYbe another Banach space. A boundedcontinuousfunction F:R× Y/mapsto→Xbelongs to AA(R×X) whenever it can be expressed as F=G+Φ,whereG∈ AA(R×X) andΦ∈PAP0(R×X). The collection of such functions will be denoted by PAA(R×X). A substantialresultisthenexttheorem,whichisduetoXiao et al. [63]. Theorem 3.10. [63]The space PAA (X)equipped with the sup norm /bardbl·/bardbl∞is a Banach space. Theorem3.11. [63]IfYisanotherBanachspace, f :R×Y/mapsto→Xbelongsto PAA (R×X) andif x/mapsto→f(t,x)isuniformlycontinuousoneachboundedsubset K of Yuniformlyint∈ R,thenthefunctiondefinedbyh (t)=f(t,ϕ(t))belongsto PAA (X)providedϕ∈PAA(Y). For more on pseudo-almost automorphic functions and their g eneralizations, we refer thereadertothe recentbookbyDiagana[ 17].8 TOKADIAGANA 4. MainResults Fixβ∈(0,1). Considerthe first-orderdi fferentialequations, dϕ dt=Aϕ+g(t),t∈R, (4.1) and dϕ dt=(A+B)ϕ+f(t,ϕ),t∈R, (4.2) whereA:D(A)⊂X/mapsto→Xis a sectorial linear operator on a Banach space X,B: C(R,D(A))/mapsto→Xis a linear operator, and g:R/mapsto→Xandf:R×X/mapsto→Xare bounded continuousfunctions. To study the existence of pseudo-almost automorphic mild so lutions to Eq. ( 4.1) (and henceEq. ( 4.2)),wewill needthefollowingassumptions, (H.1) Thelinearoperator Aissectorial. Moreover,if T(t)denotestheanalyticsemigroup associatedwithit, we supposethat T(t) ishyperbolic,thatis, σ(A)∩iR=∅. (H.2) The semigroup T(t) is not onlycompactfor t>0 but also is exponentiallystable, i.e.,thereexistsconstants N,δ>0suchthat /bardblT(t)/bardbl≤Ne−δt fort≥0. (H.3) The linear operator B:BC(R,Xβ)/mapsto→X, whereXβ:=D((−A)β), is bounded . Moreover,thefollowingholds, C0:=/bardblB/bardblB(BC(R,Xβ),X)≤1 2d(β), whered(β) :=M(β)(2δ−1)1−βΓ(1−β). (H.4) The function f:R×Xβ/mapsto→Xis pseudo-almost automorphic in the first vari- able uniformly in the second one. For each bounded subset K⊂Xβ,f(R,K) is bounded. Moreover, the function u/mapsto→f(t,u) is uniformly continuous on any boundedsubset KofXβforeacht∈R. Finally,wesupposethatthereexists L>0 suchthat sup t∈R,/bardblϕ/bardblβ≤L/vextenddouble/vextenddouble/vextenddouble/vextenddoublef(t,ϕ)/vextenddouble/vextenddouble/vextenddouble/vextenddouble≤L 2d(β). (H.5) If( un)n∈N⊂PAA(Xβ)isuniformlyboundedanduniformlyconvergentuponevery compactsubsetofR, thenf(·,un(·))isrelativelycompactin BC(R,X). Remark4.1.Notethatif(H.3)holds,thenitcanbeeasilyshownthatthel inearoperator B mapsPAA(Xβ) intoPAA(X). Definition 4.2. Under assumption(H.1), a continuousfunction ϕ:R/mapsto→Xis said to be a mildsolutiontoEq. ( 4.1)providedthat ϕ(t)=T(t−s)ϕ(s)+/integraldisplayt sT(t−τ)g(τ)dτ,∀(t,s)∈T. (4.3) Lemma 4.3. [17]Suppose assumptions (H.1)–(H.2) hold. If g :R/mapsto→Xis a bounded continuousfunction,then ϕgivenby ϕ(t) :=/integraldisplayt −∞T(t−s)g(s)ds (4.4)EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 9 forall t∈R, istheuniqueboundedmildsolutionto Eq. ( 4.1). Definition 4.4. Under assumptions (H.1), (H.2), and (H.3) and if f:R×Xβ/mapsto→Xis a boundedcontinuousfunction,thena continuousfunction ϕ:R/mapsto→Xβsatisfying ϕ(t)=T(t−s)ϕ(s)+/integraldisplayt sT(t−s)/bracketleftig Bϕ(s)+f(s,ϕ(s))/bracketrightig ds,∀(t,s)∈T (4.5) iscalledamildsolutiontoEq. ( 4.2). Underassumptions(H.1),(H.2),and(H.3)andif f:R×Xβ/mapsto→Xis aboundedcontin- uousfunction,it canbeshownthatthefunction ϕ:R/mapsto→Xβdefinedby ϕ(t)=/integraldisplayt −∞T(t−s)/bracketleftig Bϕ(s)+f(s,ϕ(s))/bracketrightig ds (4.6) forallt∈R, isa mildsolutiontoEq. ( 4.2). Definethe followingintegraloperator, (Sϕ)(t)=/integraldisplayt −∞T(t−s)/bracketleftig Bϕ(s)+f(s,ϕ(s))/bracketrightig ds. We have Lemma 4.5. Under assumptions (H.1)–(H.2)–(H.3) and if f:R×Xβ/mapsto→Xis a bounded continuous function, then the mapping S :BC(R,Xβ)/mapsto→BC(R,Xβ)is well-defined and continuous. Proof.We first show that Sis well-defined and that S(BC(R,Xβ))⊂BC(R,Xβ). Indeed, lettingu∈BC(R,Xβ),g(t) :=f(t,u(t)),andusingEq. ( 3.10),weobtain /vextenddouble/vextenddouble/vextenddoubleSu(t)/vextenddouble/vextenddouble/vextenddoubleβ≤/integraldisplayt −∞/vextenddouble/vextenddouble/vextenddoubleT(t−s)[Bu(s)+g(s)]/vextenddouble/vextenddouble/vextenddoubleβds ≤/integraldisplayt −∞M(β)e−δ 2(t−s)(t−s)1−β/bracketleftig /bardblBu(s)/bardbl+/bardblg(s)/bardbl/bracketrightig ds ≤/integraldisplayt −∞M(β)e−δ 2(t−s)(t−s)1−β/bracketleftig C0/bardblu(s)/bardblβ+/bardblg(s)/bardbl/bracketrightig ds ≤d(β)/parenleftig C0/bardblu/bardblβ,∞+/bardblg/bardbl∞/parenrightig , forallt∈R, whered:=M(β)(2δ−1)1−βΓ(1−β),andhence Su:R/mapsto→Xβisbounded. To completetheproofit remainstoshowthat Siscontinuous. Forthat,set F(s,u(s)) :=Bu(s)+g(s)=Bu(s)+f(s,u(s)),∀s∈R. Consider an arbitrary sequence of functions un∈BC(R,Xβ) that convergesuniformly tosomeu∈BC(R,Xβ),that is,/vextenddouble/vextenddouble/vextenddoubleun−u/vextenddouble/vextenddouble/vextenddoubleβ,∞→0 asn→∞.10 TOKADIAGANA Now /vextenddouble/vextenddouble/vextenddoubleSu(t)−Sun(t)/vextenddouble/vextenddouble/vextenddoubleβ=/vextenddouble/vextenddouble/vextenddouble/integraldisplayt −∞T(t−s)[F(s,un(s))−F(s,u(s))]ds/vextenddouble/vextenddouble/vextenddoubleβ ≤M(β)/integraldisplayt −∞(t−s)−βe−δ 2(t−s)/vextenddouble/vextenddouble/vextenddoubleF(s,un(s))−F(s,u(s))/vextenddouble/vextenddouble/vextenddoubleds. ≤M(β)/integraldisplayt −∞(t−s)−βe−δ 2(t−s)/vextenddouble/vextenddouble/vextenddoublef(s,un(s))−f(s,u(s))/vextenddouble/vextenddouble/vextenddoubleds +M(β)/integraldisplayt −∞(t−s)−βe−δ 2(t−s)/vextenddouble/vextenddouble/vextenddoubleB(un(s)−u(s))/vextenddouble/vextenddouble/vextenddoubleds ≤M(β)/integraldisplayt −∞(t−s)−βe−δ 2(t−s)/vextenddouble/vextenddouble/vextenddoublef(s,un(s))−f(s,u(s))/vextenddouble/vextenddouble/vextenddoubleds +d(β)C0/bardblun−u/bardblβ,∞. Using the continuity of the function f:R×Xβ/mapsto→Xand the Lebesgue Dominated ConvergenceTheoremweconcludethat /vextenddouble/vextenddouble/vextenddouble/integraldisplayt −∞T(t−s)P(s)[f(s,un(s))−f(s,u(s))]ds/vextenddouble/vextenddouble/vextenddouble→0 asn→∞. Therefore,/vextenddouble/vextenddouble/vextenddoubleSun−Su/vextenddouble/vextenddouble/vextenddoubleβ,∞→0asn→∞. Theproofis complete. /square Lemma4.6. Underassumptions (H.1)—(H.4) ,thenS(PAA(Xβ)⊂PAA(Xβ). Proof.Letu∈PAA(Xβ) and define h(s) :=f(s,u(s))+Bu(s) for alls∈R. Using (H.4) andTheorem 3.11itfollowsthatthefunction s/mapsto→f(s,u(s))belongsto PAA(X). Similarly, using Remark 4.1it follows that the function s/mapsto→Bu(s) belongs to PAA(X). In view of theabove,thefunction s/mapsto→h(s)belongsto PAA(X). Nowwrite h=h1+h2∈PAA(X)whereh1∈AA(X)andh2∈PAP0(X)andset Rhj(t) :=/integraldisplayt −∞T(t−s)hj(s)dsforallt∈R,j=1,2. Our first task consists of showing that R/parenleftbigAA(X)/parenrightbig⊂AA(Xβ). Indeed,using the fact that h1∈AA(X), for everysequence of real numbers( τ′ n)n∈Nthere exist a subsequence( τn)n∈N anda function f1suchthat f1(t) :=lim n→∞h1(t+τn) iswelldefinedforeach t∈R, and lim n→∞f1(t−τn)=h1(t) foreacht∈R. Now (Rh1)(t+τn)−(Rf1)(t)=/integraldisplayt+τn −∞T(t+τn−s)h1(s)ds−/integraldisplayt −∞T(t−s)f1(s)ds =/integraldisplayt −∞T(t−s)h1(s+τn)ds−/integraldisplayt −∞T(t−s)f1(s)ds. =/integraldisplayt −∞T(t−s)/parenleftig h1(s+τn)−f1(s)/parenrightig ds.EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 11 From Eq. ( 3.10)andthe LebesgueDominatedConvergenceTheorem,it easily follows that /vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt −∞T(t−s)/parenleftig h1(s+τn)−f1(s)/parenrightig ds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleβ≤/integraldisplayt −∞/vextenddouble/vextenddouble/vextenddouble/vextenddoubleT(t−s)/parenleftig h1(s+τn)−f1(s)/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddoubleβds ≤M(β)/integraldisplayt −∞(t−s)−βe−δ 2(t−s)/bardblh1(s+τn)−f1(s)/bardblds →0 asn→∞, andhence (Rf1)(t)=lim n→∞(Rh1)(t+τn) forallt∈R. Usingsimilar argumentsasaboveoneobtainsthat (Rh1)(t)=lim n→∞(Rf1)(t−τn) forallt∈R, whichyields, t/mapsto→(Sh1)(t) belongsto AA(Xβ). The next step consists of showing that R/parenleftbigPAP0(X)/parenrightbig⊂PAP0(Xβ). Obviously, Rh2∈ BC(R,Xβ) (see Lemma 4.5). Using the fact that h2∈PAP0(X) and Eq. ( 3.10) it can be easilyshownthat( Rh2)∈PAP0(Xβ). Indeed,for r>0, 1 2r/integraldisplayr −r/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt −∞T(t−s)h2(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleβdt≤M(β) 2r/integraldisplayr −r/integraldisplay∞ 0eδ 2ss−β/vextenddouble/vextenddouble/vextenddouble/vextenddoubleh2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubledsdt ≤M(β)/integraldisplay∞ 0eδ 2ss−β/parenleftigg1 2r/integraldisplayr −r/vextenddouble/vextenddouble/vextenddouble/vextenddoubleh2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubledt/parenrightigg ds. Usingthe factthat PAP0(X)istranslation-invariantit followsthat lim r→∞1 2r/integraldisplayr −r/vextenddouble/vextenddouble/vextenddouble/vextenddoubleh2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubledt=0, ast/mapsto→h2(t−s)∈PAP0(X)forevery s∈R. One completes the proof by using the Lebesgue Dominated Conv ergenceTheorem. In summary,( Rh2)∈PAP0(Xβ),whichcompletestheproof. /square Theorem 4.7. Suppose assumptions (H.1)—(H.5) hold, then Eq. ( 4.2) has at least one pseudo-almostautomorphicmildsolution Proof.LetBβ={u∈PAA(Xβ) :/bardblu/bardblβ≤L}. Using the proofof Lemma 4.5it followsthat Bβisa convexandclosedset. NowusingLemma 4.6it followsthat S(Bβ)⊂PAA(Xβ). Nowforall u∈Bβ, /vextenddouble/vextenddouble/vextenddoubleSu(t)/vextenddouble/vextenddouble/vextenddoubleβ≤/integraldisplayt −∞/vextenddouble/vextenddouble/vextenddoubleT(t−s)[Bu(s)+g(s)]/vextenddouble/vextenddouble/vextenddoubleβds ≤/integraldisplayt −∞M(β)e−δ 2(t−s)(t−s)−β/bracketleftig /bardblBu(s)/bardbl+/bardblf(s,u(s))/bardbl/bracketrightig ds ≤/integraldisplayt −∞M(β)e−δ 2(t−s)(t−s)−β/bracketleftig C0/bardblu(s)/bardblβ+/bardblf(s,u(s))/bardbl/bracketrightig ds ≤d(β)/parenleftigL 2d(β)+L 2d(β)/parenrightig =L forallt∈R, andhence Su∈Bβ.12 TOKADIAGANA To completetheproof,we haveto provethefollowing: a) ThatV={Su(t) :u∈Bβ}isa relativelycompactsubsetof Xβforeacht∈R; b) ThatW={Su:u∈Bβ}⊂BC(R,Xβ)isequi-continuous. To showa),fix t∈Randconsideranarbitrary ε>0. Now (Sεu)(t) :=/integraldisplayt−ε −∞T(t−s)F(s,u(s))ds,u∈Bβ =T(ε)/integraldisplayt−ε −∞T(t−ε−s)F(s,u(s))ds,u∈Bβ =T(ε)(Su)(t−ε),u∈Bβ andhence Vε:={Sεu(t) :u∈Bβ}is relativelycompactin Xβas the evolutionfamily T(ε) iscompactbyassumption. Now /vextenddouble/vextenddouble/vextenddoubleSu(t)−T(ε)/integraldisplayt−ε −∞T(t−ε−s)F(s,u(s))ds/vextenddouble/vextenddouble/vextenddoubleβ ≤/integraldisplayt t−ε/bardblT(t−s)F(s,u(s))/bardblβds ≤M(β)/integraldisplayt t−εe−δ 2(t−s)(t−s)−β/bardblF(s,u(s))/bardblds ≤M(β)/integraldisplayt t−ε(t−s)−βe−δ 2(t−s)/vextenddouble/vextenddouble/vextenddoubleg(s)/vextenddouble/vextenddouble/vextenddoubleds+M(β)/integraldisplayt t−ε(t−s)−βe−δ 2(t−s)/vextenddouble/vextenddouble/vextenddoubleBu(s)/vextenddouble/vextenddouble/vextenddoubleds ≤M(β)/integraldisplayt t−ε(t−s)−βe−δ 2(t−s)/vextenddouble/vextenddouble/vextenddoublef(s,u(s))/vextenddouble/vextenddouble/vextenddoubleds+M(β)C0/bardblu/bardblβ,∞/integraldisplayt t−ε(t−s)−βe−δ 2(t−s)ds ≤M(β)L/parenleftig d−1(β)+C0/parenrightig/integraldisplayt t−ε(t−s)−αds =M(β)L/parenleftig d−1(β)+C0/parenrightig ε1−β(1−β)−1, andhencetheset V:={Su(t) :u∈Bβ}⊂Xβisrelativelycompact. Theproofforb)followsalongthesamelinesasinLi etal.[38,Theorem31]andhence isomitted. The rest of the proof slightly follows along the same lines as in Diagana [ 18]. In- deed, since Bβis a closed convex subset of PAA(Xβ) and that S(Bβ)⊂Bβ, it follows that coS(Bβ)⊂Bβ.Consequently, S(coS(Bβ))⊂S(Bβ)⊂coS(Bβ). Further, it is not hard to see that {u(t) :u∈coS(Bβ)}is relatively compact in Xβfor each fixed t∈Rand that functions in coS(Bβ) are equi-continuouson R. Using Arzel` a- Ascoli theorem,we deducethat the restrictionof coS(Bβ) to anycompactsubset IofRis relativelycompactin C(I,Xβ). In summary, S:coS(Bβ)/mapsto→coS(Bβ) is continuousand compact. Using the Schauder fixed point it follows that Shas a fixed-point, which obviously is a pseudo-almost auto- morphicmildsolutionto Eq. ( 4.2). /squareEXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 13 Fixα∈[1 2,1). In orderto study Eq. ( 1.6),we letβ=1 2andsuppose that the following additionalassumptionholds: (H.6) Thereexistsa function ρ∈L1(R,(0,∞))with/bardblρ/bardblL1(R,(0,∞))≤1 2d(1 2)suchthat /vextenddouble/vextenddouble/vextenddouble/vextenddoubleC(t)ϕ/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE≤ρ(t)/vextenddouble/vextenddouble/vextenddoubleϕ/vextenddouble/vextenddouble/vextenddoubleE1 2 forallϕ∈E1 2andt∈R. Corollary 4.8. Under assumptions (H.1)–(H.2)–(H.4)–(H.5)–(H.6) , then Eq. ( 1.6) (and henceEq. ( 1.5)andEq. ( 1.2))hasatleast onepseudo-almostautomorphicmildsolution . Proof.It suffices to show thatAandBsatisfy similar assumptions as (H.1)–(H.2)–(H.3) andthatFsatisfies similarassumptionsas(H.4)–(H.5). Step 1. Assumption (H.6) yields Bsatisfies similar assumption as (H.3), where Bis definedby Bϕ(t) :=/integraldisplayt −∞C(t−s)ϕ(s)ds. Indeed, since the function ρis integrable, it is clear that the operator Bbelong to B(BC(R,E1 2),E) with/bardblB/bardblB(BC(R,E1 2),E)≤/bardblρ/bardblL1(R,(0,∞)). In fact, we take C0=/bardblρ/bardblL1(R,(0,∞)). The fact the function t/mapsto→Bϕ(t) is pseudo-almost automorphic for any ϕ∈PAA(E1 2) is guaranteedbyRemark 4.1. However,forthesakeofclarity,wewillshowit. Indeed,wr ite ϕ=ϕ1+ϕ2, whereϕ1∈AA(E1 2) andϕ2∈PAP0(E1 2). Using the fact that the function t/mapsto→ϕ1(t) belongs to AA(E1 2), for every sequence of real numbers ( τ′ n)n∈Nthere exist a subsequence( τn)n∈Nandafunction ψ1suchthat ψ1(t) :=lim n→∞ϕ1(t+τn) iswelldefinedforeach t∈R, and lim n→∞ψ1(t−τn)=ϕ1(t) foreacht∈R. Now Bϕ1(t+τn)−Bψ1(t)=/integraldisplayt+τn −∞C(t+τn−s)ϕ1(s)ds−/integraldisplayt −∞C(t−s)ψ1(s)ds =/integraldisplayt −∞C(t−s)ϕ1(s+τn)ds−/integraldisplayt −∞C(t−s)ψ1(s)ds =/integraldisplayt −∞C(t−s)/parenleftig ϕ1(s+τn)−ψ1(s)/parenrightig ds andhence /vextenddouble/vextenddouble/vextenddouble/vextenddoubleBϕ1(t+τn)−Bψ1(t)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE≤/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt −∞C(t−s)/parenleftig ϕ1(s+τn)−ψ1(s)/parenrightig ds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE ≤/integraldisplayt −∞/vextenddouble/vextenddouble/vextenddouble/vextenddoubleC(t−s)/parenleftig ϕ1(s+τn)−ψ1(s)/parenrightig/vextenddouble/vextenddouble/vextenddouble/vextenddoubleEds ≤/integraldisplayt −∞ρ(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleϕ1(s+τn)−ψ1(s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE1 2ds14 TOKADIAGANA whichbyLebesgueDominatedConvergenceTheoremyields lim n→∞/vextenddouble/vextenddouble/vextenddouble/vextenddoubleBϕ1(t+τn)−Bψ1(t)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE=0. Usingsimilar arguments,weobtain lim n→∞/vextenddouble/vextenddouble/vextenddouble/vextenddoubleBψ1(t−τn)−Bφ1(t)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE=0. Forr>0, 1 2r/integraldisplayr −r/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt −∞C(t−s)ϕ2(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleEdt≤1 2r/integraldisplayr −r/integraldisplay∞ 0ρ(s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleϕ2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE1 2dsdt ≤/integraldisplay∞ 0ρ(s)1 2r/integraldisplayr −r/vextenddouble/vextenddouble/vextenddouble/vextenddoubleϕ2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE1 2dtds. Now lim r→∞1 2r/integraldisplayr −r/vextenddouble/vextenddouble/vextenddouble/vextenddoubleϕ2(t−s)/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE1 2dt=0, ast/mapsto→ϕ2(t−s)∈PAP0(E1 2) forevery s∈R. Therefore, lim r→∞1 2r/integraldisplayr −r/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt −∞C(t−s)ϕ2(s)ds/vextenddouble/vextenddouble/vextenddouble/vextenddoubleEdt=0 byusingtheLebesgueDominatedConvergenceTheorem. Step 2. Clearly, the operator Asatisfies similar assumptions as (H.1)–(H.2) in the space E1 2. Indeed,forall ϕ∈D(A),we have Aϕ=∞/summationdisplay n=1AnPnϕ, where Pn:=En0 0EnandAn:=0 1 −λn−λα n,n≥1. Thecharacteristicequationfor Anisgivenby ρ2+2γλα nρ+λn=0, fromwhichwe obtainitseigenvaluesgivenby ρn 1=λα n/parenleftig −γ+/radicalig γ2−λ1−2αn/parenrightig andλn 2=λα n/parenleftig −γ−/radicalig γ2−λ1−2αn/parenrightig , andhenceσ(An)=/braceleftig ρn 1,ρn 2/bracerightig . UsingEq. ( 1.4)it followsthatthereexists ω>0 suchthatρ(A)containsthehalfplane Sω:=/braceleftig λ∈C:ℜeλ≥ω/bracerightig . Nowsinceρn 1andρn 2aredistinctandthateachofthemisofmultiplicityone,the nAnis diagonalizable. Further,it isnot di fficultto see thatAn=K−1 nJnKn, whereJn,KnandK−1 n arerespectivelygivenby Jn=ρn 10 0ρn 2,Kn=1 1 ρn 1ρn 2, andEXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 15 K−1 n=1 ρn 1−ρn 2−ρn 21 ρn 1−1. Forλ∈Sωandϕ∈E1 2,onehas R(λ,A)ϕ=∞/summationdisplay n=1(λ−An)−1Pnϕ =∞/summationdisplay n=1Kn(λ−Jn)−1K−1 nPnϕ. Hence, /vextenddouble/vextenddouble/vextenddouble/vextenddoubleR(λ,A)ϕ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 E1 2≤∞/summationdisplay n=1/vextenddouble/vextenddouble/vextenddouble/vextenddoubleKn(λ−Jn)−1K−1 n/vextenddouble/vextenddouble/vextenddouble/vextenddouble2/vextenddouble/vextenddouble/vextenddouble/vextenddoublePnϕ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 E1 2 ≤∞/summationdisplay n=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 n/vextenddouble/vextenddouble/vextenddouble/vextenddouble2/vextenddouble/vextenddouble/vextenddouble/vextenddoublePnϕ/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 E1 2. It iseasytosee that thereexist twoconstants C1,C2>0suchthat /bardblKn/bardbl≤C1|ρn 1(t)|,/bardblK−1 n/bardbl≤C2 |ρn 1|foralln≥1. Now /bardbl(λ−Jn)−1/bardbl2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1 λ−ρn 10 01 λ−ρn 2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 ≤1 |λ−ρn 1|2+1 |λ−ρn 2|2. Definethefunction Θ(λ) :=|λ| |λ−ρn 1(t)|. It isclearthatΘiscontinuousandboundedon Sω. Ifwe take C3=sup/braceleftigg|λ| |λ−λn k|:λ∈Sω,n≥1;k=1,2/bracerightigg it followsthat /bardbl(λ−Jn)−1/bardbl≤C3 |λ|, λ∈Sω. Therefore,onecanfinda constant K≥1such /bardblR(λ,A)/bardblB(E1 2)≤K |λ|, λ∈Sω, andhencetheoperator AissectorialonE1 2. SinceAissectorialinE1 2,thenitgeneratesananalyticsemigroup( T(τ))τ≥0:=(eτA)τ≥0 onE1 2givenby eτAϕ=∞/summationdisplay n=0K−1 nPneτJnPnKnPnϕ.16 TOKADIAGANA First of all, notethat the semigroup( T(τ))τ≥0ishyperbolicas σ(A)∩iR=∅. Inorder words,Asatisfies anassumptionsimilarto (H.1). Secondly,usingthe factthat Aisan operatorof compactresolventit followsthat T(τ) iscompactfor τ>0. Ontheotherhand,we have /bardbleτAϕ/bardblE1 2=∞/summationdisplay n=0/bardblK−1 nPn/bardbl/bardbleτJnPn/bardbl/bardblKnPn/bardbl/bardblPnϕ/bardblE1 2, withforeach ϕ=/parenleftiggϕ1 ϕ2/parenrightigg ∈E1 2, /bardbleτJnPnϕ/bardbl2 E1 2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleeρn 1τEn0 0eρn 2τEnϕ1 ϕ2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 E1 2 ≤ /bardbleρn 1τEnϕ1/bardbl2 1 2+/bardbleρn 2τEnϕ2/bardbl2 ≤eℜe(ρn 1)τ/bardblϕ/bardbl2 E1 2. UsingEq. ( 1.4)it followsthereexists N′≥1suchthat /bardblT(τ)/bardblB(E1 2)≤N′e−δ0τ, τ≥0, and henceT(τ) is exponentiallystable, that is, Asatisfies an assumption similar to (H.2) inE1 2. Step 3. Thefactthat Fsatisfiessimilar assumptionsas(H.4)and(H.5)isclear. /square 5. Example Inthissection,wetake α=β=1 2. LetΩ⊂RNbeanopenboundedsetwithsu fficiently smooth boundary ∂Ωand letH=L2(Ω) be the Hilbert space of all measurable functions ϕ:Ω/mapsto→Csuchthat /bardblϕ/bardblL2(Ω)=/parenleftigg/integraldisplay Ω|ϕ(x)|2dx/parenrightigg1/2 <∞. Here, we studytheexistenceof pseudo-almostautomorphics olutionsϕ(t,x) to a struc- turally damped plate-like system given by (see also Chen and Triggiani [ 13], Schnaubelt andVeraar[ 58],Triggiani[ 59]), ∂2ϕ ∂t2(t,x)−2γ∆∂ϕ ∂t(t,x)+∆2ϕ(t,x)=/integraldisplayt −∞b(t−s)ϕ(s,x)ds+f(t,ϕ(t,x)) (5.1) +ηϕ(t,x),(t,x)∈R×Ω ∆ϕ(t,x)=ϕ(t,x)=0,(t,x)∈R×∂Ω, (5.2) whereγ,η >0 are constants, b:R/mapsto→[0,∞) is a measurable function, the function f:R×L2(Ω)/mapsto→L2(Ω) is pseudo-almost automorphic in t∈Runiformly in the second variable,and∆standsfortheusualLaplaceoperatorin thespacevariable x. Setting Aϕ=∆2ϕforallϕ∈D(A)=D(∆2)=/braceleftig ϕ∈H4(Ω) :∆ϕ=ϕ=0 on∂Ω/bracerightig , Bϕ=A1 2ϕ=−∆ϕ,∀ϕ∈D(B)=H1 0(Ω)∩H2(Ω),EXISTENCE OF PSEUDO-ALMOST AUTOMORPHIC SOLUTIONS 17 C(t)ϕ=b(t)ϕforallϕ∈D(C(t))=D(∆), andf:R×/parenleftig H1 0(Ω)∩H2(Ω)/parenrightig /mapsto→L2(Ω), one can easily see that Eq. ( 1.2) is exactly the structurallydampedplate-likesystemformulatedinEqs. ( 5.1)-(5.2). HereE1 2=D(A1 2)×L2(Ω)=/parenleftig H1 0(Ω)∩H2(Ω)/parenrightig ×L2(Ω)anditisequippedwiththeinner productdefinedby /parenleftigg/parenleftiggϕ1 ϕ2/parenrightigg ,ψ1 ψ2/parenrightigg E1 2:=/integraldisplay Ω∆ϕ1∆ψ1dx+/integraldisplay Ωϕ2ψ2dx forallϕ1,ψ1∈H1 0(Ω)∩H2(Ω)andϕ2,ψ2∈L2(Ω). Similarly, D(A1 2)=H1 0(Ω)∩H2(Ω)isequippedwith thenormdefinedby /bardblϕ/bardbl1 2=/bardblA1 2ϕ/bardblL2(Ω):=/parenleftig/integraldisplay Ω|∆ϕ|2dx/parenrightig1 2 forallϕ∈H1 0(Ω)∩H2(Ω). Clearly,−Aη=−(∆2+ηI) is a sectorial operator on L2(Ω) and let ( T(t))t≥0be the analyticsemigroupassociatedwithit. It iswell-knowntha tthesemigroup T(t) isnotonly compactfor t>0butalso isexponentiallystable as /bardblT(t)/bardbl≤e−ηt forallt≥0. UsingthefacttheLaplaceoperator ∆withdomain D(∆)=H2(Ω)∩H1 0(Ω)isinvertible inL2(Ω)it followsthat /bardblϕ/bardblL2(Ω)=/bardbl∆−1∆ϕ/bardblL2(Ω) ≤ /bardbl∆−1/bardblB(L2(Ω))./bardbl∆ϕ/bardblL2(Ω) =/bardbl∆−1/bardblB(L2(Ω))./bardblA1 2ϕ/bardblL2(Ω) =/bardbl∆−1/bardblB(L2(Ω))./bardblϕ/bardbl1 2 forallϕ∈H2(Ω)∩H1 0(Ω). Ifb∈L1(R,(0,∞)),thenusingthe previousinequalityit followsthat /bardblC(t)Φ/bardblE=b(t)/bardblϕ/bardblL2(Ω) ≤b(t)/bardbl∆−1/bardblB(L2(Ω))./bardblϕ/bardbl1 2 ≤b(t)/bardbl∆−1/bardblB(L2(Ω))./bardblΦ/bardblE1 2 forallΦ=/parenleftiggϕ ψ/parenrightigg ∈E1 2andt∈R. Thissettingrequiresthefollowingassumptions, (H.7) Eq. ( 1.4)holds. 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Rhandi, Nonautonomous retarded wave equ ations.J. Math. Anal. Appl. 263(2001), no. 1, pp.14–32. 50. J.E. Mu˜ noz Rivera, E.C. Lapa, Decay rates of solutions o f an anisotropic inhomogeneous n-dimensional viscoelastic equation with polynomially decaying kernels Comm. Math. Phys. ,177(1996), pp.583–602. 51. J.E.Mu˜ nozRivera, E.C.Lapa,andR.Barreto, Decayrate s forviscoelastic plates withmemory. J.Elasticity 44(1996), no. 1,pp. 61–87. 52. J. A. Nohel, Nonlinear Volterra equations for heat flow in materials with memory. MRC Rech. Summary Report #2081, Madison, WI. 53. J.W.Nunziato, On heat conduction in materials with memo ry.Quart. Appl. Math. 29(1971), pp.187–204.20 TOKADIAGANA 54. H. Oka, Second order linear Volterra equations governed by a sine family. J. Integral Equations Appl. 8 (1996), no. 4,pp. 447–456. 55. H. Oka, Second order linear Volterra integrodi fferential equations. Semigroup Forum 53(1996), no. 1, pp. 25–43. 56. H. Oka, A class of complete second order linear di fferential equations. Proc. Amer. Math. Soc. 124(1996), no.10, pp. 3143–3150. 57. J. Pr¨ uss, Evolutionary integral equations and applica tions, Monographs in Mathematics, vol. 87, Birkh¨ auser Verlag, Basel, 1993. 58. R. Schnaubelt and M. Veraar, Structurally damped plate a nd wave equations with random point force in arbitrary space dimensions. Differential Integral Equations 23(2010), no.9-10, pp. 957–988. 59. R.Triggiani, Regularity ofsomestructurally dampedpr oblemswithpointcontrolandwithboundary control. J. Math. Anal. Appl. 161(1991), no. 2,pp. 299–331. 60. P.A.Vuillermot,Globalexponential attractors foracl assofalmost-periodic parabolicequations in RN.Proc. Amer.Math. Soc. 116(1992), no.3, pp.775–782. 61. P. A. Vuillermot, Almost-periodic attractors for a clas s of nonautonomous reaction-di ffusion equations on RN. II.Codimension-one stable manifolds. Differential Integral Equations 5(1992), no. 3,pp. 693–720. 62. P. A. Vuillermot, Almost periodic attractors for a class of nonautonomous reaction-di ffusion equations on RN. I.Global stabilization processes. J. Differential Equations 94(1991), no. 2,pp. 228–253. 63. T. J. Xiao, J. Liang, J. Zhang, Pseudo almost automorphic solutions to semilinear di fferential equations in Banach spaces. Semigroup Forum 76(2008), no. 3,pp. 518–524. 64. T. J. Xiao, X-X. Zhu, J. Liang, Pseudo-almost automorphi c mild solutions to nonautonomous di fferential equations and applications. Nonlinear Anal. 70(2009), no.11, pp. 4079–4085. 65. T. J. Xiao and J. Liang, Semigroups arising from elastic s ystems with dissipation. Comput. Math. Appl. 33 (1997), no. 10,pp. 1–9. 66. T.J. Xio and J. Liang, On complete second order linear di fferential equations in Banach spaces Pacific J. Math.,142(1990), pp. 175–195. 67. T.J.XioandJ.Liang,Anoteonthepropagatorsofsecondo rderlineardifferential equations inHilbertspaces Proc.Amer.Math. Soc. ,113(1991), pp.663–667. 68. T.J. Xiao and J. Liang, Wellposedness and exponential gr owth property of a class of complete second order linear differential equations J. Sichuan Uni. (Sichuan Daxue Xuebao) ,27(1990), pp.396–401. 69. T.J. Xio and J. Liang, Complete second order linear di fferential equations with almost periodic solutions J. Math. Anal. Appl .,163(1992), pp.136–146. 70. T.J. Xio and J. Liang, Analyticity of the propagators of s econd order linear di fferential equations in Banach spacesSemigroup Forum ,44(1992), pp.356–363. 71. G. F. Webb, An abstract semilinear Volterra integrodi fferential equation, Proc. Amer. Math. Soc. 69(1978), no.2, pp. 255–260 Departmentof Mathematics ,HowardUniversity,2441 6thStreetN.W.,Washington ,D.C.20059, USA E-mail address :tdiagana@howard.edu

2014-03-24

In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem.

Existence Results for Some Damped Second-Order Volterra Integro-Differential Equations

1403.5955v1

arXiv:2103.00461v1 [math.AP] 28 Feb 2021STABILITY FOR AN INVERSE SOURCE PROBLEM OF THE DAMPED BIHARMONIC PLATE EQUATION PEIJUN LI, XIAOHUA YAO, AND YUE ZHAO Abstract. This paper is concerned with the stability of the inverse sou rce problem for the damped biharmonic plate equation in three dimensions. The stabili ty estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source fu nction, where the latter decreases as the upper bound of the frequency increases. The stability al so shows exponential dependence on the constant damping coefficient. The analysis employs Carleman estimates and time decay estimates for the damped plate wave equation to obtain an exact observa bility bound and depends on the study of the resonance-free region and an upper bound of the r esolvent of the biharmonic operator with respect to the complex wavenumber. 1.Introduction Consider the damped biharmonic plate equation in three dime nsions ∆2upx,kq ´k2upx,kq ´ikσupx,kq “fpxq, x PR3, (1.1) whereką0 is the wavenumber, σą0 is the damping coefficient, and fPL2pR3qis a assumed to be a real-valued function with a compact support contained in BR“ txPR3:|x| ăRu, whereRą0 is a constant. Let BBRbe the boundary of BR. Since the problem is formulated in the open domain, the Sommerfeld radiation condition is imposed usually on uand ∆uto ensure the well-posedness of the problem [17]. This paper is concerned with the inverse so urce problem of determining ffrom the boundary measurements upx,kq,∇upx,kq,∆upx,kq,∇∆upx,kq, x P BBR corresponding to the wavenumber kgiven in a finite interval. In general, there is no uniqueness for the inverse source pro blems of the wave equations at a fixed frequency [2,12]. Computationally, a more serious issue is the lack of stability, i.e., a small variation of the data might lead to a huge error in the reconstruction. H ence it is crucial to examine the stability of the inverse source problems. In [2], the author s initialized the study of the inverse source problem for the Helmholtz equation by using multi-frequenc y data. Since then, it has become an active research topic on the inverse source problems via mul tiple frequency data in order to over- come the non-uniqueness issue and enhance the stability. Th e increasing stability was investigated for the inverse source problems of various wave equations wh ich include the acoustic, elastic, and electromagnetic wave equations [3–6,13,14] and the Helmho ltz equation with attenuation [8]. On the other hand, it has generated sustained interest in the ma thematics community on the boundary value problems for higher-order elliptic operators [7]. Th e biharmonic operator, which can be en- countered in models originating from elasticity for exampl e, appears as a natural candidate for such a study [15,16]. Compared with the equations involving the s econd order differential operators, the model equations with the biharmonic operators are much less studied in the community of inverse problems. We refer to [1,9–11,17] and the references cited t herein on the recovery of the lower-order coefficients by using either the far-field pattern or the Diric hlet-to-Neumann map on the boundary. 2000Mathematics Subject Classification. 35R30, 31B30. Key words and phrases. inverse source problem, the biharmonic operator, the dampe d biharmonic plate equation, stability. 12 P. LI, X. YAO, AND Y. ZHAO In a recent paper [12], the authors demonstrated the increas ing stability for the inverse source prob- lem of the biharmonic operator with a zeroth order perturbat ion by using multi-frequency near-field data. The main ingredient of the analysis relies on the study of an eigenvalue problem for the bi- harmonic operator with the hinged boundary conditions. But the method is not applicable directly to handle the biharmonic operator with a damping coefficient. Motivated by [4,8], we use the Fourier transform in time to re duce the inverse source problem into the identification of the initial data for the initial va lue problem of the damped biharmonic plate wave equation by lateral Cauchy data. The Carleman est imate is utilized to obtain an exact observability bound for the source function in the framewor k of the initial value problem for the corresponding wave equation, which connects the scatterin g data and the unknown source function by taking the inverse Fourier transform. An appropriate rat e of time decay for the damped plate wave equation is proved in order to justify the Fourier trans form. Then applying the results in [12] on the resolvent of the biharmonic operator, we obtain a reso nance-free region of the data with respect to the complex wavenumber and the bound of the analyt ic continuation of the data from the given data to the higher wavenumber data. By studying the dep endence of analytic continuation and of the exact observability bound for the damped plate wav e equation on the damping coefficient, we show the exponential dependence of increasing stability on the damping constant. The stability estimate consists of the Lipschitz type of data discrepancy and the high wavenumber tail of the source function. The latter decreases as the wavenumber of t he data increases, which implies that the inverse problem is more stable when the higher wavenumbe r data is used. But the stability deteriorates as the damping constant becomes larger. It sho uld be pointed out that due to the existence of the damping coefficient, we can not obtain a secto rial resonance-free region for the data as that in [4,13]. Instead, we choose a rectangular resonanc e-free region as that in [14], which leads to a double logarithmic type of the high wavenumber tail for t he estimate. This paper is organized as follows. In section 2, the direct s ource problem is discussed; the resolvent is introduced for the elliptic operator, and its r esonance-free region and upper bound are obtained. Section 3 is devoted to the stability analysis of t he inverse source problem by using multi- frequency data. In appendix A, we usethe Carleman estimate t o derive an exact observability bound with exponential dependence on the damping coefficient. In ap pendix B, we prove an appropriate rate of time decay for the damped plate wave equation to justi fy the Fourier transform. 2.The direct source problem In this section, we discuss the solution of the direct source problem and study the resolvent of the biharmonic operator with a damping coefficient. Theorem 2.1. LetfPL2pR3qwith a compact support. Then there exists a unique solution uof Schwartz distribution to (1.1)for every ką0. Moreover, the solution satisfies |upx,kq| ďCpk,fqe´cpk,σq|x| as|x| Ñ 8,whereCpk,fqandcpk,σqare positive constants depending on k,fandk,σ, respectively. Proof.Taking the Fourier transform of upx,kqformally with respect to the spatial variable x, we define u˚px,kq “ż R3eix¨ξ1 |ξ|4´k2´ikσˆfpξqdξ, x PR3, where ˆfpξq “1 p2πq3ż R3fpxqe´ix¨ξdx. It follows from the Plancherel theorem that for each ką0 we have that u˚p¨,kq PH4pR3qand satisfies the equation (1.1) in the sense of Schwartz distrib ution.AN INVERSE SOURCE PROBLEM 3 Denote Gpx,kq “ż R3eix¨ξ1 |ξ|4´k2´ikσdξ. By a direct calculation we can write u˚px,kqas u˚px,kq “ pG˚fqpxq “1 2κ2ż R3´eiκ|x´y| 4π|x´y|´e´κ|x´y| 4π|x´y|¯ fpyqdy, (2.1) whereκ“ pk2`ikσq1 4such that ℜκą0 andℑκą0. Since fhas a compact support, we obtain from (2.1) that the solution u˚px,kqsatisfies the estimate |u˚px,kq| ďCpk,fqe´cpk,σq|x| as|x| Ñ 8, whereCpk,fqandcpk,σqare positive constants dependingon k,fandk,σ, respectively. Bydirectcalculations, wemayalsoshowthat ∇u˚and∆u˚havesimilarexponential decayestimates. Next is show the uniqueness. Let ˜ u˚px,kqbe another Schwartz distributional solution to (1.1). Clearly we have p∆2´k2´ikσqpu˚´˜u˚q “0. Taking the Fourier transform on both sides of the above equat ion yields p|ξ|4´k2´ikσqp{u˚´˜˚uqpξq “0. Notice that for ką0 we have |ξ|4´k2´ikσ‰0 for all ξPR3. Taking the generalized inverse Fourier transform gives u˚´˜u˚“0, which proves the uniqueness. /square To study the resolvent we let u˚px,κq:“upx,kq, κ “ pk2`ikσq1 4, whereℜκą0 andℑκą0. By (1.1), u˚satisfies ∆2u˚´κ4u˚“f. Denote by R“ tzPC:pδ,`8q ˆ p´ d,dquthe infinite rectangular slab, where δis any positive constant and d!1. ForkPR, denote the resolvent Rpkq:“ p∆2´k2´ikσq´1. Then we have Rpκq “ p∆2´κ4q´1. Hereafter, the notation aÀbstands for aďCb,whereCą0 is a generic constant which may change step by step in the proo fs. Lemma 2.2. For each kPRandρPC8 0pBRqthe resolvent operator Rpkqis analytic and has the following estimate: }ρRpkqρ}L2pBRqÑHjpBRqÀ |k|j 2e2Rpσ`1q|k|1 2, j “0,1,2,3,4. Proof.It is clear to note that for a sufficiently small d, the set tpk2`ikσq1 4:kPRubelongs to the first quadrant. Consequently, pk2`ikσq1 4is analytic with respect to kPR. By [12, Theorem 2.1], the resolvent Rpκqis analytic in Czt0uand the following estimate holds: }ρRpκqρ}L2pBRqÑHjpBRqÀ |κ|´2xκyjpe2Rpℑκq´`e2Rpℜκq´q, j “0,1,2,3,4,(2.2) wherex´:“maxt´x,0uand xκy “ p1` |κ|2q1{2. On the other hand, letting k“k1`ik2, we have from a direct calculation that k2`ikσ“k2 1´k2 2´k2σ` p2k1k2`k1σqi. It is easy to see that if dis sufficiently small, which gives that |k2|is sufficiently small, there is a positive lower bound for |k2`ikσ|withkPRand then |κ| ącfor some positive constant c. The proof is completed by replacing κwith pk2`ikσq1 4in (2.2). /square4 P. LI, X. YAO, AND Y. ZHAO 3.The inverse source problem In this section, we address the inverse source problem of the damped biharmonic plate equation and present an increasing stability estimate by using multi -frequency scattering data. Denote }upx,kq}2 BBR:“ż BBR´ pk4`k2q|upx,kq|2`k2|∇upx,kq|2 ` pk2`1q|∆upx,kq|2` |∇∆upx,kq|2¯ dspxq. The following lemma provides a relation between the unknown source function and the boundary measurements. Hereafter, by Remark B.3, we assume that fPHnpBRqwhereně4. Lemma 3.1. Letube the solution to the direct scattering problem (1.1). Then }f}2 L2pBRqÀ2eCσ2ż`8 0}upx,kq}2 BBRdk. Proof.Consider the initial value problem for the damped biharmoni c plate wave equation # B2 tUpx,tq `∆2Upx,tq `σBtUpx,tq “0, px,tq PBRˆ p0,`8q, Upx,0q “0,BtUpx,0q “fpxq, x PBR.(3.1) We define Upx,tq “0 whentă0 and denote UTpx,tq “Upx,tqχr0,Tsptqand xUTpx,kq “żT 0Upx,tqeiktdt. By the decay estimate (B.2) we have that Upx,tq PL2 tp0,`8qand lim TÑ8UTpx,tq “Upx,tqin L2 tpRquniformly for all xPR3. It follows from the Plancherel Theorem that xUTalso converges in L2 kpRqto a function u˚px,kq PL2 kpRquniformly for all xPR3, which implies that u˚px,kqis the Fourier transform of Upx,tq. Denoteby x¨,¨yandStheusualscalarinnerproductof L2pR3qandthespaceofSchwartzfunctions, respectively. We take u˚px,kqas a Schwartz distribution such that u˚px,kqpϕq “ xu˚,ϕyfor each ϕPS. In what follows, we show that u˚px,kqsatisfies the equation (1.1) in the sense of Schwartz distribution. First we multiply both sides of the wave equation (3.1) by a Sc hwartz function ϕand take inte- gration over R3. Using the wave equation (3.1) and the integration by parts w ith respect to the t variable over r0,TsforTą0, we obtain 0“żT 0xB2 tU`∆2U`σBtU,ϕyeiktdt “eikTxBtUpx,Tq,ϕy ´ikeikTxUpx,Tq,ϕy `σeikTxUpx,Tq,ϕy ´ xBtUpx,0q,ϕy `AżT 0p∆2U´k2U´ikσU qeiktdt,ϕE . (3.2) Itfollowsfromthedecay estimate(B.2)that |BtUpx,tq|,|Upx,tq| À p1`tq´3 4uniformlyforall xPR3, which give lim TÑ8eikTxBtUpx,Tq,ϕy “lim TÑ8ikeikTxUpx,Tq,ϕy “lim TÑ8σeikTxUpx,Tq,ϕy “0.AN INVERSE SOURCE PROBLEM 5 On the other hand, we have from the integration by parts that AżT 0p∆2U´k2U´ikσU qeiktdt,ϕE “AżT 0Udt,∆2ϕE `AżT 0p´k2U´ikσU qeiktdt,ϕE . (3.3) Since lim TÑ`8xUTpx,kq “u˚px,kqinL2 kpRquniformly for xPR3, we can choose a positive sequence tTnu8 n“1such that lim nÑ8Tn“ `8and lim nÑ8yUTnpx,kq “u˚px,kqpointwisely for a.e. kPRand uniformly for all xPR3. Define a sequence of Schwartz distributions tDnu8 n“1ĂS1as follows Dnpϕq:“ xyUTn,ϕy, ϕ PS. Since lim nÑ8yUTnpx,kq “u˚px,kqfor a.e.kPRand uniformly for all xPR3, we have lim nÑ8Dnpϕq “ xu˚,ϕy. Consequently, replacing TbyTnin (3.3) and letting nÑ 8, we get lim nÑ8´@żTn 0Udt,∆2ϕD `@żTn 0p´k2U´ikσU qeiktdt,ϕD¯ “u˚p∆2ϕq ´k2u˚pϕq ´ikσu ˚pϕq “ p∆2´k2´ikσqu˚pϕq, which further implies by (3.2) that p∆2´k2´ikσqu˚pϕq “ xf,ϕy for every ϕPS. Then u˚px,kqis a solution to the equation (1.1) as a Schwartz distributio n. Furthermore, it follows from the uniqueness of the direct pr oblem that we obtain u˚px,kq “upx,kq, which gives that upx,kqis the Fourier transform of Upx,tq. By Theorem B.1, we have the estimates |B2 tU|,|BtU|,|Bt∇U|,|Bt∆U|,|∆U|,|∇∆U| À p1`tq´3 4. Moreover, they are continuous and belong to L2 tpRquniformly for all xPR3. Similarly, we may show that yB2 tU“ ´k2u,yBtU“iku, {Bt∇U“ik∇u, {Bt∆U“ik∆u,y∆U“∆u,{∇∆U“∇∆u. It follows from Plancherel’s theorem thatż`8 0´ |B2 tU|2` |BtU|2` |Bt∇U|2` |Bt∆U|2` |∆U|2` |∇∆U|2¯ dt “ż`8 ´8´ |k2u|2` |ku|2` |k∇u|2` |k∆u|2` |∆u|2` |∇∆u|2¯ dk. (3.4) By (3.4) and the exact observability bounds (A.1), we obtain }f}2 L2pBRqÀeCσ2ż`8 ´8ż BBR´ pk4`k2q|upx,kq|2`k2|∇upx,kq|2 ` pk2`1q|∆upx,kq|2` |∇∆upx,kq|2¯ dspxqdk ÀeCσ2ż8 ´8}upx,kq}2 BBRdk.6 P. LI, X. YAO, AND Y. ZHAO Sincefpxqis real-valued, we have upx,kq “upx,´kqforkPRand then ż8 ´8}upx,kq}2 BBRdk“2ż8 0}upx,kq}2 BBRdk, which completes the proof. /square Letδbe a positive constant and define Ipkq “żk δ}upx,ωq}2 BBRdspxqdω. The following lemma gives a link between the values of an anal ytical function for small and large arguments (cf. [14, Lemma A.1]). Lemma 3.2. Letppzqbe analytic in the infinite rectangular slab R“ tzPC:pδ,`8q ˆ p´ d,dqu, whereδis a positive constant, and continuous in Rsatisfying# |ppzq| ďǫ1, z P pδ,Ks, |ppzq| ďM, z PR, whereδ,K,ǫ1andMare positive constants. Then there exists a function µpzqwithzP pK,`8q satisfying µpzq ě64ad 3π2pa2`4d2qeπ 2dpa 2´zq, wherea“K´δ, such that |ppzq| ďMǫµpzq@zP pK,`8q. Lemma 3.3. Letfbe a real-valued function and }f}L2pBRqďQ. Then there exist positive constants dandδ,Ksatisfying 0ăδăK, which do not depend on fandQ, such that |Ipkq| ÀQ2e4Rpσ`2qκǫ2µpkq 1 @kP pK,`8q and ǫ2 1“żK δż BBR}upx,kq}2 BBRdspxqdk, µ pkq ě64ad 3π2pa2`4d2qeπ 2dpa 2´kq, wherea“K´δ. Proof.Let I1pkq “żk δż BBR´ pω4`ω2qupx,ωqupx,´ωq `ω2∇upx,ωq ¨∇upx,´ωq ` pω2`1q∆upx,ωq∆upx,´ωq `∇∆upx,ωq ¨∇∆upx,´ωq¯ dspxqdω, wherekPR. Following similar arguments as those in the proof of Lemma 2 .2, we may show that Rp´kqis also analytic for kPR. Sincefis real-valued, we have upx,kq “upx,´kqforkPR, which gives I1pkq “Ipkq, k ą0. It follows from Lemma 2.2 that |I1pkq| ÀQ2eCσ2e4Rpσ`1q|k|, k PR, which gives e´4Rpσ`2q|k||I1pkq| ÀQ2eCσ2, k PR.AN INVERSE SOURCE PROBLEM 7 An application of Lemma 3.2 leads to ˇˇe´4Rpσ`2q|k|IpkqˇˇÀQ2ǫ2µpkq@kP pK,`8q, where µpkq ě64ad 3π2pa2`4d2qeπ 2dpa 2´kq, which completes the proof. /square Here we state a simple uniqueness result for the inverse sour ce problem. Theorem 3.4. LetfPL2pBRqandIĂR`be an open interval. Then the source function fcan be uniquely determined by the multi-frequency Cauchy data tupx,kq,∇upx,kq,∆upx,kq,∇∆upx,kq: xP BBR,kPIu. Proof.Letupx,kq “∇upx,kq “∆upx,kq “∇∆upx,kq “0 for all xP BBRandkPI. It suffices to prove that fpxq “0. By Lemma 2.2, upx,kqis analytic in the infinite slab Rfor anyδą0, which implies that upx,kq “∆upx,kq “0 for all kPR`. We conclude from Lemma 3.1 that f“0./square The following result concerns the estimate of upx,kqfor high wavenumbers. Lemma 3.5. LetfPHnpBRqand }f}HnpBRqďQ. Then the following estimate holds: ż8 s}upx,kq}2 BBRdkÀ1 sn´3}f}2 HnpBRq. Proof.Recall the identity ż8 s}upx,kq}2 BBRdk“ż8 sż BBR´ pk4`k2q|upx,kq|2`k2|∇upx,kq|2 ` pk2`1q|∆upx,kq|2` |∇∆upx,kq|2¯ dspxqdk. (3.5) Using the decomposition Rpκq “ p∆2´κ4q´1“1 2κ2“ p´∆´κ2q´1´ p´∆`κ2q´1‰ , we obtain upxq “ż BR1 2κ2´eiκ|x´y| 4π|x´y|´e´κ|x´y| 4π|x´y|¯ fpyqdy, x P BBR. For instance, we consider one of the integrals on the right-h and side of (3.5) J:“ż8 sk4|upx,kq|dk “ż8 sk4ˇˇˇż BR1 2κ2´eiκ|x´y| 4π|x´y|´e´κ|x´y| 4π|x´y|¯ fpyqdyˇˇˇ2 dk. Using the spherical coordinates r“ |x´y|originated at y, we have J“1 8πż8 sż BBRk2ˇˇˇż2π 0dθżπ 0sinϕdϕż8 0peiκr´e´κrqfrdrˇˇˇ2 dspxqdk. By the integration by parts and noting xP BBRand supp fĂBˆRĂBRfor some ˆRăR, we obtain J“1 4πż8 sż BBRk2ˇˇˇż2π 0dθżπ 0sinϕdϕż2R R´ˆR´eiκr piκqn´e´κr p´κqn¯Bnpfrq Brndrˇˇˇ2 dspxqdk.8 P. LI, X. YAO, AND Y. ZHAO SincexP BBRand |κ| ěk1{2forką0, we get from direction calculations that JÀ }f}2 HnpBRqż8 sk2´ndkÀ1 sn´3}f}2 HnpBRq. Theotherintegrals ontheright-handsideof (3.5)can beest imated similarly. Thedetails areomitted for brevity. /square Define a real-valued function space CQ“ tfPHnpBRq:ně4,}f}HnpBRqďQ,suppfĂBˆRĂBR, f:BRÑRu, whereˆRăR. Now we are in the position to present the main result of this p aper. Theorem 3.6. Letupx,κqbe the solution of the scattering problem (1.1)corresponding to the source fPCQ. Then for ǫsufficiently small, the following estimate holds: }f}2 L2pBRqÀeCσ2´ ǫ2`Q2 K1 2pn´3qpln|lnǫ|q1 2pn´3q¯ , (3.6) where ǫ:“żK 0}upx,kq}2 BBRdk“żδ 0}upx,kq}2 BBRdk`ǫ2 1. Proof.We can assume that ǫďe´1, otherwise the estimate is obvious. First, we link the data Ipkqfor large wavenumber ksatisfying kďLwith the given data ǫ1of small wavenumber by using the analytic continuation in Lemm a 3.3, where Lis some large positive integer to be determined later. It follows from Lemma 3.3 tha t Ipkq ÀQ2ec|κ|ǫµpκq 1 ÀQ2exptcκ´c2a a2`c3ec1pa 2´κq|lnǫ1|u ÀQ2expt´c2a a2`c3ec1pa 2´κq|lnǫ1|p1´c4κpa2`c3q aec1pκ´a 2q|lnǫ1|´1qu ÀQ2expt´c2a a2`c3ec1pa 2´Lq|lnǫ1|p1´c4Lpa2`c3q aec1pL´a 2q|lnǫ1|´1qu ÀQ2expt´b0e´c1L|lnǫ1|p1´b1Lec1L|lnǫ1|´1qu, wherec,ci,i“1,2 andb0,b1are constants. Let L“#” 1 2c1ln|lnǫ1|ı , k ď1 2c1ln|lnǫ1|, k, k ą1 2c1ln|lnǫ1|. IfKď1 2c1ln|lnǫ1|, we obtain for sufficiently small ǫ1that Ipkq ÀQ2expt´b0e´c1L|lnǫ1|p1´b1Lec1L|lnǫ1|´1qu ÀQ2expt´1 2b0e´c1L|lnǫ1|u. Notinge´xďp2n`3q! x2n`3forxą0, we have IpLq ÀQ2ep2n`3qc1L|lnǫ1|´p2n`3q.AN INVERSE SOURCE PROBLEM 9 TakingL“1 2c1ln|lnǫ1|, combining the above estimates, Lemma 3.1 and Lemma 3.5, we g et }f}2 L2pBRqÀeCσ2´ ǫ2`IpLq `ż8 Lż BBR}upx,kq}2 BBRdk¯ ÀeCσ2´ ǫ2`Q2ep2n`3qc1L|lnǫ1|´p2n`3q`Q2 Ln´3¯ ÀeCσ2´ ǫ2`Q2´ |lnǫ1|2n`3 2|lnǫ1|´p2n`3q` pln|lnǫ1|q3´n¯ ¯ ÀeCσ2´ ǫ2`Q2´ |lnǫ1|´2n`3 2` pln|lnǫ1|q3´n¯ ¯ ÀeCσ2´ ǫ2`Q2pln|lnǫ1|q3´n¯ ÀeCσ2´ ǫ2`Q2 K1 2pn´3qpln|lnǫ1|q1 2pn´3q¯ ÀeCσ2´ ǫ2`Q2 K1 2pn´3qpln|lnǫ|q1 2pn´3q¯ , where we have used |lnǫ1|1{2ěln|lnǫ1|for sufficiently small ǫ1and ln |lnǫ1| ěln|lnǫ|. IfKą1 2c1ln|lnǫ1|, we have from Lemma 3.5 that }f}2 L2pBRqÀeCσ2´ ǫ2`ż8 Kż BBR}upx,kq}2 BBRdk¯ ÀeCσ2´ ǫ2`Q2 Kn´3¯ ÀeCσ2´ ǫ2`Q2 K1 2pn´3qpln|lnǫ|q1 2pn´3q¯ , which completes the proof. /square It can be observed that for a fixed damping coefficient σ, the stability (3.6) consists of two parts: the data discrepancy and the high frequency tail. The former is of the Lipschitz type. The latter decreases as Kincreases which makes the problem have an almost Lipschitz s tability. But the stability deteriorates exponentially as the damping coeffic ientσincreases. Appendix A.An exact observability bound Consider the initial value problem for the damped biharmoni c plate wave equation # B2 tUpx,tq `∆2Upx,tq `σBtUpx,tq “0, px,tq PBRˆ p0,`8q, Upx,0q “0,BtUpx,0q “fpxq, x PBR.(A.1) The following theorem presents an exact observability boun d for the above equation. The proof follows closely from that in [8, Theorem 3.1]. Theorem A.1. Let the observation time 4p2R`1q ăTă5p2R`1q. Then there exists a constant Cdepending on the domain BRsuch that }f}2 L2pBRqďCeCσ2` }B2 tU}2 L2pBBRˆp0,Tqq` }BtU}2 L2pBBRˆp0,Tqq` }Bt∇U}2 L2pBBRˆp0,Tqq ` }Bt∆U}2 L2pBBRˆp0,Tqq` }∆U}2 L2pBBRˆp0,Tqq` }∇∆U}2 L2pBBRˆp0,Tqq˘ .(A.2)10 P. LI, X. YAO, AND Y. ZHAO Before showing the proof, we introduce the energies Eptq “1 2ż Ω` |BtUpx,tq|2` |∆Upx,tq|2` |Upx,tq|2˘ dx, E0ptq “1 2ż Ω` |BtUpx,tq|2` |∆Upx,tq|2˘ dx, and denote F2“ż BΩˆpt1,t2q` |B2 tUpx,tq|2` |BtUpx,tq|2` |Bt∇Upx,tq|2 ` |Bt∆Upx,tq|2` |∆Upx,tq|2` |∇∆Upx,tq|2˘ dspxqdt. Lemma A.2. LetUbe a solution of the damped biharmonic plate wave equation (A.1)with the initial value fPH1pBRq,suppf ĂBR. Let0ďt1ăt2ďTand1ď2σ. Then the following estimates holds: Ept2q ďe4pt2´t1q2p2Ept1q `F2q, (A.3) Ept2q ďep2σ`4pt2´t1qqpt2´t1qpEpt2q `F2q. (A.4) Proof.Multiplying both sides of (A.1) by pBtUqeθtand integrating over Ω ˆ pt1,t2qgive ż Ωˆppt1,t2q´1 2BtpBtUq2`∆2UBtU`σpBtUq2¯ eθtdxdt“0. Using the integration ∆2UBtUby parts over Ω and noting ∆ UBtp∆Uq “1 2Bt|∆U|2, we obtain żt2 t1pBtE0ptqqeθtdt`ż Ωˆpt1,t2qσpBtUq2eθtdxdt `ż BΩˆpt1,t2qpBνp∆UqBtU´∆UBtpBνUqqeθtdspxqdt“0. Hence, E0pt2qeθt2´E0pt1qeθt1“ż Ωˆpt1,t2q´θ 2ppBtUq2` |∆U|2q ´σpBtUq2¯ eθtdxdt ´ż BΩˆpt1,t2qpBνp∆UqBtU´∆UBtpBνUqqeθtdspxqdt“0. Lettingθ“0, using Schwartz’s inequality, and noting σą0, we get E0pt2q ďE0pt1q `ż Ωˆpt1,t2qp´σqpBtUq2dxdt `1 2ż BΩˆpt1,t2q´ pBtUq2` pBtpBνUqq2¯ dspxqdt `1 2ż BΩˆpt1,t2q´ p∆Uq2` pBνp∆Uqq2¯ dspxqdt ďE0pt1q `F2.AN INVERSE SOURCE PROBLEM 11 Similarly, letting θ“2σ, we derive E0pt1qe2σt1ďE0pt2qe2σt2`ż Ωˆpt2,t1q´σp∆Uq2dxdt `1 2ż BΩˆpt1,t2q´ pBtUq2` pBtpBνUqq2¯ e2σtdspxqdt `1 2ż BΩˆpt1,t2q´ p∆Uq2` pBνp∆Uqq2¯ e2σtdspxqdt ďE0pt2qe2σt2`1 2ż BΩˆpt1,t2q´ pBtUq2` pBtpBνUqq2¯ e2σtdspxqdt `1 2ż BΩˆpt1,t2q´ p∆Uq2` pBνp∆Uqq2¯ e2σtdspxqdt. which gives E0pt1q ďe2σpt2´t1qpE0pt2q `F2q. The proof is completed by following similar arguments as tho se in [8, Lemma 3.2]. /square Now we return to the proof of Theorem A.2 Proof of Theorem A.2. Letϕpx,tq “ |x´a|2´θ2pt´T 2q2, where dist pa,Ωq “1,θ“1 2. Using the Carleman-type estimate in [18], we obtain τ6ż Q|U|2e2τϕdxdt`τ3ż Q|BtU|2e2τϕdxdt`τż Q|∆U|2e2τϕdxdt Àż QppB2 t`∆2qUq2e2τϕdxdt `ż BQτ6p|Bν∆U|2` |Bt∆U|2` |B2 tUq|2qe2τϕdspxqdt. (A.5) It is easy to see that 1 ´θ2ε2 0ďϕon Ω ˆ t|t´T 2| ăε0ufor some positive εă1. Then we have from A.4 that τ6ż Q|U|2e2τϕdxdt`τ3ż Q|BtU|2e2τϕdxdt`τż Q|∆U|2e2τϕdxdt ěτ6ż ΩˆpT 2´ε0,T 2`ε0q|U|2e2τp1´θ2ε2 0qdxdt`τ3ż ΩˆpT 2´ε0,T 2`ε0q|BtU|2e2τp1´θ2ε2 0qdxdt `τż ΩˆpT 2´ε0,T 2`ε0q|∆U|2e2τp1´θ2ε2 0qdxdt ěτe2τp1´θ2ε2 0qż ΩˆpT 2´ε0,T 2`ε0qEptqdt ěτe2τp1´θ2ε2 0qε0p2e´p2σ`4TqTEp0q ´F2q. (A.6) Moreover, it follows from (A.4) and ϕď p2R`1q2´θ2T2{4 on Ω ˆ p0,Tqthat τ6ż Q|U|2e2τϕdxdt`τ3ż Q|BtU|2e2τϕdxdt`τż Q|∆U|2e2τϕdxdt ďτ6e2τpp2R`1q2´θ2T2{4qpEp0q `EpTqq ďτ6e2τpp2R`1q2´θ2T2{4qppe4T2`1qEp0q `e4T2F2q.12 P. LI, X. YAO, AND Y. ZHAO By (A.5) and (A.6), we obtain τe2τp1´θ2ε2 0qε0e´p2σ`1`4TqTEp0q `τ6ż Q|U|2e2τϕdxdt`τ3ż Q|BtU|2e2τϕdxdt`τż Q|∆U|2e2τϕdxdt ď´ σ2ż Q|BtU|2e2τϕdxdt`ż BQτ6p|Bν∆U|2` |Bt∆U|2` |B2 tUq|2qe2τϕdspxqdt ` pτe2τp1´θ2ε2 0q`τ6e2τpp2R`1q2´θ2T2{4qe4T2qF2`τ6e2τpp2R`1q2´θ2T2{4qe4T2Ep0q¯ .(A.7) Choosing τsufficiently large, we may remove the first integral on the righ t hand side of (A.7). We also choose T2“4p2R`1q2 θ2`4ε2 0andτ“ p2σ`8TqT`lnp2pε0q´1Cq `Cσ2. Noting τ5e´τď5!, we have τ5e2τpp2R`1q2´θ2T2{4´1`θ2ε2 0q`p2σ`8TqT“τ5e´2τ`p2σ`8TqT ď5!e´τ`p2σ`8TqTďε0 2C. In addition, since Tď5p2R`1q, it follows that τ5e2τpp2R`1q2´1`θ2ε2 0`p2σ`4TqTqďτ5e2pp2σ`8TqT`Cσ2`Cqp2R`1q2`p2σ`4TqTďCeCσ2. Using the above inequality and the inequality ϕă p2R`1q2onQand dividing both sides in (A.7) by the factor of Ep0qon the left hand side, we obtain Ep0q ďCeCσ2F2. Sincefis supported in Ω, there holds }U}L2pBBRˆp0,TqqďC}BtU}L2pBBRˆp0,Tqq, which completes the proof. /square Appendix B.A decay estimate We prove a decay estimate for the solution of the initial valu e problem of the damped plate wave equation # B2 tUpx,tq `∆2Upx,tq `σBtUpx,tq “0, px,tq PR3ˆ p0,`8q, Upx,0q “0,BtUpx,0q “fpxq, x PR3,(B.1) wherefpxq PL1pR3q XHspR3q. By the Fourier transform, the solution Upx,tqof (B.1) is given as Upx,tq “F´1pmσpt,ξqˆfpξqqpxq, whereF´1denotes the inverse Fourier transform, mσpt,ξq “e´σ 2t a σ2´4|ξ|4´ e1 2t? σ2´4|ξ|4´e´1 2t? σ2´4|ξ|4¯ , andˆfpξqis the Fourier transform of f, i.e., ˆfpξq “1 p2πq3ż R3e´ix¨ξfpxqdx. Leta σ2´4|ξ|4“ia 4|ξ|4´σ2when |ξ|4ąσ2 4. Then we have mσpt,ξq “$ ’& ’%e´σ 2tsinh pt 2? σ2´4|ξ|4q? σ2´4|ξ|4, |ξ|4ăσ2 4, e´σ 2tsinpt 2? 4|ξ|4´σ2q? 4|ξ|4´σ2, |ξ|4ąσ2 4.AN INVERSE SOURCE PROBLEM 13 It is clear to note from the representation of mσpt,ξqthat the solution Upx,tqdepends on both of the low and high frequency of ξ. In fact, the solution Upx,tqbehaves as a “parabolic type” of e´t∆2f for the low frequency part, while for the high frequency part it behaves like a “dispersive type” of eit∆2f. Theorem B.1. LetUpx,tqbe the solution of (B.1). ThenUpx,tqsatisfies the decay estimate supxPR3|Bα xBj tUpx,tq| À p1`tq´3`|α| 4}f}L1pR3q`e´ct}f}HspR3q, (B.2) wherejPN,αis a multi-index vector in N3such that Bα“ Bα1x1Bα2x2Bα3x3,są2j` |α| ´1 2andcą0 is some positive constant. In particular, for |α| “s“0, the following estimate holds: supxPR3|Upx,tq| À p1`tq´3 4p}f}L1pR3q` }f}L2pR3qq. (B.3) Remark B.2. The estimate (B.3)provides a time decay of the order Opp1`tq´3 4qforUpx,tq uniformly for all xPR3, which gives sup xPR3ż8 0|Upx,tq|2dtÀż8 0p1`tq´3{2dtă `8. Hence, let Upx,tq “0whentă0, thenUpx,tqhas a Fourier transform ˆUpx,kq PL2pRqfor each xPR3. Moreover, the following Plancherel equality holds: ż`8 0|Upx,tq|2dt“ż`8 ´8|ˆUpx,kq|2dk. Remark B.3. To study the inverse source problem, it suffices to assume that fPH4pR3q. In this case, it follows from the above theorem that both B2 tUpx,tqand∆2Upx,tqare continuous functions. Moreover, we have from (B.2)that the following estimate holds: supxPR3|Bj tUpx,tq| À p1`tq´3 4}f}L1pR3q`e´ct}f}HspR3q, j “1,2, supxPR3|Bα xUpx,tq| À p1`tq´3`|α| 4}f}L1pR3q`e´ct}f}HspR3q,|α| ď4. Proof.Without loss of generality, we may assume that σ“1, and then mσpt,ξq “e´1 2t a 1´4|ξ|4´ e1 2t? 1´4|ξ|4´e´1 2t? 1´4|ξ|4¯ . First we prove (B.2) for j“0. Choose χPC8 0pR3qsuch that supp χĂBp0,1 2qandχpξq “1 for |ξ| ď1 4. Let Upx,tq “F´1pmpt,ξqχpξqˆfq `F´1pmpt,ξqp1´χpξqqˆfq :“U1px,tq `U2px,tq. ForU1px,tq, sincea 1´4|ξ|4ď1´2|ξ|4when 0 ď |ξ| ď1 2, we have for |ξ| ď1 2that mpt,ξq “1a 1´4|ξ|4e´t 2p1˘? 1´4|ξ|4qď2e´t|ξ|4, t ě0. For each xPR3we have BαU1px,tq “ż R3eix¨ξpiξqαmpt,ξqχpξqˆfpξqdξ, which gives sup xPR3|Bα xU1px,tq| ďż |ξ|ď1 2|ξ|αe´t|ξ|4|ˆfpξq|dξÀ }ˆf}L8pR3qż |ξ|ď1 2|ξ|αe´t|ξ|4dξ.14 P. LI, X. YAO, AND Y. ZHAO Since ż |ξ|ď1 2|ξ|αe´t|ξ|4dξď# C, 0ďtď1, t´3`|α| 4, t ě1, and }ˆf}L8pR3qď }f}L1pR3q, we obtain sup xPR3|Bα xU1px,tq| À p1`tq´3`|α| 4|f}L1pR3q@αPN3. (B.4) To estimate U2px,tq, noting p1´∆qp 2U2px,tq “ż R3eix¨ξp1` |ξ|2qp 2mpt,ξqp1´χpξqqˆfpξqdξ, we have from Plancherel’s theorem that ż R3|p1´∆qp 2U2px,tq|2dx“ż R3p1` |ξ|2qp|mpt,ξqp1´χpξqqˆfpξq|2dξ. (B.5) It holds that |mpt,ξq| ď$ ’’’’& ’’’’%te´t 2p1´? 1´4|ξ|4qˇˇˇ1´e´t? 1´4|ξ|4 t? 1´4|ξ|4ˇˇˇÀe´t 8,1 2ă |ξ| ď? 2 2, 1 2e´t 2sint 2? 4|ξ|4´1 t 2? 4|ξ|4´1Àe´t 8,? 2 2ă |ξ| ď1, e´t 2? 4|ξ|4´1|sint 2a 4|ξ|4´1| ďe´t 2? 4|ξ|4´1, |ξ| ą1. Hence, when |ξ| ě1 2we have |p1` |ξ|2qmpt,ξq| Àe´t 8. It follows from (B.5) that }U2px,tq}2 HppR3qďż |ξ|ě1 2|p1` |ξ|2qp 2mpt,ξqˆfpξq|2dξ ďe´t 4ż R3|p1` |ξ|2q´1`p 2ˆfpξq|2dξ“e´t 4}f}2 Hp´2pR3q. On the other hand, by Sobolev’s theorem, we have for pą3 2that sup xPR3|U2px,tq| ď }U2p¨,tq}HppR3qÀe´t 8}f}Hp´2pR3q. More generally, for any αPN3it holds that p1´∆qp 2Bα xU2px,tq “F´1pp1` |ξ|2qp 2mpt,ξqp1´χpξqqyBαfq, which leads to sup xPR3|Bα xU2px,tq| Àe´t 8}Bαf}Hp´2pR3qÀe´t 8}f}HspR3q. (B.6) Heres“p´2` |α| ą |α| ´1 2by choosing pą3 2. Combining the estimate (B.4) with (B.6) yields (B.2) for j“0. Next we consider the general case with j‰0. Noting Bj tUpx,tq “ż R3eix¨ξBj tmpt,ξqˆfpξqdξ,AN INVERSE SOURCE PROBLEM 15 we obtain from direct calculations that Bj tmpt,ξq “ Bj´e´1 2t a 1´4|ξ|4´ e1 2t? 1´4|ξ|4´e´1 2t? 1´4|ξ|4¯¯ “jÿ l“02´jpa 1´4|ξ|4ql´1e´t 2´ e1 2t? 1´4|ξ|4` p´1ql`1e´1 2t? 1´4|ξ|4¯ :“jÿ l“0mlpt,ξq. Hence we can write Bj tUpx,tqas Bj tUpx,tq “jÿ l“0ż R3eix¨ξmlpt,ξqˆfpξqdξ:“jÿ l“0Wlpx,tq. (B.7) For each 0 ďlďj, j‰0, using similar arguments for the case j“0 we obtain sup xPR3|Bα xWlpx,tq| ď p1`tq´3`|α| 4}f}L1pR3q`e´t 8}f}HspR3q (B.8) forsą2l` |α| ´1 2. Combining (B.7) and (B.8), we obtain the general estimate ( B.2). /square Remark B.4. For the damped biharmonic plate wave equation, besides the d ecay estimate (B.2), we can deduce other decay estimates of the Lp-Lqtype and time-space estimates by more sophisticated analysis for the Fourier multiplier mpt,ξq. For example, it can be proved that }Upx,tq}LqpR3qÀ p1`tq´3 4p1 p´1 qq}f}LppR3q`e´ct}f}Wq,spR3q, where1ăpďqă `8andsě3p1 q´1 2q ´2. We hope to present the proofs of these Lp-Lqestimates and their applications elsewhere. Acknowledgement We would like to thank Prof. Masahiro Yamamoto for providing the reference [18] on Carleman estimates of the Kirchhoff plate equation. The research of PL is supported in part by the NSF grant DMS-1912704. The research of XY is supported in part by NSFC ( No. 11771165). The research of YZ is supported in part by NSFC (No. 12001222). References [1] T. Aktosun and V. Papanicolaou, Time evolution of the sca ttering data for a fourth-order linear differential operator, Inverse Problems, 24 (2008), 055013. [2] G. Bao, J. Lin, and F. Triki, A multi-frequency inverse so urce problem, J. Differential Equations, 249 (2010), 3443–3465. [3] G. Bao, P. Li, and Y. Zhao, Stability for the inverse sourc e problems in elastic and electromagnetic waves, J. Math. Pures Appl., 134 (2020), 122–178. [4] J. Cheng, V. Isakov, and S. Lu, Increasing stability in th e inverse source problem with many frequencies, J. Differential Equations, 260 (2016), 4786–4804. [5] M. Entekhabi and V. Isakov, On increasing stability in th e two dimensional inverse source scattering problem with many frequencies, Inverse Problems, 34 (2018), 055005 . [6] M. Entekhabi and V. Isakov, Increasing stability in acou stic and elastic inverse source problems, SIAM J. Appl. Math., 52 (2020), 5232–5256. [7] F. Gazzola, H.-C. Grunau, and G. Sweers, Polyharmonic bo undary value problems, Springer-Verlag Berlin Hei- delberg, 2010. [8] V. Isakov and S. Lu, Increasing stability in the inverse s ource problem with attenuation and many frequencies, SIAM J. Appl. Math., 78 (2018), 1–18. [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 [10] K. Krupchyk, M. Lassas, and G. Uhlmann, Inverse boundar y value problems for the perturbed poly-harmonic operator, Trans. Amer. Math. Soc., 366 (2014), 95–112. [11] M. Lassas, K. Krupchyk and G. Uhlmann, Determining a firs t order perturbation of the biharmonic operator by partial boundary measurements, J. Funct. Anal., 262 (2012) , 1781–1801. [12] P. Li, X. Yao, and Y. Zhao, Stability for an inverse sourc e problem of the biharmonic operator, arXiv:2102.04631. [13] P. Li and G. Yuan, Increasing stability for the inverse s ource scattering problem with multifrequencies, Inverse Problems Imag., 11 (2017), 745–759. [14] P. Li, J. Zhai, and Y. Zhao, Stability for the acoustic in verse source problem in inhomogeneous media, SIAM J. Appl. Math., to appear. [15] N.V. Movchan, R.C. McPhedran, A.B. Movchan, and C.G. Po ulton, Wave scattering by platonic grating stacks, Proc. R. Soc. A, 465 (2009), 3383–3400. [16] J. Rousseau and L. Robbiano, Spectral inequality and re solvent estimate for the bi-harmonic operator, J. Eur. Math. Soc., 22 (2020), 1003–1094. [17] T. Tyni and V. Serov, Scattering problems for perturbat ions of the multidimensional biharmonic operator, Inverse Problems and Imaging, 12 (2018), 205–227. [18] G. Yuan and M. Yamamoto, Lipschitz stability in inverse problems for a Kirchhoff plate equation, Asymptotic Analysis, 53 (2007), 29–60. Department of Mathematics, Purdue University, West Lafaye tte, Indiana 47907, USA Email address :lipeijun@math.purdue.edu School of Mathematics and Statistics, China Central Normal University, Wuhan, Hubei, China Email address :yaoxiaohua@mail.ccnu.edu.cn School of Mathematics and Statistics, China Central Normal University, Wuhan, Hubei, China Email address :zhaoyueccnu@163.com

2021-02-28

This paper is concerned with the stability of the inverse source problem for the damped biharmonic plate equation in three dimensions. The stability estimate consists of the Lipschitz type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The stability also shows exponential dependence on the constant damping coefficient. The analysis employs Carleman estimates and time decay estimates for the damped plate wave equation to obtain an exact observability bound and depends on the study of the resonance-free region and an upper bound of the resolvent of the biharmonic operator with respect to the complex wavenumber.

Stability for an inverse source problem of the damped biharmonic plate equation

2103.00461v1

Ultrathin ferrimagnetic GdFeCo lms with very low damping Lakhan Bainsla*,1,a)Akash Kumar,1Ahmad A. Awad,1Chunlei Wang,2Mohammad Zahedinejad,1Nilamani Behera,1Himanshu Fulara,1Roman Khymyn,1Afshin Houshang,1Jonas Weissenrieder,2and J. Akerman1,b) 1)Physics Department, University of Gothenburg, 412 96 Gothenburg, Sweden. 2)Department of Applied Physics, KTH Royal Institute of Technology, 106 91 Stockholm, Sweden Ferromagnetic materials dominate as the magnetically active element in spintronic devices, but come with drawbacks such as large stray elds, and low operational frequencies. Compensated ferrimagnets provide an alternative as they combine the ultrafast magnetization dynamics of antiferromagnets with a ferromagnet-like spin-orbit-torque (SOT) behavior. However to use ferrimagnets in spintronic devices their advantageous prop- erties must be retained also in ultrathin lms ( t<10 nm). In this study, ferrimagnetic Gd x(Fe87:5Co12:5)1x thin lms in the thickness range t= 2{20 nm were grown on high resistance Si(100) substrates and studied using broadband ferromagnetic resonance measurements at room temperature. By tuning their stoichiometry, a nearly compensated behavior is observed in 2 nm Gd x(Fe87:5Co12:5)1xultrathin lms for the rst time, with an eective magnetization of Me= 0.02 T and a low eective Gilbert damping constant of = 0.0078, comparable to the lowest values reported so far in 30 nm lms. These results show great promise for the development of ultrafast and energy ecient ferrimagnetic spintronic devices. I. INTRODUCTION Spintronic devices utilize the spin degree of freedom for data storage, information processing, and sensing1,2with commercial applications such as hard drives, magnetic random access memories, and sensors. Besides conven- tional memory applications based on quasi-static opera- tion of magnetic tunnel junctions, high frequency spin- tronic oscillators3,4have recently been demonstrated for analog computing applications such as bio-inspired neu- romorphic computing5,6, logic operations, energy har- vesting and Ising Machines.7For the rst time, such oscil- lators are now used in commercial magnetic hard drives to facilitate writing to the disc.8The key challenges in developing such devices is to nd material combinations which allow for fast operation, low-power consumption, non-volatility, and high endurance. Due to their nat- ural spin polarization and easy manipulation, ferromag- netic materials (FM) dominate as active elements in these devices.4However, FMs come with drawbacks such as: (i) large magnetic stray elds aecting the operation of neighbouring devices; (ii) limited scalability of magnetic bits in memory devices; (iii) the operating frequency of spin-based oscillators limited by ferromagnetic resonance frequency, and (iv) slow synchronization of such oscilla- tors. These shortcomings drive researchers to nd more suitable materials for future spintronic devices. Very recently, the interest in antiferromagnetic (AFM) spintronics9{11increased rapidly, as AFM materials have no stray elds and can oer ultrafast spin dynamics, in- cluding AFM resonance frequencies in the THz region. It was theoretically shown that such high-frequency ex- citations are possible to achieve without any applied magnetic eld by injecting spin currents into AFM a)Electronic mail: lakhan.bainsla@physics.gu.se b)Electronic mail: johan.akerman@physics.gu.sematerials.12{15Experiments have since demonstrated possible THz writing/reading capabilities.16However, the absence of a net magnetic moment in AFMs leads to diculties in the read-out of the spin dynamics, in- cluding any microwave output signal from the AFM oscillators.13{15 A possible solution is presented by ferrimagnets (FiMs), which combine the properties of FMs and AFMs. FiMs posses magnetic sub-lattices in the same way as AFMs do, but their sub-lattices are inequivalent. The magnetic sub-lattices in FiMs often consist of dierent magnetic ions, such as rare earth (e.g. Gd) and transi- tion metal (e.g. Fe, Co) alloys (RE-TM) such as CoGd, and as a result, a large residual magnetization remains despite the two opposing sub-magnetizations. The tem- perature dependence of RE and TM sub-magnetizations in FiM can be quite dierent which result in magneti- zations that can increase, and even change sign, with temperature17,18, in stark contrast to the non-monotonic decreasing temperature dependence for FMs and AFMs. Similar eects could also be seen by varying the com- position of ferrimagnetic alloys instead of changing the temperature.19In addition, the dierent properties of the two magnetic sub-lattices also results in two compen- sation points, namely the magnetization compensation pointTmand the angular compensation point Ta. AtTm, the two magnetic sub-lattices cancel each other, which re- sults in a zero net magnetic moment, while at T a, their net angular momentum vanishes, as in AFMs. Therefore, atTa, FiMs can have a near-THz resonance as in AFMs, while still having a net magnetic moment which can lead to strong read-out signals, including ecient microwave signal output from FiM-based oscillators20, as well as ef- cient control and excitation. FiMs also show high spin polarization which also make them suitable candidate for ecient magnetic tunnel junctions.21 Due to these unique properties, research in FiMs for spintronic applications is intensifying22, focusing mainly on RE-TM based systems such as CoTb23, CoGd24, andarXiv:2111.08768v1 [cond-mat.mtrl-sci] 16 Nov 20212 Figure 1. (a) Schematic illustration of the coplanar waveguide (CPW), the thin lm sample and its orientation, the directions of the applied magnetic eld H, the microwave eld hrf, and the eective magnetic eld Heduring FMR measurements. Inset shows the lm stack. (b) FMR response (derivative of the FMR absorption) for a 10 nm Gd 12:5Fe76:1Co11:4lm (S2) recorded at dierent frequencies and tted (solid lines) to Eq. 1. While FMR curves were recorded at 1 GHz frequency intervals throughout this study, gure (b) only shows curves with
f= 2 GHz for clarity. GdFeCo25and Mn 3xPtxGa26,27based Heusler alloy. Among these, GdFeCo has been studied the most with demonstrations of fast domain wall motion28and ultra- fast spin dynamics17nearTa, large spin-orbit torques and their sign reversal,25,29low magnetic damping in thick 30 nm lms,30and sub-picosecond magnetization reversal,31to name a few. What is missing, however, is a demonstration that these unique material properties per- sist down to much thinner lms, which will ultimately be needed if FiMs are to be used in spin-Hall nano oscillators (SHNOs).4 In the present study, we systematically study the growth and functional properties of ultrathin ferrimag- netic Gd x(Fe87:5Co12:5)1xthin lms [referred to as Gdx(FeCo) 1xhereafter]. GdFeCo thin lms in the thickness range of 2{20 nm were grown on high resis- tance silicon (HR-Si) substrate. The atomic composi- tion of Gd x(FeCo) 1xwas controlled using co-sputtering and determined using inductively coupled plasma optical emission spectroscopy (ICP-OES). The magnetic prop- erties and Gilbert damping were studied using broad- band ferromagnetic resonance (FMR) measurements. We also demonstrate ultra low Gilbert damping for 2 nm GdFeCo, near the compensation point of Gd x(FeCo) 1x. These results paves the way for integration of FiMs into various spintronic devices and applications. II. RESULTS AND DISCUSSION The growth conditions for GdFeCo were rst optimized by growing four 10 nm thick Gd 12:5Fe76:1Co11:4lms on HR-Si (100) substrates using dierent MgO seed layer thicknesses: 0 nm (S1), 6 nm (S2), 10 nm (S3 & S4);in S4, the seed was annealed at 600C for 1 hour prior to GdFeCo deposition to check the eect of MgO crys- tallinity. MgO was chosen as seed since it is insulating and therefore will not contribute any spin sinking to the magnetic damping.32 A. Seed layer dependence on 10nm thick Gd12:5Fe76:1Co11:4lms Further details of the growth conditions are given in the experimental section. FMR measurements, on 6 3 mm2rectangular pieces cut from these lms, were then performed using a NanOsc PhaseFMR-40 FMR Spec- trometer. The sample orientation on the coplanar waveg- uide (CPW), together with the directions of the applied eld, the microwave excitation eld hrf, and the eec- tive magnetic eld He, are shown in Fig. 1(a). Typical (derivative) FMR absorption spectra obtained for S2 are shown in gure 1(b) together with ts to a sum of sym- metric and anti-symmetric Lorentzian derivatives:33 dP dH(H) =8C1
H(HHR) [
H2+ 4(HHR)2]2+2C2(
H24(HHR)2) [
H2+ 4(HHR)2]2 (1) whereHR,
H,C1, andC2represent the resonance eld, the full width at half maximum (FWHM) of the FMR ab- sorption, and the symmetric and anti-symmetric tting parameters of the Lorentzian derivatives, respectively. The extracted values of HRvs.fare shown in gure 2 (b) together with ts to Kittel's equation:34 f= 0 2q (HRHk)(HRHk+Meff) (2)3 Figure 2. (a) Seed layer dependence of frequency vs resonance eld of the 10 nm thick Gd 12:5Fe76:1Co11:4lms, here solid symbols and solid lines are the experimental data points and tting with equation (2), respectively. (b) Resonance linewidth (
H)vs.frequency of the 10 nm thick Gd 12:5Fe76:1Co11:4lms, here solid symbols and solid lines are the experimental data points and tting with equation (3), respectively. The eective Gilbert damping constant values of all the samples are given in gure 2 (b). The black and violet dotted lines in gure 2(b) shows the tting of equation (3) in low and high frequency regions, respectively. where, ,HkandMeare the gyromagnetic ratio, the in-plane magnetic anisotropy eld, and the eective mag- netization of the sample, respectively, all allowed to be free tting parameters. Values for andHkonly showed minor variation between the four samples, with =2= 29.4-30.0 GHz/T and Hk= 66-104 Oe. Mevaried more strongly, with values of 0.79, 1.19, 0.71 and 0.76 T ob- tained for S1, S2, S3 and S4, respectively. The eective Gilbert damping constant can then be obtained from ts of
Hvs.fto:35
H=
H0+4f 0(3) where the oset
H0represents the inhomogeneous broadening. Equation (3) is well tted to the experimen- tal values, using
H0andas adjustable tting param- eters for all the four samples, as shown in the gure 2(b).
H0= 2{4 mT is essentially sample independent within the measurement accuracy. In contrast, the obtained val- ues ofvary quite strongly and are given inside gure 2(b). The GdFeCo grown with 6 nm MgO seed layer (S2) clearly shows the lowest value of = 0:0055, although this might be aected by the slight non-linear behavior around 10 to 15 GHz. However, when only the high-eld data is tted, the extracted damping of = 0.0076 is still the lowest and at all frequencies the linewidth of S2 lies well below all the other samples. As damping is one of the most important parameters for spintronic devices, we hence chose the growth conditions of S2 for all subse- quent lms in this study.B. Thickness dependence on Gd 12:5Fe76:1Co11:4lms After optimizing the growth conditions for Gd12:5Fe76:1Co11:4, the thickness dependence of the lms was studied with the same composition using the growth conditions of sample S2. The FMR linewidth
Hvs. f is shown in gure 3(a) and exhibits a relatively strong dependence on thickness. It is noteworthy that the 4 nm lm shows the narrowest linewidth at all frequencies, clearly demonstrating that very low damping can be achieved also in ultra-thin GdFeCo. The extracted Meandare shown vs.thickness in gure 3(b), both showing a strong thickness dependence. Damping as low as = 0:0055 is obtained for the 10 nm thick lms. If only the high-eld portion of the data is tted, the extracted damping increases to 0.0076, which is still about an order of magnitude lower than any literature value on 10 or 30 nm lms.19,36Both the 10 and 20 nm lms showed a minor nonlinearity in
Hvs.fdata and were therefore analysed by tting the data in both the low and the high eld regions separately, as shown by the dotted lines in gure 3(a). Thevalue for the 20 nm lm increased slightly from 0.0098 to 0.0109 if only high eld data is used for analysis. The relatively higher damping for the 20 nm lm might be due to the radiative damping mechanism which increases proportionally with magnetic layer thickness.37We conclude that 2 nm ultrathin lms can indeed be grown with reasonably low damping. Since the damping is strongly thickness dependent in this regime, the optimum thickness for devices may likely be found in the 2{4 nm range.4 Figure 3. (a) FMR linewidth
Hvs.ffor four Gd 12:5Fe76:1Co11:4lms with dierent thicknesses, together with linear ts to equation (3). The dotted lines show ts for the 20 nm lm in its low and high frequency regions, respectively. (b) Eective magnetization and eective Gilbert damping constant vs.thickness; lines are guides to the eye. C. Composition dependence on 2nm thick lms To nally investigate whether we can achieve a com- pensated ferrimagnetic behavior also in ultra-thin lms, we grew 2 nm Gd x(FeCo) 1xlms in the composition range 12{27 at.% Gd. The lms were characterized using FMR spectrometry as described above and the extracted results are shown in gure 4. The extracted Meandfollow a similar trend as re- ported earlier for one order of magnitude thicker GdFeCo lms characterized using an all-optical pump-probe tech- nique.17We rst note that we can indeed reach an es- sentially fully compensated antiferromagnetic behavior in two lms around a composition of 25 at.% Gd. We have marked this compensation point with xmand a dashed line in gure 4 (c). Both lms show very low damping of 0.0078 and 0.009 respectively. However, just below this composition, the damping shows a peak, which is con- sistent with an angular compensation point, which we denote byxa. It is noteworthy that the extracted damp- ing value of = 0.0142 is still more than an order of magnitude lower than = 0.45 of 30 nm lms measured using FMR spectrometry19and= 0.20 of 20 nm lms measured using an optical pump-probe technique.17 III. CONCLUSION In view of the potential application of compensated ferrimagnets to spintronic devices, we prepared ferri- magnetic thin lms of Gd x(FeCo) 1xon high resistance Si(100) substrates and studied them using the FMR mea- surements. Their growth conditions were optimized us- ing 10 nm thick Gd 12:5Fe76:1Co11:4lms, after which thickness dependent studies were done on the same com- position in the thickness range of 2{20 nm. Composi- tion dependence studies were nally done on 2 nm thick Gdx(FeCo) 1xlms and an essentially compensated fer-rimagnetic behavior was observed for the rst time in ultrathin 2 nm lms. The angular momentum compensa- tion and magnetic compensation points observed in this work are very close to those reported earlier on much thicker lms in the literature. A record low value of about 0.0078 is obtained near the magnetic compensa- tion point, which is an order of magnitude lower than the values reported in the literature using similar analysis methods. The observation of compensated ferrimagnetic behavior in ultrathin lms together with very low value ofare promising results for the future development of ultrafast and energy ecient ferrimagnetic spintronic de- vices. EXPERIMENTAL SECTION A. Thin lms growth and composition analysis All the samples were prepared on high resistivity Si(100) substrates using a magnetron sputtering sys- tem with a base pressure of less than 2 108torr. Thin lms of Gd x(FeCo) 1xwere deposited using the co-sputtering of high purity (more than 99.95%) Gd and Fe 87:5Co12:5targets, and composition analysis was done using the inductively coupled plasma mass spectroscopy (ICP-MS). Thin lms stacking structure of Si(100)/MgO(t)/Gd 12:5Fe76:1Co11:4(10)/SiO 2(4) were used for seed layer dependence studies, here, the number in the bracket is the thickness of the layer in nm, where t=0, 6 and 10 nm. Four sam- ples, namely S1 to S4 were prepared to obtain the best conditions to grow Gd 12:5Fe76:1Co11:4(10) lms. For S1, Gd 12:5Fe76:1Co11:4(10) was grown directly over HR-Si (100) substrates, while in both S2 and S3 Gd 12:5Fe76:1Co11:4were grown with MgO seed layer of 6 and 10 nm, respectively. All the lay- ers in S1-S3 were grown at room temperature and5 Figure 4. (a) Frequency vs.resonance eld and (b) resonance linewidth vs.frequency, of 2 nm thick Gd x(FeCo) 1xlms as a function of Gd content in atomic %. (c) Eective magnetization and eective Gilbert damping constant vs.Gd content. Solid symbols represent the values obtained by tting the experimental FMR data in (a) and (b) using the equation (2) and (3), respectively; solid lines in (c) are guides to the eye. xaand xmshow the angular and magnetic compensation points, respectively, obtained from the literature17,19. no further heat treatment was given to them. In S4, 10 nm MgO seed layer were grown over HR Si(100) substrates at RT and followed by a in-situ post-annealing at 600C for 1 hour, and after that Gd12:5Fe76:1Co11:4were deposited. The stacking struc- ture of Si(100)/MgO(6)/Gd 12:5Fe76:1Co11:4(m)/SiO 2(4) were used for thickness dependence studies, where m is the thickness of Gd 12:5Fe76:1Co11:4layer, and varied from 2 to 20 nm. For composi- tion dependence studies, stacking structure of Si(100)/MgO(6)/Gd x(FeCo) 1x(2)/SiO 2(4) were used, where xvaried from 12.5 to 26.7. The composition of Gdx(FeCo) 1xlms was varied by changing the sput- tering rate of Fe 87:5Co12:5target, while keeping the Gd sputtering rate xed for most lms. All the samples for thickness dependence and composition dependence were grown at room temperature and no post-annealing was used. Layer thicknesses were determined by estimating the growth rate using the Dektak proler on more than 100 nm thick lms.B. Inductively coupled plasma mass spectroscopy (ICP-MS) measurements The elemental composition (Co, Fe, and Gd) of the thin lm samples was determined by inductively coupled plasma optical emission spectroscopy (ICP-OES) using a Thermo Fisher Scientic iCAP 6000 Series spectrometer. Each thin lm sample was exhaustively extracted in 5 mL HNO3 (65%, Supelco, Merck KgaA, Sigma-Aldrich) for a duration of 30 min. 5 mL ultrapure MilliQ-water (18 M cm) was added to the solution and the extract was al- lowed to rest for 30 minutes. The extract was transferred to a 100 mL volumetric ask. The extracted sample was then rinsed for several cycles in ultrapure water. The water used for rinsing was transferred to the same volu- metric ask. The extract was diluted to 100 mL for ICP analysis. ICP check standards were prepared from stan- dard solutions (Co and Fe: Merck, Germany; Ga: Accu- standard, USA). The relative standard deviation (from three individual injections) were within 1%.6 Table I. The obtained values of eective Gilbert damping constant at room temperature (RT) in this work and comparison with the lowest values reported so far in the literature at RT and also at their respective angular momentum compensation (Ta) and magnetic compensation (T m) points. Film composition Film thickness  Measurement technique Analysis method Reference Gd23:5Fe68:9Co7:6 30 0.45 (at RT) FMR Kittel's FMR19 0.35 (at RT) Pump-probe Gd22Fe74:6Co3:4 20 0.21 (at T a) Pump-probe -do-17 0.13 (at T m) Gd25Fe65:6Co9:4 10 0.07 (at RT) Spin torque FMR -do-36 0.01 (at RT) Spin torque FMR Ferrimagnetc resonance Gd23:5Fe66:9Co9:6 30 0.0072 (at RT) Domain wall (DW) Field driven DW30 motion mobility Gd12:5Fe76:1Co11:4 10 0.0055 Broadband FMR Kittel's FMR This work 0.0076 (HF data) -do- -do- This work Gd12:5Fe76:1Co11:4 4 0.0064 -do- -do- This work Gd12:5Fe76:1Co11:4 2 0.0101 -do- -do- This work Gd23:4Fe67:0Co9:6 2 0.0141 -do- -do- This work Gd24:4Fe66:1Co9:5 2 0.0078 -do- -do- This work C. Ferromagnetic resonance (FMR) measurements Rectangular pieces of about 6 3 mm2were cut from the blanket lms and broadband FMR spectroscopy was performed using a NanOsc Phase FMR (40 GHz) system with a co-planar waveguide for microwave eld excita- tion. Microwave excitation elds hrfwith frequencies up to 30 GHz were applied in the lm plane, and perpendic- ular to the applied in-plane dc magnetic eld H. All the FMR measurements were performed at the room tem- perature. The schematic of FMR measurement setup is shown in 1(a), and further details about the measure- ments are given in Section 2 (results and discussions). SUPPORTING INFORMATION Supporting Information is available from the Wiley Online Library or from the corresponding author.ACKNOWLEDGEMENTS Lakhan Bainsla thanks MSCA - European Commission for Marie Curie Individual Fellowship (MSCA-IF Grant No. 896307). This work was also partially supported by the Swedish Research Council (VR Grant No. 2016- 05980) and the Horizon 2020 research and innovation programme (ERC Advanced Grant No. 835068 "TOP- SPIN"). CONFLICT OF INTEREST The authors declare no con ict of interest. AUTHOR CONTRIBUTIONS L.B. and J. A. planned the study. L.B. grew the lms, performed the FMR measurements and analysed the ob- tained FMR data. J.W. helped with ICP-MS measure- ments and analysis. L.B. wrote the original draft of the paper. J. A. coordinated and supervised the work. All authors contributed to the data analysis and co-wrote the manuscript.7 DATA AVAILABILITY STATEMENT The data that support the ndings of this study are available from the corresponding author on reasonable request. REFERENCES 1S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, v. S. von Moln ar, M. Roukes, A. Y. Chtchelkanova, and D. Treger, science 294, 1488 (2001). 2J.Akerman, science 308, 508 (2005). 3V. Demidov, S. Urazhdin, A. Zholud, A. Sadovnikov, and S. Demokritov, Appl. Phys. Lett. 105, 172410 (2014). 4T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli, A. Houshang, A. A. Awad, P. Durrenfeld, B. G. Malm, A. Rusu, and J.Akerman, Proc. IEEE 104, 1919 (2016). 5M. Romera, P. Talatchian, S. Tsunegi, F. A. Araujo, V. Cros, P. Bortolotti, J. Trastoy, K. 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2021-11-16

Ferromagnetic materials dominate as the magnetically active element in spintronic devices, but come with drawbacks such as large stray fields, and low operational frequencies. Compensated ferrimagnets provide an alternative as they combine the ultrafast magnetization dynamics of antiferromagnets with a ferromagnet-like spin-orbit-torque (SOT) behavior. However to use ferrimagnets in spintronic devices their advantageous properties must be retained also in ultrathin films (t < 10 nm). In this study, ferrimagnetic Gdx(Fe87.5Co12.5)1-x thin films in the thickness range t = 2-20 nm were grown on high resistance Si(100) substrates and studied using broadband ferromagnetic resonance measurements at room temperature. By tuning their stoichiometry, a nearly compensated behavior is observed in 2 nm Gdx(Fe87.5Co12.5)1-x ultrathin films for the first time, with an effective magnetization of Meff = 0.02 T and a low effective Gilbert damping constant of {\alpha} = 0.0078, comparable to the lowest values reported so far in 30 nm films. These results show great promise for the development of ultrafast and energy efficient ferrimagnetic spintronic devices.

Ultrathin ferrimagnetic GdFeCo films with very low damping

2111.08768v1

Magnetic moment of inertia within the breathing model Danny Thonig,Manuel Pereiro, and Olle Eriksson Department of Physics and Astronomy, Material Theory, University Uppsala, S-75120 Uppsala, Sweden (Dated: June 20, 2021) An essential property of magnetic devices is the relaxation rate in magnetic switching which strongly depends on the energy dissipation and magnetic inertia of the magnetization dynamics. Both parameters are commonly taken as a phenomenological entities. However very recently, a large eort has been dedicated to obtain Gilbert damping from rst principles. In contrast, there is no ab initio study that so far has reproduced measured data of magnetic inertia in magnetic materials. In this letter, we present and elaborate on a theoretical model for calculating the magnetic moment of inertia based on the torque-torque correlation model. Particularly, the method has been applied to bulk bcc Fe, fcc Co and fcc Ni in the framework of the tight-binding approximation and the numerical values are comparable with recent experimental measurements. The theoretical results elucidate the physical origin of the moment of inertia based on the electronic structure. Even though the moment of inertia and damping are produced by the spin-orbit coupling, our analysis shows that they are caused by undergo dierent electronic structure mechanisms. PACS numbers: 75.10.-b,75.30.-m,75.40.Mg,75.78.-n,75.40.Gb The research on magnetic materials with particular fo- cus on spintronics or magnonic applications became more and more intensied, over the last decades [1, 2]. For this purpose, \good" candidates are materials exhibiting thermally stable magnetic properties [3], energy ecient magnetization dynamics [4, 5], as well as fast and stable magnetic switching [6, 7]. Especially the latter can be induced by i)an external magnetic eld, ii)spin polar- ized currents [8], iii)laser induced all-optical switching [9], or iv)electric elds [10]. The aforementioned mag- netic excitation methods allow switching of the magnetic moment on sub-ps timescales. The classical atomistic Landau-Lifshitz-Gilbert (LLG) equation [11, 12] provides a proper description of mag- netic moment switching [13], but is derived within the adiabatic limit [14, 15]. This limit characterises the blurry boundary where the time scales of electrons and atomic magnetic moments are separable [16] | usually between 10100 fs. In this time-scale, the applicabil- ity of the atomistic LLG equation must be scrutinized in great detail. In particular, in its common formula- tion, it does not account for creation of magnetic inertia [17], compared to its classical mechanical counterpart of a gyroscope. At short times, the rotation axis of the gyroscope do not coincide with the angular momentum axis due to a \fast" external force. This results in a superimposed precession around the angular-momentum and the gravity eld axis; the gyroscope nutates. It is expected for magnetisation dynamics that atomic mag- netic moments behave in an analogous way on ultrafast timescales [17, 18] (Fig. 1). Conceptional thoughts in terms of \magnetic mass" of domain walls were already introduced theoretically by D oring [19] in the late 50's and evidence was found ex- perimentally by De Leeuw and Robertson [20]. More recently, nutation was discovered on a single-atom mag- netic moment trajectory in a Josephson junction [21{23] B precession conenutation cone mFIG. 1. (Color online) Schematic gure of nutation in the atomistic magnetic moment evolution. The magnetic moment m(red arrow) evolves around an eective magnetic eld B (gray arrow) by a superposition of the precession around the eld (bright blue line) and around the angular momentum axis (dark blue line). The resulting trajectory (gray line) shows an elongated cycloid. due to angular momentum transfer caused by an elec- tron spin ip. From micromagnetic Boltzman theory, Ciornei et al. [18, 24] derived a term in the extended LLG equation that addresses \magnetic mass" scaled by the moment of inertia tensor . This macroscopic model was transferred to atomistic magnetization dynamics and applied to nanostructures by the authors of Ref. 17, and analyzed analytically in Ref. 25 and Ref. 26. Even in the dynamics of Skyrmions, magnetic inertia was observed experimentally [27]. Like the Gilbert damping , the moment of inertia ten- sorhave been considered as a parameter in theoretical investigations and postulated to be material specic. Re- cently, the latter was experimentally examined by Li et al. [28] who measured the moment of inertia for Ni 79Fe21 and Co lms near room temperature with ferromagnetic resonance (FMR) in the high-frequency regime (aroundarXiv:1607.01307v1 [cond-mat.mtrl-sci] 5 Jul 20162 200 GHz). At these high frequencies, an additional sti- ening was observed that was quadratic in the probing fre- quency!and, consequently, proportional to the moment of inertia=
. Here, the lifetime of the nutation  was determined to be in the range of = 0:120:47 ps, depending not only on the selected material but also on its thickness. This result calls for a proper theoretical description and calculations based on ab-initio electronic structure footings. A rst model was already provided by Bhattacharjee et al. [29], where the moment of inertia was derived in terms of Green's functions in the framework of the linear response theory. However, neither rst-principles electronic structure-based numerical values nor a detailed physical picture of the origin of the inertia and a poten- tial coupling to the electronic structure was reported in this study. In this Letter, we derive a model for the moment of inertia tensor based on the torque-torque cor- relation formalism [30, 31]. We reveal the basic electron mechanisms for observing magnetic inertia by calculat- ing numerical values for bulk itinerant magnets Fe, Co, and Ni with both the torque-torque correlation model and the linear response Green's function model [29]. In- terestingly, our study elucidate also the misconception about the sign convention of the moment of inertia [32]. The moment of inertia is dened in a similar way as the Gilbert damping within the eective dissipation eldBdiss[30, 33]. This ad hoc introduced eld is ex- panded in terms of viscous damping @m=@tand magnetic inertia@2m=@t2in the relaxation time approach [32, 34] (see Supplementary Material). The o-equilibrium mag- netic state induces excited states in the electronic struc- ture due to spin-orbit coupling. Within the adiabatic limit, the electrons equilibrate into the ground state at certain time scales due to band transitions [35]. If this relaxation time is close to the adiabatic limit, it will have two implications for magnetism: i)magnetic mo- ments respond in a inert fashion, due to formation of magnetism, ii)the kinetic energy is proportional to mu2=2 with the velocity u=@m=@tand the \mass" m of mag- netic moments, following equations of motion of classical Newtonian mechanics. The inertia forces the magnetic moment to remain in their present state, represented in the Kambersky model by =
(Ref. 32 and 34); theraison d'etre of inertia is to behave opposite to the Gilbert damping. In experiments, the Gilbert damping and the moment of inertia are measurable from the diagonal elements of the magnetic response function via ferromagnetic res- onance [31] (see Supplementary Material) =!2 0 !Mlim !!0=? !(1) =1 2!2 0 !Mlim !!0@!<? !1 !0; (2) where!M= Band!0= B0are the frequencies re-lated to the internal eective and the external magnetic eld, respectively. Thus, the moment of inertia is equal to the change of the FMR peak position, say the rst derivative of the real part of with respect to the prob- ing frequency [29, 36]. Alternatively, rapid external eld changes induced by spin-polarized currents lead also to nutation of the macrospin [37]. Settingonab-initio footings, we use the torque- torque correlation model, as applied for the Gilbert damping in Ref. 30 and 35. We obtain (see Supplemen- tary Material) =g msX nmZ T nm(k)T nm(k)Wnmdk (3) =g~ msX nmZ T nm(k)T nm(k)Vnmdk; (4) where; =x;y;z andmsis the size of the mag- netic moment. The spin-orbit-torque matrix elements Tnm=hn;kj[;Hsoc]jm;ki| related to the commuta- tor of the Pauli matrices and the spin-orbit Hamilto- nian | create transitions between electron states jn;ki andjm;kiin bandsnandm. This mechanism is equal for both, Gilbert damping and moment of inertia. Note that the wave vector kis conserved, since we neglect non- uniform magnon creation with non-zero wave vector. The dierence between moment of inertia and damping comes from dierent weighting mechanism Wnm;Vnm: for the dampingWnm=R (")Ank(")Amk(")d"where the elec- tron spectral functions are represented by Lorentzian's Ank(") centred around the band energies "nkand broad- ened by interactions with the lattice, electron-electron interactions or alloying. The width of the spectral func- tion provides a phenomenological account for angular momentum transfer to other reservoirs. For inertia, how- ever,Vnm=R f(") (Ank(")Bmk(") +Bnk(")Amk(")) d" whereBmk(") = 2(""mk)((""mk)232)=((""mk)2+2)3 (see Supplementary Material). Here, f(") and(") are the Fermi-Dirac distribution and the rst derivative of it with respect to ". Knowing the explicit form of Bmk, we can reveal particular properties of the moment of inertia: i)for !0 (!1 ),Vnm=2=("nk"mk)3. Sincen=m is not excluded, !1 ; the perturbed electron system will not relax back into the equilibrium. ii)In the limit !1 (!0), the electron system equilibrates imme- diately into the ground state and, consequently, = 0. These limiting properties are consistent with the expres- sion=
. Eq. (4) also indicates that the time scale is dictated by ~and, consequently, on a femto-second time scale. To study these properties, we performed rst- principles tight binding (TB) calculations [38] of the torque-correlation model as described by Eq. (4) as well as for the Green's function model reported in Ref. 29. The materials investigated in this letter are bcc Fe, fcc Co, and fcc Ni. Since our magnetic moment is xed3 -1·10−3-5·10−405·10−41·10−3−ι(fs) 10−110+0 Γ (eV)Fe Co NiTorque Green 10−21α10−410−21Γ (eV) FIG. 2. (Color) Moment of inertia as a function of the band width for bcc Fe (green dotes and lines), fcc Co (red dotes and lines), and fcc Ni (blue dotes and lines) and with two dierent methods: i)the torque-correlation method (lled triangles) and the ii)Greens function method [29](lled cir- cles). The dotted gray lines indicating the zero level. The insets show the calculated Gilbert damping as a function of . Lines are added to guide the eye. Notice the negative sign of the moment of inertia. in thezdirection, variations occur primarily in xory and, consequently, the eective torque matrix element is T=hn;kj[;Hsoc]jm;ki, where=xiy. The cubic symmetry of the selected materials allows only di- agonal elements in both damping and moment of inertia tensor. The numerical calculations, as shown in Fig. 2, give results that are consistent with the torque-torque correlation model predictions in both limits, !0 and !1 . Note that the latter is only true if we assume the validity of the adiabatic limit up to = 0. It should also be noted that Eq. (4) is only valid in the adiabatic limit (>10 fs). The strong dependency on indicates, however, that the current model is not a parameter-free approach. Fortunately, the relevant parameters can be extracted from ab-initio methods: e.g., is related ei- ther to the electron-phonon self energy [39] or to electron correlations [40]. The approximation =
derived by F ahnle et al. [32] from the Kambersk y model is not valid for all . It holds for <10 meV, where intraband transi- tions dominate for both damping and moment of inertia; bands with dierent energies narrowly overlap. Here, the moment of inertia decreases proportional to 1=4up to a certain minimum. Above the minimum and with an ap- propriate large band width , interband transitions hap- pen so that the moment of inertia approaches zero for high values of . In this range, the relation =
 used by Ciornei et al [18] holds and softens the FMR res- onance frequency. Comparing qualitative the dierence 10−410−310−210−1−ι(fs)/α 510+02510+12510+22 τ(fs) 5·10−310−22·10−23·10−2 Γ (eV)−ι αFIG. 3. (Color online) Gilbert damping (red dashed line), moment of inertia (blue dashed line), and the resulting nu- tation lifetime ==(black line) as a function of in the intraband region for Fe bulk. Arrows indicating the ordinate belonging of the data lines. Notice the negative sign of the moment of inertia. between the itinerant magnets Fe, Co and Ni, we obtain similar features in andvs. , but the position of the minimum and the slope in the intraband region varies with the elements: min= 5:9
103fs1at = 60 meV for bcc Fe, min= 6:5
103fs1at = 50 meV for fcc Co, andmin= 6:1
103fs1at = 80 meV for fcc Ni. The crossing point of intra- and interband transitions for the damping was already reported by Gilmore et al. [35] and Thonig et al. [41]. The same trends are also repro- duced by applying the Green's function formalism from Bhattacharjee et al. [29] (see Fig. 2). Consequently, both methods | torque-torque correlation and the linear re- sponse Green's function method | are equivalent as it can also be demonstrated not only for the moment of inertia but also for the Gilbert damping (see Supple- mentary Material)[41]. In the torque-torque correlation model (4), the coupling denes the width of the en- ergy window in which transitions Tnmtake place. The Green function approach, however, provides a more ac- curate description with respect to the ab initio results than the torque-torque correlation approach. This may be understood from the fact that a nite broadens and slightly shifts maxima in the spectral function. In par- ticular, shifted electronic states at energies around the Fermi level causes dierences in the minimum of in both models. Furthermore, the moment of inertia can be re- solved by an orbital decomposition and, like the Gilbert damping, scales quadratically with the spin-orbit cou- pling, caused by the torque operator ^Tin Eq. (4). Thus, one criteria for nding large moments of inertia is by hav- ing materials with strong spin-orbit coupling. In order to show the region of where the approxi- mation=
holds, we show in Fig. 3 calculated values of,, and the resulting nutation lifetime for a selection of that are below min. According to the data reported in Ref. 28, this is a suitable regime accessible4 for experiments. To achieve the room temperature mea- sured experimental values of = 0:120:47 ps, we have furthermore to guarantee that  >>  . An appropriate experimental range is 510 meV, which is realistic and caused, e.g., by the electron-phonon coupling. A nu- tation lifetime of 0:250:1 ps is revealed for these values of (see Fig. 3), a value similar to that found in ex- periment. The aforementioned electron-phonon coupling, however, is underestimated compared to the electron- phonon coupling from a Debye model ( 50 meV) [42]. In addition, eects on spin disorder and electron corre- lation are neglected, that could lead to uncertainties in and hence discrepancies to the experiment. On the other hand, it is not excluded that other second order energy dissipation terms, Bdiss, proportional to ( @e=@t)2 will also contribute [32] (see Supplementary material). The derivation of the moment of inertia tensor from the Kambersk y model and our numerics corroborates that recently observed properties of the Gilbert damping will be also valid for the moment of inertia: i)the moment of inertia is temperature dependent [41, 43] and decays with increasing phonon temperature, where the later usu- ally increase the electron-phonon coupling in certain temperature intervals [42]; ii)the moment of inertia is a tensor, however, o-diagonal elements for bulk mate- rials are negligible small; iii)it is non-local [36, 41, 44] and depends on the magnetic moment [45{47]. Note that the sign change of the moment of inertia also eects the dynamics of the magnetic moments (see Supplementary Material). The physical mechanism of magnetic moment of inertia becomes understandable from an inspection of the elec- tron band structure (see Fig. 4 for fcc Co, as an example). The model proposed here allows to reveal the inertia k- and band-index nresolved contributions (integrand of Eq. (4)). Note that we analyse for simplicity and clarity only one contribution, AnBm, in the expression for Vnm. As Fig. 4 shows the contribution to Vnmis signicant only for specic energy levels and specic k-points. The g- ure also shows a considerable anisotropy, in the sense that magnetisations aligned along the z- or y-directions give signicantly dierent contributions. Also, a closer in- spection shows that degenerate or even close energy levels nandm, which overlap due to the broadening of energy levels, e.g. as caused by electron-phonon coupling, , ac- celerate the relaxation of the electron-hole pairs caused by magnetic moment rotation combined with the spin orbit coupling. This acceleration decrease the moment of inertia, since inertia is the tendency of staying in a constant magnetic state. Our analysis also shows that the moment of inertia is linked to the spin-polarization of the bands. Since, as mentioned, the inertia preserves the angular momentum, it has largest contributions in the electronic structure, where multiple electron bands with the same spin-polarization are close to each other (cf. Fig. 4 c). However, some aspects of the inertia, -4-3-2-10E−EF(eV) -4-3-2-10E−EF(eV) ι<0 ι>0 -4-3-2-10E−EF(eV) Γ H N k(a−1 0)(a) (b) (c)y z FIG. 4. (Color online) Moment of inertia in the electron band structure for bulk fcc Co with the magnetic moment a) in y direction and b) in zdirection. The color and the intensity indicates the sign and value of the inertia contribution (blue - <0; red - >0; yellow - 0). The dotted gray line is the Fermi energy and is 0 :1 eV. c) Spin polarization of the electronic band structure (blue - spin down; red - spin up; yellow - mixed states). e.g. being caused by band overlaps, is similar to the Gilbert damping [48], although the moment of inertia is a property that spans over the whole band structure and not only over the Fermi-surface. Inertia is relevant in the equation of motion [17, 35] only for &0:1 ps and particularly for low dimensional systems. Nevertheless, in the literature there are measurements, as reported in Ref. 37, where the inertia eects are present. In summary, we have derived a theoretical model for the magnetic moment of inertia based on the torque- torque correlation model and provided rst-principle properties of the moment of inertia that are compared to the Gilbert damping. The Gilbert damping and the moment of inertia are both proportional to the spin- orbit coupling, however, the basic electron band struc-5 ture mechanisms for having inertia are shown to be dif- ferent than those for the damping. We analyze details of the dispersion of electron energy states, and the fea- tures of a band structure that are important for having a sizable magnetic inertia. We also demonstrate that the torque correlation model provides identical results to those obtained from a Greens functions formulation. Furthermore, we provide numerical values of the moment of inertia that are comparable with recent experimen- tal measurements[28]. The calculated moment of inertia parameter can be included in atomistic spin-dynamics codes, giving a large step forward in describing ultrafast, sub-ps processes. Acknowledgements The authors thank Jonas Frans- son and Yi Li for fruitful discussions. The support of the Swedish Research Council (VR), eSSENCE and the KAW foundation (projects 2013.0020 and 2012.0031) are acknowledged. The computations were performed on re- sources provided by the Swedish National Infrastructure for Computing (SNIC). danny.thonig@physics.uu.se [1] S. S. P. Parkin, J. X., C. Kaiser, A. Panchula, K. Roche, and M. Samant, Proceedings of the IEEE 91, 661 (2003). [2] Y. Xu and S. Thompson, eds., Spintronic Materials and Technology , Series in Materials Science and Engineering (CRC Press, 2006). [3] T. Miyamachi, T. Schuh, T. M arkl, C. Bresch, T. Bal- ashov, A. St ohr, C. Karlewski, S. Andr e, M. Marthaler, M. Homann, et al., Nature 503, 242 (2013). [4] K.-W. Kim and H.-W. Lee, Nature Mat. 14, 253 (2016). [5] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nature Phys. 11, 453 (2015). [6] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann, J. St ohr, G. Ju, B. Lu, and D. Weller, Nature 428, 831 (2004). [7] A. Chudnovskiy, C. 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2016-07-05

An essential property of magnetic devices is the relaxation rate in magnetic switching which strongly depends on the energy dissipation and magnetic inertia of the magnetization dynamics. Both parameters are commonly taken as a phenomenological entities. However very recently, a large effort has been dedicated to obtain Gilbert damping from first principles. In contrast, there is no ab initio study that so far has reproduced measured data of magnetic inertia in magnetic materials. In this letter, we present and elaborate on a theoretical model for calculating the magnetic moment of inertia based on the torque-torque correlation model. Particularly, the method has been applied to bulk bcc Fe, fcc Co and fcc Ni in the framework of the tight-binding approximation and the numerical values are comparable with recent experimental measurements. The theoretical results elucidate the physical origin of the moment of inertia based on the electronic structure. Even though the moment of inertia and damping are produced by the spin-orbit coupling, our analysis shows that they are caused by undergo different electronic structure mechanisms.

Magnetic moment of inertia within the breathing model

1607.01307v1

arXiv:2006.14949v1 [math.AP] 24 Jun 2020STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING AND NON-SMOOTH COEFFICIENT AT INTERFACE FATHI HASSINE Abstract. In this paper, we study the stability problem of a star-shape d network of elastic strings with a local Kelvin-Voigt damping. Under the assumption tha t the damping coefficients have some singularities near the transmission point, we prove that th e semigroup corresponding to the system is polynomially stable and the decay rates depends on the spe ed of the degeneracy. This result improves the decay rate of the semigroup associated to the sy stem on an earlier result of Z. Liu and Q. Zhang in [20] involving the wave equation with local Ke lvin-Voigt damping and non-smooth coefficient at interface. Contents 1. Introduction 1 2. Well-posedness 4 3. Strong stability 5 4. Polynomial stability 7 References 14 1.Introduction We consider one-dimensional wave propagation through N+1 edges (with N≥1) consisting of an elastic and a Kelvin-Voigt medium all connected to one tra nsmission point. The later material is a viscoelastic material having the properties both of ela sticity and viscosity. More precisely we consider the following initial and boundary-value problem (1.1) ¨u0(x,t)−u′′ 0(x,t) = 0 ( x,t)∈(0,ℓ0)×(0,+∞), ¨uj(x,t)−/bracketleftbig u′ j(x,t)+dj(x)˙u′ j(x,t)/bracketrightbig′= 0 (x,t)∈(0,ℓ1)×(0,+∞), j= 1,...,N, uj(ℓj,t) = 0 t∈(0,+∞), j= 0,...,N, u0(0,t) =···=uN(0,t) t∈(0,+∞), u′ 0(0,t)+N/summationdisplay j=1u′ j(0,t)+dj(0)˙u′ j(0,t) = 0t∈(0,+∞), uj(x,0) =u0 j(x),˙uj(x,0) =u1 j(x) x∈(0,ℓj), j= 0,...,N, where the point stands for the time derivative and the prime s tands for the space derivative, uj: [0,ℓj]×[0,+∞[−→Rforj= 0,...,Nare the displacement of the of the string of length ℓjand the coefficient damping djis assumed to be a non-negative function. The natural energy of system (1.1) is given by E(t) =1 2N/summationdisplay j=0/integraldisplayℓj 0/parenleftbig |˙uj(x,t)|2+|u′ j(x,t)|2/parenrightbig dx and it is dissipated according to the following law d dtE(t) =−N/summationdisplay j=1/integraldisplayℓj 0dj(x)|˙u′ j(x,t)|2dx,∀t >0. 2010Mathematics Subject Classification. 35B35, 35B40, 93D20. Key words and phrases. Network of strings, Kelvin-Voigt damping, non-smooth coeffi cient. 12 FATHI HASSINE The stability of this model was intensively studied in this l ast two decades: On high dimensional case: Liu and Rao [19] proved the exponen tial decay of the energy providing that the damping region is a neighborhood of the whole bounda ry, and further restrictions are imposed on the damping coefficient. Next, this result was gene ralized and improved by Tebou [25] when damping is localized in a suitable open subset, of the do main under consideration, which satisfies the piecewise multipliers condition of Liu. He sho ws that the energy of this system decays polynomially when the damping coefficient is only boundedmea surable, and it decays exponentially when the damping coefficient as well as its gradient are bounde d measurable, and the damping coefficient further satisfies a structural condition. Recent ly, Using Carleman estimates Ammari et al. [2] show a logarithmic decay rate of the semigroup associ ated to the system when the damping coefficient is arbitrary localized. Next, Burq generalized t his result in [7]. This author shows also a polynomial decay rate of the semigroup when the damping regi on verifying some geometric control condition of order equal to1 2and of order equal to1 4for a cubic domain with eventual degeneracy on the coefficient damping in case of dimension 2. In case of the interval (or when N= 1): It is well known that the Kelvin-Voigt damping is much stronger than the viscous damping in the sens that if the enti re medium is of the Kelvin-Voigt type, thedampingforthewave equation not onlyinducesexponenti al energydecay butalsotheassociated to semigroup is analytic [14]; while the entire medium of the viscous type, the associate semigroup is only exponential stable [8] and does not have any smoothin g property since the spectrum of the semigroup generator has a vertical asymptote on the left han d side of the imaginary axes of the complex plane. When the damping is localized (i.e., distrib uted only on the proper subset of the spatial domain), such a comparison is not valid anymore. Whi le the local viscous damping still enjoys the exponential stability as long as the damped regio n contains an interval of any size in the domain, the local KelvinVoigt damping doesn’t follow the sa me analogue. Chen et al. [9] proved lack of the exponential stability when the damping coefficien t is a step function. This unexpected result reveals that the KelvinVoigt damping does not follow the ”geometric optics” condition (see [5]). And Liu and Rao [18] proved that the solution of this mod el actually decays at a rate of t−2. The optimality of this order was proven by Alves et al. in [1] . In 2002, it was shown in [16] that exponential energy decay still holds if the damping coe fficient is smooth enough. Later, the smoothness condition was weakened in [26] and satisfies the f ollowing condition a′(0) = 0,/integraldisplayx 0|a′(s)|2 a(s)ds≤C|a′(x)| ∀x∈[0,1]. Thisindicates that theasymptotic behaviorof thesolution dependsontheregularity ofthedamping coefficient function, which is not the case for the viscous dam ping model. Renardy [24] in 2004 proved that thereal partof theeigenvalues arenot boundedb elow ifthedampingcoefficient behaves likexαwithα >1near theinterface x= 0. Underthis samecondition Liuet al. [17] proved that the solution of the system is eventually differentiable which als o guarantees the exponential stability since there is no spectrum on the imaginary axis and the syste m is dissipative. In 2016, Under the assumption that the damping coefficient has a singularity at the interface of the damped and undamped regions and behaves like xαwithα∈(0,1) near the interface, [20] proved that the semigroup corresponding to the system is polynomially stab le and the decay rate depends on the parameter α. In case of multi-link structure: In [10] we proved that the se migroup is polynomially stable when the coefficient damping is piecewise function with an opt imal decay rate equal to 2 (we refer also to [11] for the case of transmission Euler-Bernoulli pl ate and wave equation with a localized Kelvin-Voigt damping). Recently, Ammari et al. [3] conside r the tree of elastic strings with local Kelvin-Voigt damping. They proved under some assumptions o n the smoothness of the damping coefficients, say W2,∞, and some other considerations that the semigroup is expone ntially stable if the damping coefficient is continuous at every node of the tr ee and otherwise it is polynomially stable with a decay rate equal to 2. In light of all the above results it is obvious to say that the a symptotic behavior of the solution to system (1.1) depends on the regularity on the damping coeffi cient function. In this paper we want 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 damping coefficient has a singularity at the interface and beh aves like xαwithα∈(0,1). Precisely, we make the following assumptions: For every j= 1,...,N •There exist aj, bj∈[0,ℓj] withaj< bjsuch that (A 1.2) [ aj,bj]⊂supp(dj) and dj∈L∞(0,ℓj). •There exist αj∈(0,1) andκj≥0 such that (A 1.3) lim x→0+dj(x) xαj=κj. •There exists ηj∈[0,1) such that (A 1.4) lim x→0+xd′ j(x) dj(x)=ηj. Remark 1.1. The typical example of functions djthat satisfies assumptions (A 1.2),(A 1.3)and (A 1.4), is when dj(x) =xαjwithαj∈(0,1)for every j= 1,...,N. An interesting example too is when we take dj(x) =xα′ j|ln(x)|βjwithα′ j∈(0,1)andβj>0in this case assumptions (A 1.2), (A 1.3)and(A 1.4)are satisfied with κj= 0,αj=α′ j−εjfor all0< εj< α′ jandηj=α′ j. LetH=V×N/productdisplay j=0L2(0,ℓj) be the Hilbert space endowed with the inner product define fo r (u,v) = ((uj)j=0,...,N,(vj)j=0,...,N)∈ Hand (˜u,˜v) = ((˜uj)j=0,...,N,(˜vj)j=0,...,N)∈ Hby /an}bracketle{t(u,v),(˜u,˜v)/an}bracketri}htH=N/summationdisplay j=0/integraldisplayℓj 0u′ j(x).˜u′ j(x)dx+/integraldisplayℓj 0vj(x).˜vj(x)dx, whereVis the Hilbert space defined by V= (uj)j=0,...,N∈N/productdisplay j=0H1(0,ℓj) :uj(ℓj) = 0∀j= 0,...,N;u0(0) =···=uN(0) . By setting U(t) = ((uj(t))j=0,...,N,(vj(t))j=0,...,N) andU0=/parenleftig (u0 j)j=0,...,N,(u1 j)j=0,...,N/parenrightig we can rewrite system (1.1) as a first order differential equation as f ollows (1.11) ˙U(t) =AU(t), U(0) =U0∈ D(A), where A((uj)j=0,...,N,(vj)j=0,...,N) =/parenleftbig (vj)j=0,...,N,(u′′ 0,[u′ 1+d1v′ 1]′,...,[u′ N+dNv′ N]′)/parenrightbig , with D(A) =/braceleftbigg ((uj)j=0,...,N,(vj)j=0,...,N)∈ H: (vj)j=0,...,N∈V;u′′ 0∈L2(0,ℓ0); [u′ j+djv′ j]′∈L2(0,ℓj)∀j= 1,...,N;u′ 0(0)+N/summationdisplay j=1u′ j(0)+dj(0)v′ j(0) = 0/bracerightbigg . Theorem 1.1. Assume that for j= 1,...,Nthe coefficient functions dj∈ C([0,ℓj])∩ C1((0,ℓj)) are such that conditions (A 1.2),(A 1.3)and(A 1.4)hold. Then, the semigroup etAassociated to system(1.1)(see Proposition 2.1) is polynomially stable precisely we h ave: There exists C >0such that /bardbletAU0/bardblH≤C (t+1)2−α 1−αk/vextenddouble/vextenddoubleU0/vextenddouble/vextenddouble D(Ak)∀U0=/parenleftbig (u0 j)j=0,...,N,(u1 j)j=0,...,N/parenrightbig ∈ D(Ak)∀t≥0, whereα= min{α1,...,α N}.4 FATHI HASSINE Remark 1.2. This theorem reveals that the stability order of the semigrou petAassociated to problem (1.1)depends on the behavior of the damping coefficients djdescribed by the parameters αjfor j= 1,...,N. This result improves the decay rate of the energy given in [20]from1 1−αto2−α 1−α and make it more meaningful in fact, as αgoes to1−the order of polynomial stability2−α 1−αgoes to ∞which is consistent with the exponential stability when α= 1(see[3, 20, 16, 20] ) and as αgoes to0+the order of polynomial stability2−α 1−αgoes to2which is consistent with the optimal order stability when α= 0(see[1, 10, 18] ). Remark 1.3. When the coefficient functions djbehave polynomially near 0asxαjwith0< αj<1 then from Theorem 1.1 the semigroup etAdecays polynomially with the decay rate given above in the theorem. Moreover, when djdecay faster than xαjand slower than xαj+εfor allε >0this can may be seen for instance with the example of dj(x) =xαj|ln(x)|βjwithβj>0then according to Remark 1.1 the semigroup etAdecays polynomially with the decay rate equal to2−α+ε 1−α+εfor each ε∈(0,α)whereα= min{α1,...,α N}. Consequently, this decay rate is worse than if the coefficient functions djbehave near 0likexαjand better than if the coefficient functions djbehave near 0like xα′ jfor anyα′ j>0such that α′ j< αj. This article is organized as follows. In section 2, we prove t he well posedness of system (1.1). In section 3, we show that the semigroup associated to the gener atorAis strongly stable . In section 4, we prove the polynomial decay rate given by Theorem 1.1. 2.Well-posedness In this section we use the semigroup approach to prove the wel l-posedness of system (1.1). Proposition 2.1. Assume that condition (A 1.2)holds. Then Agenerates a C0-semigroup of contractions etAon the Hilbert space H. Proof.For any ( u,v) =/parenleftbig (uj)j=0,...,N,(vj)j=0,...,N/parenrightbig ∈ D(A), we have /an}bracketle{tA(u,v),(u,v)/an}bracketri}htH=−N/summationdisplay j=1/integraldisplayℓj 0dj(x)|∂xvj(x)|2dx. This shows that the operator Ais dissipative. Given (f,g) =/parenleftbig (fj)j=0,...,N,(gj)j=0,...,N/parenrightbig ∈ Hwe look for ( u,v) =/parenleftbig (uj)j=0,...,N,(vj)j=0,...,N/parenrightbig ∈ D(A) such that A(u,v) = (f,g), this is also written vj=fj∀j= 0,...,N, u′′ 0=g0, (u′ j+djv′ j)′=gj∀j= 1,...,N, uj(ℓj) = 0∀j= 0,...,N, u0(0) =···=uN(0), u′ 0(0)+N/summationdisplay j=1u′ j(0)+dj(0)v′ j(0) = 0, or equivalently (2.2) vj=fj∀j= 0,...,N, u′′ 0=g0, u′′ j=gj−(djf′ j)′∀j= 1,...,N, uj(ℓj) = 0∀j= 0,...,N, u0(0) =···=uN(0), u′ 0(0)+N/summationdisplay j=1u′ j(0)+dj(0)f′ j(0) = 0.STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 5 For this aim we set the continuous coercive and bi-linear for m inV L(u,˜u) =N/summationdisplay j=0/integraldisplayℓj 0u′ j.˜u′ jdx. By Lax-Milligram theorem there exists a unique element ( uj)j=0,...,N∈Vsuch that (2.3)N/summationdisplay j=0/integraldisplayℓj 0u′ j.˜u′ jdx=−N/summationdisplay j=1/integraldisplayℓj 0djf′ j.˜u′ jdx−N/summationdisplay j=0/integraldisplayℓj 0gj.˜ujdx. It follows that by taking (2.3) in the sens of distribution th at∂2 xu0=g0inL2(0,ℓ0) andu′′ j= gj−[djf′ j]′inL2(0,ℓj) for all j= 1,...,N. Back again to (2.3) and integrating by parts we find thatu′ 0(0)+N/summationdisplay j=1u′ j(0)+dj(0)f′ j(0) = 0. This prove that the operator Ais surjective. Moreover, by multiplying the second line by u1of (2.2) and the third line by ujand integrating over (0 ,ℓ0) and (0,ℓj) respectively and summing up then by Poincar´ e inequality a nd Cauchy-Schwarz inequality we find that there exists a constant C >0 such that N/summationdisplay j=0/integraldisplayℓj 0|u′ j|2dx≤C N/summationdisplay j=0/integraldisplayℓj 0|f′ j|2dx+N/summationdisplay j=0/integraldisplayℓj 0|gj|2dx which combined with the first line of (2.2) leads to N/summationdisplay j=0/integraldisplayℓj 0|u′ j|2dx+N/summationdisplay j=0/integraldisplayℓj 0|vj|2dx≤C N/summationdisplay j=0/integraldisplayℓj 0|f′ j|2dx+N/summationdisplay j=0/integraldisplayℓj 0|gj|2dx . This implies that 0 ∈ρ(A) and by contraction principle, we easily get R(λ−A) =Hfor sufficient smallλ >0. Since D(A) is dense in Hthen thanks to Lumer-Phillips theorem [22, Theorem 1.4.3], Agenerates a C0-semi-group of contractions on H. /square As a consequence of Proposition 2.1 we have the following wel l-posedness result of system (1.1). Corollary 2.1. For any initial data U0∈ H, there exists a unique solution U(t)∈ C([0,+∞[,H) to the problem (1.11). Moreover, if U0∈ D(A), then U(t)∈ C([0,+∞[,D(A))∩C1([0,+∞),H). 3.Strong stability The aim of this section is to prove that the semi-group genera ted by the operator Ais strongly stable. In another words this means that the energy of system (1.1) degenerates over the time to zero. Lemma 3.1. Assume that condition (A 1.2)holds. Then for every λ∈Rthe operator (iλ−A)is injective. Proof.Since 0∈ρ(A) (according to the proof of Theorem 2.3), we only need to chec k that for every λ∈R∗we have ker( iλI−A) ={0}. Letλ/ne}ationslash= 0 and ( u,v) =/parenleftbig (uj)j=0,...,N,(vj)j=0,...,N/parenrightbig ∈ D(A) such that (3.2) A(u,v) =iλ(u,v) Taking the real part of the inner product in Hof (3.2) with ( u,v) and using the dissipation of A, we get Reiλ/bardbl(u,v)/bardbl2 H= Re/an}bracketle{tA(u,v),(u,v)/an}bracketri}htH=−N/summationdisplay j=1/integraldisplayℓj 0dj(x)|v′ j(x)|2dx= 0, which implies that (3.3) djv′ j= 0 inL2(0,ℓj),∀j= 1,...,N.6 FATHI HASSINE Inserting (3.3) into (3.2), we obtain (3.4) iλuj=vj in (0,ℓj), j= 0,...,N, λ2uj+u′′ j= 0 in (0 ,ℓj), j= 0,...,N, uj(ℓj) = 0 j= 0,...,N, u1(0) =···=uN(0) N/summationdisplay j=0u′ j(0) = 0 Combining (3.3) with the first line of (3.4), we get u′ j= 0 a.e in [ aj,bj]∀j= 1,...,N. Sinceuj∈H2(0,ℓj) then following to the embedding H1(0,ℓj)֒→C0(0,ℓj) we have (3.5) u′ j≡0 in [aj,bj]∀j= 1,...,N. Which by the second line of (3.4) leads to uj≡0 in [aj,bj]∀j= 1,...,N. So for every j= 1,...,N,ujandvjare solution to the following problem iλuj=vj in (0,ℓj), λ2uj+u′′ j= 0 in (0 ,ℓj), uj(aj) =u′ j(aj) = 0, and this clearly gives that uj=vj≡0 in (0,ℓj) for every j= 1,...,N. Following to system (3.4) we have then (3.6) iλu0=v0 in (0,ℓ0), λ2u0+u′′ 0= 0 in (0 ,ℓ0) u0(ℓ0) =u′ 0(ℓ0) = 0 which gives also that u0=v0≡0 in (0,ℓ0). This shows that ( u,v) = (0,0) and consequently (iλI−A) is injective for all λ∈R. /square Lemma 3.2. Assume that condition (A 1.2)holds. Then for every λ∈Rthe operator (iλ−A)is surjective. Proof.Since 0∈ρ(A) (see proof of Theorem 2.3), we only need to check that for eve ryλ∈R∗we haveR(iλI−A) =H. Letλ/ne}ationslash= 0, then given ( f,g) =/parenleftbig (fj)j=0,...,N,(gj)j=0,...,N/parenrightbig ∈ Hwe are looking for (u,v) =/parenleftbig (uj)j=0,...,N,(vj)j=0,...,N/parenrightbig D(A) such that (3.8) ( iλI−A)(u,v) = (f,g), or equivalently (3.9) vj=iλuj−fj in (0,ℓj), j= 0,...,N, −λ2u0−u′′ 0=iλf0+g0 in (0,ℓ0), −λ2uj−(u′ j+iλdjuj′)′=iλfj+gj−(djf′ j)′in (0,ℓj), j= 1,...,N. We define for all u= ((uj)j=0,...,N/parenrightbig ∈Vthe operator Au=/parenleftbig −u′′ 0,−(u′ 1+iλdju′ 1)′,...,−(u′ N+iλdNu′ N)′/parenrightbig . Thanks to Lax-Milgram’s theorem [15, Theorem 2.9.1], it is e asy to show that Ais an isomorphism fromVintoV′(whereV′is the dual space of Vwith respect to the pivot space H). Then the second ant the third line of (3.9) can be written as follows (3.10) u−λ2A−1u=A−1/parenleftbig iλf0+g0,iλf1+g1−(d1f′ 1)′,...,iλf N+gN−(dNf′ N)′/parenrightbig . Ifu∈ker(I−λA−1), then we obtain (3.11)/braceleftbiggλ2u0+u′′ 0= 0 in (0 ,ℓ0), λ2uj+(u′ j+iλdju′ j)′= 0 in (0 ,ℓj), j= 1,...,N.STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 7 Forj= 1,...,Nwe multiply each line of (3.11) by ujand integrating over (0 ,ℓj) and summing up (3.12) λ2N/summationdisplay j=0/integraldisplayℓj 0|uj|2dx−N/summationdisplay j=0/integraldisplayℓj 0|u′ j|2dx−iλN/summationdisplay j=1/integraldisplayℓj 0dj|u′ j|2dx= 0. By taking the imaginary part of (3.12) we get N/summationdisplay j=1/integraldisplayℓj 0dj|u′ j|2dx= 0. This means that dju′ j= 0 in (0 ,ℓj) for alli= 1,...,N, which inserted into (3.11) one gets λ2uj+u′′ j= 0 in (0 ,ℓj), j= 0,...,N. Then using the same arguments as proof of Lemma 3.1 we find that u= 0. Hence, we proved that ker( I−λ2A−1) ={0}. Besides, thanks to the compact embeddings V ֒→HandH ֒→V′ the operator A−1is compact in V. So that, following to Fredholm’s alternative, the operato r (I−λ2A−1) is invertible in V. Therefore, equation (3.10) have a unique solution in V. Thus, the operator iλI−Ais surjective. This completes the proof. /square Thanks to Lemmas 3.1 and 3.2 and the closed graph theorem we ha veσ(A)∩iR=∅. This with Arendt and Batty [4] result following to which a C0-semi-group of contractions in a Banach space is strongly stable, if ρ(A)∩iRcontains only a countable number of continuous spectrum of Alead to the following Theorem 3.1. Assume that condition (A 1.2)holds. Then the semigroup (etA)t≥0is strongly stable in the energy space Hi.e., lim t→+∞/bardbletAU0/bardblH= 0,∀U0∈ H. 4.Polynomial stability In this section, we prove Theorem 1.1. The idea is to estimate the energy norm and boundary terms at the interface by the local viscoelastic damping. Th e difficulty is to deal with the higher orderboundarytermattheinterfacesothattheenergyon(0 ,ℓ0)canbecontrolledbytheviscoelastic damping on (0 ,ℓj) for every j= 1,...,N. Our proof is based on the following result Proposition 4.1. [6, Theorem 2.4] LetetBbe a bounded C0-semi-group on a Hilbert space Xwith generator Bsuch that iR∈ρ(A). ThenetBis polynomially stable with order1 γi.e. there exists C >0such that /bardbletBu/bardblX≤C t1 γ/bardblu/bardblD(B)∀u∈ D(B)∀t≥0, if and only if limsup |λ|→∞/bardblλ−γ(iλI−B)−1/bardblX<∞. According to Proposition 4.1 we shall verify that for α= min{α1,...,α N}andγ=1−α 2−αthere existsC0>0 such that (4.1) inf /bardbl((uj)j=0,...,N,(vj)j=0,...,N)/bardblH=1 λ∈Rλγ/bardbliλ((uj)j=0,...,N,(vj)j=0,...,N)−A((uj)j=0,...,N,(vj)j=0,...,N)/bardblH≥C0. Suppose that (4.1) fails then there exist a sequence of real n umbersλnand a sequence of functions (un,vn)n∈N=/parenleftbig (u0,n,...,u N,n),(v0,n,...,vN,n)/parenrightbig n∈N⊂ D(A) such that λn−→ ∞asn−→ ∞, (4.2)/vextenddouble/vextenddouble(un,vn)/vextenddouble/vextenddouble= 1, (4.3) λγ n/bardbliλn(un,vn)−A(un,vn)/bardblH=o(1). (4.4)8 FATHI HASSINE Since, we have (4.5) λγ nRe/an}bracketle{tiλ(un,vn)−A(un,vn),(un,vn)/an}bracketri}ht=λγ nn/summationdisplay j=1/integraldisplayℓj 0dj|v′ j,n|2dx then using (4.3) and (4.4) we obtain (4.6)n/summationdisplay j=1/bardbld1 2 jv′ j,n/bardblL2(0,ℓj)=o(λ−γ 2n). Following to (4.4) we have λγ n(iλnuj,n−vj,n) =fj,n−→0 inH1(0,ℓj), j= 0,...,N, (4.7) λγ n(iλnv0,n−u′′ 0,n) =g0,n−→0 inL2(0,ℓ0), (4.8) λγ n(iλnvj,n−T′ j,n) =gj,n−→0 inL2(0,ℓj), j= 1,...,N, (4.9) with the transmission conditions u0,n(0) =···=uN,n(0), (4.10) u′ 0,n(0)+N/summationdisplay j=1Tj,n(0) = 0, (4.11) where for j= 1,...,Nwe have denoted by (4.12) Tj,n=u′ j,n+djv′ j,n= (1+iλndj)u′ j,n−λ−γ ndjf′ j,n. By (4.6) and (4.7) we find (4.13)n/summationdisplay j=1/bardbld1 2 ju′ j,n/bardblL2(0,ℓj)=o(λ−γ 2−1 n). One multiplies (4.8) by ( x−ℓ0)u′ 0,nintegrating by parts over the interval (0 ,ℓ0) and use (4.3) and (4.7) we get (4.14)/integraldisplayℓ0 0/parenleftbig |u′ 0,n|2+|v0,n|2/parenrightbig dx−ℓ0/parenleftbig |u′ 0,n(0)|2+|v0,n(0)|2/parenrightbig =o(1). We multiply (4.9) by vj,nforj= 1,...,Nand (4.8) by v0,nthen integrating over (0 ,ℓj) forj= 0,...,Nand summing up to get (4.15) iλγ+1 nN/summationdisplay j=0/bardblvj,n/bardbl2 L2(0,ℓj)+λγ nN/summationdisplay j=0/an}bracketle{tu′ j,n,v′ j,n/an}bracketri}htL2(0,ℓj)+λγ nN/summationdisplay j=1/bardbld1 2 jv′ j,n/bardblL2(0,ℓj)=o(1). We take the inner product of (4.7) with uj,ninH1(0,ℓj) forj= 0,...,Nand summing up, (4.16) iλγ+1 nN/summationdisplay j=0/bardblu′ j/bardbl2 L2(0,ℓj)−λγ nN/summationdisplay j=0/an}bracketle{tv′ j,n,u′ j,n/an}bracketri}htL2(0,ℓj)=o(1). Adding (4.15) and (4.16) and taking the imaginary part of the equality then by (4.6) we arrive at (4.17)N/summationdisplay j=0/parenleftig /bardblu′ j,n/bardblL2(0,ℓj)−/bardblvj,n/bardblL2(0,ℓj)/parenrightig =o(1). At this stage we recall the following Hardy type inequalitie s Lemma 4.1. [21, Theorem 3.8] LetL >0anda: [0,L]→R+be such that a∈ C([0,L])∩C1((0,L]) and satisfying lim x→+∞xa′(x) a(x)=η∈[0,1).STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 9 Then there exists C(η,L)>0such that for all locally continuous function zon[0,L]satisfying z(0) = 0 and/integraldisplayL 0a(x)|z′(x)|2dx <∞ the following inequality holds /integraldisplayL 0a(x) x2|z(x)|2dx≤C(η,L)/integraldisplayL 0a(x)|z′(x)|2dx. Lemma 4.2. [20, Lemma 2.2] LetL >0andρ1, ρ2>0be two weight functions defied on (0,L). Then the following conditions are equivalent: (4.18)/integraldisplayL 0ρ1(x)|Tf(x)|2dx≤C/integraldisplayL 0ρ2(x)|f(x)|2dx, and K= sup x∈(0,L)/parenleftbigg/integraldisplayL−x 0ρ1(x)dx/parenrightbigg/parenleftbigg/integraldisplayL L−x[ρ2(x)]−1dx/parenrightbigg <∞ whereTf(x) =/integraldisplayx 0f(s)dx. Moreover, the best constant Cin(4.18)satisfies K≤C≤2K. Letβsuch that1−αj 2< β <1 for all j= 1,...,Nandδjare positive numbers that will be specified later. Following to Lemmas 4.1 and 4.2 and assumpti ons (A 1.3) and (A 1.4) then for n large enough /bardblvj,n/bardbl L2/parenleftBigg/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg/parenrightBigg≤max x∈/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg/braceleftigg x1−β dj(x)1 2/bracerightigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1 2 j x(xβvj,n)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2/parenleftBigg/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg/parenrightBigg ≤Cλ−δj(1−β−αj 2) n/parenleftigg/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1 2 jxβv′ j,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2([0,ℓj])+β/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1 2 jxβ−1vj,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2([0,ℓj])/parenrightigg ≤Cλ−δj(1−β−αj 2) n/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1 2 jv′ j,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2([0,ℓj]). (4.21) Performing the following calculation and uses (4.21) one fin ds (4.22) min x∈/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg{|vj,n(x)|+|Tj,n(x)|} ≤√ 2λδj 2n /bardblvj,n/bardbl L2/parenleftBigg/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg/parenrightBigg+/bardblTj,n/bardbl L2/parenleftBigg/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg/parenrightBigg ≤√ 2λδj 2n /bardblvj,n/bardbl L2/parenleftBigg/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg/parenrightBigg+/bardblu′ j,n/bardbl L2/parenleftBigg/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg/parenrightBigg+/bardbldjv′ j,n/bardbl L2/parenleftBigg/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg/parenrightBigg ≤Cλδj 2n/parenleftigg λ−δj(1−β−αj 2) n/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1 2 jv′ j,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2([0,ℓj])+ max x∈/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg{dj(x)−1 2}/bardbld1 2 ju′ j,n/bardblL2([0,ℓj]) + max x∈/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg{dj(x)1 2}/bardbld1 2 jv′ j,n/bardblL2([0,ℓj])/parenrightigg ≤Cλδj 2n/parenleftigg λ−δj(1−β−αj 2) n/vextenddouble/vextenddouble/vextenddouble/vextenddoubled1 2 jv′ j,n/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2([0,ℓj])+λδjαj 2n/bardbld1 2 ju′ j,n/bardblL2([0,ℓj])+λ−δjαj 2n/bardbld1 2 jv′ j,n/bardblL2([0,ℓj])/parenrightigg .10 FATHI HASSINE Inserting (4.6), (4.13) into (4.22), then we obtain min x∈/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg{|vj,n(x)|+|Tj,n(x)|}=/parenleftbigg λ−δj(1−αj 2−β)−γ 2n +λδj 2(αj+1)−γ 2−1 n +λδj 2(1−αj)−γ 2n/parenrightbigg o(1), for every βsuch that1−αj 2< β <1 for every j= 1,...,N. Then we can choose βsuch that −δj(1−αj 2−β)−γ 2<0 for every j= 1,...,N. Hence, as long as we choose δj>0 andγ >0 such that (4.23)δj 2(αj+1)−γ 2−1≤0 andδj 2(1−αj)−γ 2≤0∀j= 1,...,N. the following estimate holds (4.24) min x∈/bracketleftBigg λ−δj n 2,λ−δj n/bracketrightBigg{|vj,n(x)|+|Tj,n(x)|}=o(1). And consequently, we are able to find ξj,n∈/bracketleftbigg λ−δj n 2,λ−δjn/bracketrightbigg such that (4.25) |vj,n(ξj,n)|=o(1) and |Tj,n(ξj,n)|=o(1). We set z± j,n(x) =iλn/radicalbig 1+λndj(x)/integraldisplayξj,n xvj,n(τ)dτ±vj,n(x),∀x∈[0,ξj,n]. Then we have (4.26)z± ′ j,n(x) =−iλnd′ j(x) 4(1+λndj(x))(z+ j,n(x)+z− j,n(x))∓iλn/radicalbig 1+λndj(x)z± j,n(x) ∓λ2 n 1+λndj(x)/integraldisplayξj,n xvj,n(τ)dτ±v′ j,n(x). Combining (4.7) and (4.12) we have ±v′ j,n(x) =±iλnu′ j,n(x)∓λ−γ nf′ j,n(x) =±iλn 1+iλndj(x)Tj,n(x)±iλ1−γ ndj(x) 1+iλndjf′ j,n(x)∓λ−γ nf′ j,n(x) =±iλn 1+iλndjTj,n(x)∓λ−γ n 1+iλndj(x)f′ j,n(x). (4.27) Integrating (4.9) over ( x,ξj,n) and multiplying by ±iλn 1+iλndj(x)then we get (4.28) ∓λ2 n 1+iλndj(x)/integraldisplayξj,n xvj,n(τ)dτ=±iλn 1+iλndj(x)(Tj,n(ξj,n)−Tj,n(x))±iλ1−γ n 1+iλndj(x)/integraldisplayξj,n xgj,n(τ)dτ. Inserting (4.27) and (4.28) into (4.26), we find (4.29) z± ′ j,n(x) =∓iλn/radicalbig 1+λndj(x)z± j,n(x)−iλnd′ j(x) 4(1+λndj(x))(z+ j,n(x)+z− j,n(x))±iλn 1+iλndj(x)Tj,n(ξj,n)±Fj,n(x) where Fj,n(x) =−λ−γ n 1+iλndj(x)f′ j,n(x)+iλ1−γ n 1+iλndj(x)/integraldisplayξj,n xgj,n(τ)dτ.STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 11 Solving (4.29), then for every x∈[0,ξj,n], one gets (4.30) z± j,n(x) =vj,n(ξj,n)e∓(qj,n(x)−qj,n(ξj,n))−/integraldisplayx ξj,ne∓(qj,n(x)−qj,n(s))iλnd′ j(s) 4(1+λndj(s))(z+ j,n(s)+z− j,n(s))ds ±Tj,n(ξj,n)/integraldisplayx ξj,ne∓(qj,n(x)−qj,n(s))iλn 1+iλndj(s)ds±/integraldisplayx ξj,ne∓(qj,n(x)−qj,n(s))Fj,n(s)ds where qj,n(x) =iλn/integraldisplayx 0ds/radicalbig 1+iλndj(s). For allx∈[0,ξj,n] we have qj,n(x) =iλn/integraldisplayx 0eiϕj,n(s) (1+(λj,ndj(s))2)1 4ds where ϕj,n(s) =−1 2arg(1+iλndj,n(s)). Consequently, Re(qj,n(x)) =±λn/integraldisplayx 0sin(ϕj,n(s)) (1+(λj,ndj(s))2)1 4ds. Whens∈[0,ξj,n] and from assumption we have (4.31)1 (1+(λj,ndj(s))2)1 4=/braceleftiggO(1) if δjαj≥1 O(λδjαj−1 2n) ifδjαj<1 and |sin(ϕj,n(s))|=/radicaligg 1 2−1 2/radicalbig 1+(λndj(s))2 =/braceleftbigg O(λ1−δjαjn) ifδjαj>1 O(1) if δjαj≤1.(4.32) With 0≤x≤s≤ξj,n, from (4.31) and (4.32) we see that |Re(qj,n(x)−qj,n(s))| ≤λn/integraldisplays x|sin(ϕj,n(τ))| (1+(λndj(τ))2)1 4dτ ≤sup τ∈(x,s)/braceleftigg |sin(ϕj,n(τ))| (1+(λndj(τ))2)1 4/bracerightigg λn(s−x) = O(λ−(αj+1)δj+2 n ) =o(1) if δjαj>1 O(λ1−δjn) =O(λ1−1 αjn) =o(1) ifδjαj= 1 O(λδj(αj−2)+1 2n ) if δjαj<1. This implies that |e±(qj,n(x)−qj,n(s))| ≤1 (4.33) providing that δj>0 satisfying (4.23) and (4.34) δj≥1 αjorδj≤1 2−αj∀j= 1,...,N.12 FATHI HASSINE From (4.31) for every x∈[0,ξj,n], we have /integraldisplayξj,n xds |1+iλndj(s)|=/braceleftbiggO(1)(ξj,n−x) if δjαj≥1 O(λαjδj−1 n)(ξj,n−x) ifδjαj<1 =/braceleftigg O(λ−δjn) if δjαj≥1 O(λ(αj−1)δj−1 n ) ifδjαj<1(4.35) and /parenleftbigg/integraldisplayξj,n xds |1+iλndj(s)|2/parenrightbigg1 2 = O(λ−δj 2n) if δjαj≥1 O(λ(αj−1 2)δj−1 n ) ifδjαj<1.(4.36) Therefore, when δjandγsatisfy (4.23) and (4.34), form (4.25), (4.33) and (4.35) we obtain |Tj,n(ξj,n)|./vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayx ξj,ne∓(qj,n(x)−qj,n(s))iλn 1+iλndj(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ |Tj,n(ξj,n)|./integraldisplayξj,n xλn |1+iλndj(s)|ds =o(1)∀x∈[0,ξj,n]. (4.37) Moreover, underthesameconditionson δjandγ, dueto(4.7), (4.33)and(4.36)theCauchy-Schwarz inequality leads to /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayx ξj,ne∓(qj,n(x)−qj,n(s))λ−γ n 1+iλndj(s)f′ j,n(s)ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/integraldisplayξj,n xλ−γ n |1+iλndj(s)||f′ j,n(s)|ds ≤λ−γ n/bardblf′ j,n/bardblL2(0,ℓj)/parenleftbigg/integraldisplayξj,n xds |1+iλndj(s)|2/parenrightbigg1 2 = o(λ−δj 2−γ n) if δjαj≥1 o(λδj(αj−1 2)−γ−1 n ) ifδjαj<1 =o(1)∀x∈[0,ξj,n], (4.38) and due to (4.9), (4.33) and (4.35), we have /integraldisplayx ξj,ne∓(qj,n(x)−qj,n(s))iλ1−γ n 1+iλndj(s)/integraldisplayξj,n sgj,n(τ)dτds =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayξj,n x/integraldisplayτ xe∓(qj,n(x)−qj,n(s))λ1−γ n 1+iλndj(s)gj,n(τ)dτds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤λ1−γ n/integraldisplayξj,n x/integraldisplayτ x|gj,n(τ)| |1+iλndj(s)|dτds ≤λ1−γ n/integraldisplayξj,n xds |1+iλndj(s)|/integraldisplayξj,n x|gj,n(τ)|dτ = o(λ1−γ−3δj 2n) ifδjαj≥1 o(λδj(αj−3 2)−γ n ) ifδjαj<1 =o(1)∀x∈[0,ξj,n]. (4.39) Combining (4.38) and (4.39) yields (4.40)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayx ξj,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 From assumption (A 1.3) we have d′ j(s)>0 near 0, then by Cauchy-Schwarz inequality and as- sumption (A 1.4) we find /integraldisplayξj,n xλnd′ j(s) |1+iλndj(s)|ds≤/parenleftigg/integraldisplayξj,n xλ2 nd′ j(s) 1+(λndj(s))2ds/parenrightigg1 2 ./parenleftbigg/integraldisplayξj,n xd′ j(s)ds/parenrightbigg1 2 ≤/parenleftbig arctan(λ2 ndj(ξj,n))−arctan(λ2 ndj(x))/parenrightbig1 2.(dj(ξj,n)−dj(x))1 2 =O(λ−αjδj 2n)∀x∈[0,ξj,n]. (4.43) Inserting (4.25), (4.33), (4.37), (4.40) and (4.43) into (4 .30), then we get |z± j,n(x)| ≤o(1)(mj,n+1)∀x∈[0,ξj,n], where mj,n= max x∈[0,ξj,n]{|z+ j,n(x)|+|z− j,n(x)|}, which leads to (4.44) mj,n=o(1). Since we can write vj,n(x) =1 2(z+ j,n(x)−z− j,n(x)) then we follow from (4.44) that (4.45) /bardblvj,n/bardblL2(0,ξj,n)=1 2/bardblz+ j,n−z− j,n/bardblL2(0,ξj,n)≤mj,n 2/radicalbig ξj,n=o(λ−δj 2n) and (4.46) |vj,n(0)|=1 2|z+ j,n(0)−z− j,n(0)| ≤mj,n 2=o(1). Integrating (4.9) over (0 ,ξj,n), (4.47) iλn/integraldisplayξj,n 0vj,n(s)ds−Tj,n(ξj,n)+Tj,n(0) =λ−γ n/integraldisplayξj,n 0gj,n(s)ds. Due to (4.44) and the fact that a(0) = 0, we have (4.48)/vextendsingle/vextendsingle/vextendsingle/vextendsingleiλn/integraldisplayξj,n 0vj,nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1 2|z+ j,n(0)−z− j,n(0)|=o(1). Substituting (4.9), (4.24) and (4.48) into (4.47) yields (4.49) |Tj,n(0)|=o(1). Substituting (4.46), (4.49) into (4.14), by the transmissi on conditions (4.10) and (4.11) we con- clude (4.50)/integraldisplayℓ0 0/parenleftbig |u′ 0,n|2+|v0,n|2/parenrightbig dx=o(1). Forj= 1,...,Nwe have /bardbld1 2 ju′ j/bardblL2(0,ℓj)≥ /bardbld1 2 ju′ j/bardblL2(ξj,n,ℓj)≥min x∈(ξj,n,ℓj)/parenleftbigg/radicalig dj(x)/parenrightbigg /bardblu′ j,n/bardblL2(ξj,n,ℓj) ≥/radicalig dj(ξj,n)/bardblu′ j,n/bardblL2(ξj,n,ℓj)≥Cξαj 2 j,n/bardblu′ j,n/bardblL2(ξj,n,ℓj)≥Cλ−δjαj 2n/bardblu′ j,n/bardblL2(ξj,n,ℓj). (4.51) From (4.23) we haveαjδj 2≤γ 2+1 for every j= 1,...,N, then by combining (4.13) and (4.51) one gets (4.52) /bardblu′ j,n/bardblL2(ξj,n,ℓj)=o(λαjδj 2−γ 2−1 n ).14 FATHI HASSINE Therefore by the trace formula (4.53) |uj,n(ξj,n)|=o(λαjδj 2−γ 2−1 n ). By trace formula and (4.12) we have (4.54) |uj,n(0)| ≤C/bardblu′ j,n/bardblL2(0,ℓj)=O(λαjδj 2−γ 2−1 n ). From (4.12) and (4.7) one has (4.55) λn/bardbluj,n/bardblL2(0,ℓj)=O(1). Multiplying (4.9) by λ−γ nuj,nand integrating over (0 ,ξj,n) then by integrating by parts we arrive (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′ j,n/bardblL2(0,ξj,n) +Tj,n(0)uj,n(0)−Tj,n(ξj,n)uj,n(ξj,n) =o(λ−γ n). Inserting (4.25), (4.45), (4.49), (4.53), (4.54) and (4.55 ) into (4.56) one finds (4.57) /bardblu′ j,n/bardblL2(0,ξj,n)=o(1),∀j= 1,...,N. So that, adding (4.52) and (4.57), we obtain (4.58) /bardblu′ j,n/bardblL2(0,ℓj)=o(1),∀j= 1,...,N. From (4.17), (4.50) and (4.58) leads to (4.59) /bardblvj/bardblL2(0,ℓj)=o(1),∀j= 1,...,N. This conclude the prove since now we have proved that /bardbl(un,vn)/bardblH=o(1) from (4.50), (4.58) and (4.59) providing that (4.23) and (4.34) hold true. Finally, notingthat thebest γandδj, intermofmaximization of γ, that satisfies (4.23)and(4.34) are where γ= max/braceleftbigg1−αj 2−αj, j= 1,...,N/bracerightbigg andδj=1 2−αjforj= 1,...,N. This completes the proof. Acknowledgments. The author thanks professor Ka¨ ıs Ammari for his advices, hi s suggestions and for all the discussions. References [1]M. Alves, J.M. Rivera, M. Sep `ulveda, O.V. Villagr `an, and M.Z. Garay, The asymptotic behavior of the linear transmission problem in viscoelasticity, Math. Nachr. ,287(2014), 483–497. [2]K. Ammari, F. Hassine and L. Robbiano, Stabilization for the wave equation with singular Kelvin-V oigt damping, Archive for Rational Mechanics and Analysis ,236(2020), 577–601. [3]K. Ammari, Z. Liu and F. Shel, Stabilization for the wave equation with singular Kelvin-V oigt damping, Semigroup Forum, 100(2020), 364–382. [4]W. Arendt and C.J. Batty, Tauberian theorems and stability of one-parameter semigro ups,Trans. Amer. Math. Soc., 306(1988), 837–852. [5]C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, an d stabilization of waves from the boundary, SIAM J. Control Optim., 30(1992), 1024–1065. [6]A. Borichev and Y. 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Hassine, Logarithmic stabilization of the Euler-Bernoulli transmi ssion plate equation with locally distributed Kelvin-Voigt damping, J. Math. Anal. Appl., 455(2017), 1765–1782.STABILITY OF A STAR-SHAPED NETWORK WITH LOCAL KELVIN-VOIGT DAMPING 15 [13]F. Huang, Characteristic conditions for exponential stability of li near dynamical systems in Hilbert space, Ann. Differential Equations., 1(1985), 43–56. [14]F. Huang, On the mathematical model for linear elastic systems with an alytic damping, SIAM J. Control Optim., 26(1988), 714–724. [15]J.L. Lions and E. Magenes, Probl` emes aux limites non homog` enes et applications , Dunod, Paris, 1968. [16]K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys.,53(2002), 265–280. [17]K. Liu, Z. Liu and Q. Zhang, Eventual differentiability of a string with local KelvinVoi gt damping, ESAIM Control Optim. Calc. Var., 23(2017), 443–454. [18]K. S. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Zeitschrift f¨ ur Angewandte Mathematik und Physik (ZAMP), 56(2005), 630–644. [19]K. S. Liu and B. Rao, Exponential stability for wave equations with local Kelvin -Voigt damping, Zeitschrift f¨ ur Angewandte Mathematik und Physik (ZAMP), 57(2006), 419–432. [20]Z. Liu and Q. Zhang, Stability of string with local Kelvin-Voigt dampingand non smooth coefficient at interface, SIAM J. control optim., 54(4) (2016), 1859–1871. [21]P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat eq uations, J. evol. equ., 6(2006), 325–362. [22]A. Pazy, Semigroups of linear operators and applications to partial differential equations , Springer, New York, 1983. [23]J. Pr¨uss,On the spectrum of C0-semigroups, Trans. Amer. Math. Soc., 284(1984), 847–857. [24]M. Renardy, On localized KelvinVoigt damping, Z. Angew. Math. Mech, 84(2004), 280–283. [25]L. Tebou, Stabilization of some elastic systems with localized Kelvi n-Voigt damping, Discrete and continuous dynamical systems, 36(2016), 7117–7136. [26]Q. Zhang, Exponential stability of an elastic string with local Kelvi nVoigt damping, Z. Angew. Math. Phys., 61(2010), 1009–1015. UR Analysis and Control of PDEs, UR 13ES64, Department of Mat hematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia E-mail address :fathi.hassine@fsm.rnu.tn

2020-06-24

In this paper, we study the stability problem of a star-shaped network of elastic strings with a local Kelvin-Voigt damping. Under the assumption that the damping coefficients have some singularities near the transmission point, we prove that the semigroup corresponding to the system is polynomially stable and the decay rates depends on the speed of the degeneracy. This result improves the decay rate of the semigroup associated to the system on an earlier result of Z.~Liu and Q.~Zhang in \cite{LZ} involving the wave equation with local Kelvin-Voigt damping and non-smooth coefficient at interface.

Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface

2006.14949v1

arXiv:1704.03326v1 [cond-mat.mtrl-sci] 11 Apr 2017CoFeAlB alloy with low damping and low magnetization for spi n transfer torque switching A. Conca,1,∗T. Nakano,2T. Meyer,1Y. Ando,2and B. Hillebrands1 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany 2Department of Applied Physics, Tohoku University, Japan (Dated: June 29, 2021) We investigate the effect of Al doping on the magnetic propert ies of the alloy CoFeB. Comparative measurements of the saturation magnetization, the Gilbert damping parameter αand the exchange constantasafunctionoftheannealingtemperature forCoFe B andCoFeAlBthinfilmsare presented. Our results reveal a strong reduction of the magnetization f or CoFeAlB in comparison to CoFeB. If the prepared CoFeAlB films are amorphous, the damping para meterαis unaffected by the Al doping in comparison to the CoFeB alloy. In contrast, in the c ase of a crystalline CoFeAlB film, α is found to be reduced. Furthermore, the x-ray characteriza tion and the evolution of the exchange constant with the annealing temperature indicate a similar crystallization process in both alloys. The data proves the suitability of CoFeAlB for spin torque sw itching properties where a reduction of the switching current in comparison with CoFeB is expecte d. The alloy CoFeB is widely used in magnetic tunnel- ing junctions in combination with MgO barriers due to the large magnetoresistance effect originating in the spin filtering effect [1–4]. For the application in magnetic ran- dom accessmemories, the switching ofthe magnetization of the free layer via spin transfer torque (STT) with spin polarised currents is a key technology. However, the re- quired currents for the switching process are still large and hinder the applicability of this technique. The criti- cal switching current density for an in-plane magnetized system is given by [5] Jc0=2eαMStf(HK+Hext+2πMS) /planckover2pi1η(1) whereeis the electron charge, αis the Gilbert damping parameter, MSis the saturation magnetization, tfis the thickness of the free layer, Hextis the external field, HK is the effective anisotropy field and ηis the spin transfer efficiency. Fromtheexpressionitisclearthat, concerning material parameters, Jc0is ruled by the product αM2 S. For out-of-plane oriented layers, the term 2 πMSvanishes and then JC0is proportional to αMS[6]. Even in the case of using pure spin currents created by the Spin Hall effect, the required currents are proportional to factors of the form αnMSwithn= 1,1/2 [7]. A proper strat- egy to reduce the critical switching currents is then de- fined by reducing the saturation magnetization. This can be achieved by the development of new materials or the modification of known materials with promising prop- erties. Since the compatibility with a MgO tunneling barrier and the spin filtering effect must be guaranteed together with industrial applicability, the second option is clearly an advantage by reducing MSin the CoFeB al- loy. In this case, a critical point is that this reduction must not be associated with an increase of the damping parameter α.In the last years, several reports on doped CoFeB al- loys have proven the potential of this approach. The introduction of Cr results in a strong reduction of MS [8–10], however, it is sometimes also causing an increase of the damping parameter [8]. The reduction of MSby doping CoFeB with Ni is smaller compared to a doping with Cr but it additionally leads to a reduction of α[8]. FIG. 1. (Color online) θ/2θ-scans for 40 nm thick films of Co40Fe40B20(top) and Co 36Fe36Al18B10(bottom) showing the evolution of crystallization with the annealing temper a- ture.2 FIG. 2. (Color online) Evolution of the saturation magnetiz a- tion for CoFeB and CoFeAlB with the annealing temperature Tann. In constrast, the reduction of magnetization with V is comparable to Cr [9] but to our knowledge no values for αhave been published. In the case of doping of CoFeB by Cr or by V, a reduction of the switching current has been shown [8, 9]. In this Letter, we report on results on Al doped CoFeB alloy thin films characterized by ferromagnetic resonance spectroscopy. The dependence of MS, the Gilbert damp- ing parameter αand the exchange constant on the an- nealing temperature is discussed together with the crys- talline structure of the films and the suitability for STT switching devices. The samples are grown on Si/SiO 2substrates us- ing DC (for metals) and RF (for MgO) sput- tering techniques. The layer stack of the sam- ples is Si/SiO 2/Ta(5)/MgO(2)/FM(40)/MgO(2)/Ta(5) where FM = Co 40Fe40B20(CoFeB) or Co 36Fe36Al18B10 (CoFeAlB). Here, the values in brackets denote the layer thicknesses in nm. In particular, the FM/MgO interface is chosen since it is widely used for STT devices based on MTJs. This interface is also required to promote the correct crystallizationof the CoFeB layerupon annealing since the MgO layer acts as a template for a CoFe bcc (100)-oriented structure [1–3] with consequent B migra- tion. The dynamic properties and material parameters were studied by measuring the ferromagnetic resonance using a strip-line vector network analyzer (VNA-FMR). For this, the samples were placed face down and the S 12 transmission parameter was recorded. A more detailed description of the FMR measurement and analysis pro- cedure is shown in previous work [11, 12]. Brillouin light spectroscopy (BLS) was additionally used for the mea- surement of the exchange constant. The crystalline bulk properties of the films were studied by X-ray diffractom- etry (XRD) using the Cu-K αline. Figure 1 shows the θ/2θ-scans for CoFeB (top) andFIG. 3. (Color online) Linewidth at a fixed frequency of 18 GHz (a) and Gilbert dampingparameter αdependenceon the annealing temperature T ann(b). The αvalue for Tann= 500◦ is only a rough estimation since the large linewidth value do es not allow for a proper estimation. The inset shows the linear dependence of the linewidth on the frequency exemplarily fo r CoFeAlB annealed at 350◦C and 400◦C. The red lines are a linear fit. CoFeAlB (bottom) samples annealed at different tem- peratures T ann. The appearance of the CoFe diffractions peaks, as shown by the arrowsin Fig. 1 indicate the start of crystallization at high annealing temperatures of more than 400◦C. In the case of lower annealing temperatures or the as-deposited samples, the FM layer is in an amor- phous state. The first appearance of the (200) diffraction peak occurs at the same point for both alloys showing a verysimilarthermalevolution. Thissimplifiesasubstitu- tion ofCoFeB by the Al alloyin tunneling junctions since the same annealing recipes can be applied. This is criti- cal since the used values must be also optimized for the quality of the tunneling barrier itself or the perpendicu- lar anisotropy induced by the FM/MgO interface. The (110) CoFe peak is also present for both material compo- sitions owing to a partial texturing of the film. However, the larger intensity of the (200) peak is not compatible with a random crystallite orientation but with a domi- nant (100) oriented film [13, 14]. This is needed since the spin filtering effect responsible for the large magnetore- sistance effect in MgO-based junctions requires a (100)3 FIG.4. (Color online)Dependenceoftheproduct αM2 Sonthe annealing temperature T annfor CoFeB and CoFeAlB. This quantityisrulingtheswitchingcurrentinin-planemagnet ized STT devices as shown in Eq. 1. orientation. The dependence of the FMR frequency on the external magnetic field is described by Kittel’s formula [15]. The value ofMeffextracted from the Kittel fit is related with the saturation magnetization of the sample and the in- terfacial properties by Meff=MS−2K⊥ S/µ0MSdwhere K⊥ Sis the interface perpendicular anisotropy constant. For the thickness used in this work (40 nm) and physi- cally reasonable K⊥ Svalues, the influence of the interface is negligible and therefore Meff≈MS. For details about the estimation of Meffthe reader is referred to [12]. Figure 2 shows the obtained values for MSfor all sam- ples. AstrongreductionforCoFeAlBin comparisonwith standard CoFeB is observed and the relative difference is maintained for all T ann. The evolution with annealing is very similar for both alloys. Significantly, the increase in MSstartsforvaluesofT annlowerthan expectedfromthe appearance of the characteristic CoFe diffraction peaks in the XRD data (see Fig. 1). This shows that the mea- surement of MSis the more sensitive method to probe the change of the crystalline structure. For CoFeB a saturationvalue around MS≈1500kA/m is reached at T ann= 450◦C. This is compatible with val- ues reported for CoFe (1350-1700 kA/m) [16, 17] and CoFeB (1350-1500 kA/m) [17, 18]. On the contrary, for CoFeAlB the introduction of Al reduces the magnetiza- tion of the samples and the annealing does not recover to CoFe-like values. Figure 3(a) shows the dependence of the magnetic field linewidth on T annmeasured at a fixed frequency of 18 GHz. From the linear dependence of this linewidth on the FMR frequency, the Gilbert damping parameter is extracted (as exemplarily shown for the CoFeAlB al- loy in the inset in Fig. 3(b)) and the results are shown in Fig. 3(b). For T annvalues up to 350◦C, where theFIG. 5. (Color online) Dependence of the exchange con- stantAexon the annealing temperature T annfor CoFeB and CoFeAlB. The top panels show typical BLS spectra for ma- terials (see text). amorphousphaseisstill dominating, almostnodifference between both alloys is observed. With increasing tem- perature the damping increases for both alloys but the evolution is different. For CoFeAlB the increase starts almost abruptly at T ann= 400◦C, reaches a maximum aroundα= 0.02 and then decreases again to α= 0.012 for Tann= 500◦C. In contrast, the increase for CoFeB is more smoothly with T annand increases stadily with higher T ann. In fact, due to the large linewidths reached for Tann= 500◦C, the value of αcannot be properly estimated and only a lower limit of 0.03-0.04 can be given. This situation is represented by the dashed line in Fig. 3(b). It is important to note here that when the crystallization process is fulfilled (i.e. for T ann= 500◦C) αis much lower for the Al doped alloy. This is rele- vant for the application in tunneling junctions where a full crystallization is required for the presence of the spin filtering effect originating large magnetoresistance values in combination with MgO barriers [4]. For further comparison of both alloys, the quantity αM2 Shasbeen calculatedand plotted in Fig. 4. As shown in Eq. 1, this value is ruling the critical switching current in in-plane magnetized systems. We observe for the al- loys showing a mostly amorphous phase (T ann<400◦C) a slight improvement for CoFeAlB in comparison with CoFeB due to the lower MS. However, for fully crys- talline films (T ann= 500◦C), the CoFeAlB shows a much smaller value for αM2 S. Since a full crystalline phase is needed for any application of this alloy in MTJ-based de- vices, this denotes a major advantage of this compound compared to standard. The exchange constant Aexis a critical parameter that is strongly influenced by the introduction of Al. Its esti- mationinrequiredformodelingthespintorqueswitching behaviorofthe alloys. The accessto the constantisgiven4 by the dependence of the frequency of the perpendicular standing spin-wave (PSSW) modes on the external static magnetic field [19]. As shown in previous works [12, 20], itispossibletoobservethePSSWmodesinmetallicfilms with a standard VNA-FMR setup. However, the signal is strongly reduced compared to the FMR peak. For the samples presented in this paper, the PSSW peak could not be observed for T ann>400◦C since the increased damping leads to a broadening and lowering of the peak which prevents the estimation of Aex. For this reason, BLS spectroscopy is used for the measurement of the fre- quency position of the PSSW modes. This technique has alargersensitivityforthePSSWmodesthanVNA-FMR. Figure 5(c) shows the evolution of Aexupon annealing for both alloys. For the films dominated by the amor- phous phase the value is much lower for CoFeAlB which is also compatible with the lower magnetization. How- ever, asthe crystallizationevolves,theexchangeconstant increases stronger than for CoFeB and the same value is obtained for the fully crystallized films. This fact points to a similar role of Al and B during the crystallization process: when the CoFe crystallitesform, the light atoms are expelled forming a Al-B-rich matrix embedding the magnetic crystallites. This explains also the similar evo- lution observed in the XRD data shown in Fig. 1. The lower maximal magnetization obtained for the CoFeAlB can be explained by the reduced CoFe content but also a certain number of residual Al and B atoms in the crys- tallites, which may differ for both alloys. TheAexvalues for as-deposited CoFeB films are very similar to previous reports [12, 20, 21]. Concerning the values for the crystallized samples, since the properties are strongly dependent on the B content and of the ra- tio between Co and Fe as well as on the exact annealing conditions, a comparison with literature has to be made carefully. Nevertheless, the maximal value and the evo- lution with T annfor CoFeB is similar to the one reported by some of the authors [12]. Also results for alloys with the same B content arecompatiblewith ourdata [22, 23]. CoFeB films with reduced B content show larger values [17], the same is true for CoFe alloys with values between 3.84-2.61 ×1011J/m depending on the exact stoichiome- try [16, 17]. This may again be a hint that a rest of Al or B is present in the CoFe crystallites. In summary, the presented experimental results show that CoFeAlB is a good candidate as alternative to CoFeB for spin torque switching devices due to the re- duction of the factor αM2 Swhich dominates the critical switching current. This reduction was found to originate from a strong reduction of the saturation magnetization andadecreaseddampingparameter αforfullycrystalline CoFeAlB films. Furthermore, the results reveal a larger thermal stability of the damping properties in CoFeAlB compared to CoFeB. The absolute values of MSand the exchange constant Aexfor crystalline films point to a for- mation of CoFe crystallines with a non-vanishing contentof the lights atoms embedded in a B or Al matrix. Financial support by M-era.Net through the HEUMEM project, the DFG in the framework of the research unit TRR 173 Spin+X and by the JSPS Core-to-Core Program is gratefully acknowledged. ∗conca@physik.uni-kl.de [1] S.YuasaandD.D.Djayaprawira, J.Phys.D:Appl.Phys. 40, R337-R354 (2007). [2] S. Yuasa,Y. Suzuki, T. Katayama, and K. Ando, Appl. Phys. Lett. 87, 242503 (2005). [3] Y. S. Choi, K. Tsunekawa, Y. Nagamine, and D. Djayaprawira, J. Appl. Phys. 101, 013907 (2007). [4] X.-G. Zhang and W. H. Butler, J. Phys.: Condens. Mat- ter15R1603, (2003). [5] Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L.-C. Wang, and Y. Huai, J. Phys. D: Appl. Phys. 19, 165209 (2007). [6] K. L. Wang, J. G. Alzate, and P.K. Amiri, J. Phys. D: Appl. Phys. 46, 074003 (2013). [7] T. Taniguchi, S. Mitani, and M. Hayashi, Phys. Rev. B 92, 024428 (2015). [8] K. Oguz, M. Ozdemir, O. Dur, and J. M. D. Coey, J. Appl. Phys. 111, 113904 (2012). [9] H. Kubota, A. Fukushima, K. Yakushiji, S. Yakata, S. Yuasa, K. Ando, M. Ogane, Y.Ando, andT. Miyazaki, J. Appl. Phys. 105, 07D117 (2009). [10] Y. Cui, M. Ding, S. J. Poon, T. P. Adl, S. Keshavarz, T. Mewes, S. A. Wolf, and J. Lu, J. Appl. Phys. 114, 153902 (2013). [11] A. Conca, S. Keller, L. Mihalceanu, T. Kehagias, G. P. Dimitrakopulos, B. Hillebrands, and E. Th. Pa- paioannou, Phys. Rev. B 93, 134405 (2016). [12] A. Conca, E. Th. Papaioannou, S. Klingler, J. Greser, T. Sebastian, B. Leven, J. L¨ osch, and B. Hillebrands, Appl. Phys. Lett. 104, 182407 (2014). [13] G. Concas, F. Congiu, G. Ennas, G. Piccaluga, and G. Spano. J. of Non-Crystalline Solids 330, 234 (2003). [14] C. Y. You, T. Ohkubo, Y. K. Takahashi, and K. Hono, J. Appl. Phys. 104, 033517 (2008). [15] C. Kittel, Phys. Rev. 73, 155 (1948). [16] X. Liu, R. Sooryakumar, C. J. Gutierrez, and G. A. Prinz, J. Appl. Phys. 75, 7021 (1994). [17] C. Bilzer, T. Devolder, J.-V. Kim, G. Counil, C. Chap- pert, S. Cardoso, and P. P. Freitas, J. Appl. Phys. 100, 053903 (2006). [18] X. Liu, W. Zhang, M. J. Carter, and G. Xiao, J. Appl. Phys.110, 033910 (2011). [19] S. O. Demokritov, B. Hillebrands, Spin Dynamics in Confined Magnetic Structures I , Springer, Berlin, (2002). [20] A. Conca, J. Greser, T. Sebastian, S. Klingler, B. Obry, B. Leven, and B. Hillebrands, J. Appl. Phys. 113, 213909 (2013). [21] J. Cho, J.Jung, K.-E.Kim, S.-I.Kim, S.-Y.Park, andM.- H. Jung, C.-Y. You, J. of Magn and Magn. Mat. 339, 36 (2013). [22] A. Helmer, S. Cornelissen, T. Devolder, J.-V. Kim, W. van Roy, L. Lagae, and C. Chappert, Phys. Rev. B 81, 094416 (2010). [23] H.Sato, M.Yamanouchi, K.Miura, S.Ikeda, R.Koizumi,5 F. Matsukura, and H. Ohno, IEEE Magn. Lett., 3, 3000204 (2012).

2017-04-11

We investigate the effect of Al doping on the magnetic properties of the alloy CoFeB. Comparative measurements of the saturation magnetization, the Gilbert damping parameter $\alpha$ and the exchange constant as a function of the annealing temperature for CoFeB and CoFeAlB thin films are presented. Our results reveal a strong reduction of the magnetization for CoFeAlB in comparison to CoFeB. If the prepared CoFeAlB films are amorphous, the damping parameter $\alpha$ is unaffected by the Al doping in comparison to the CoFeB alloy. In contrast, in the case of a crystalline CoFeAlB film, $\alpha$ is found to be reduced. Furthermore, the x-ray characterization and the evolution of the exchange constant with the annealing temperature indicate a similar crystallization process in both alloys. The data proves the suitability of CoFeAlB for spin torque switching properties where a reduction of the switching current in comparison with CoFeB is expected.

CoFeAlB alloy with low damping and low magnetization for spin transfer torque switching

1704.03326v1

Gilbert damping of high anisotropy Co/Pt multilayers Thibaut Devolder Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Université Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France S. Couet, J. Swerts, and G. S. Kar imec, Kapeldreef 75, 3001 Heverlee, Belgium (Dated: June 16, 2021) Using broadband ferromagnetic resonance, we measure the damping parameter of [Co(5 Å)/Pt(3 Å)] 6mul- tilayers whose growth was optimized to maximize the perpendicular anisotropy. Structural characterizations in- dicate abrupt interfaces essentially free of intermixing despite the miscible character of Co and Pt. Gilbert damp- ing parameters as low as 0.021 can be obtained despite a magneto-crystalline anisotropy as large as 106J/m3. The inhomogeneous broadening accounts for part of the ferromagnetic resonance linewidth, indicating some structural disorder leading to a equivalent 20 mT of inhomogenity of the effective field. The unexpectedly rel- atively low damping factor indicates that the presence of the Pt heavy metal within the multilayer may not be detrimental to the damping provided that intermixing is avoided at the Co/Pt interfaces. I. INTRODUCTION Thanks to their large perpendicular magnetic anisotropy, their confortable magneto-optical signals and their easy growth by physical vapor deposition1, the [Co/Pt] multilay- ers are one of the most popular system in spintronics. Early in spintronics history this model system was used to study the physics of domain wall propagation2, for the develop- ment of advanced patterning techniques3and for the assess- ment of micromagnetic theories4. More recently they have been extensively used as high quality fixed layers in per- pendicularly magnetized tunnel junctions, in particular in the most advanced prototypes of spin-transfer-torque magnetic random access memories memories5. Despite the widespread use of Co/Pt multilayers, their high frequency properties, and in particular their Gilbert damping parameter remains largely debated with experimental values that can differ by orders of magnitude from60.02 to 20 times larger7and the- oretical calculations from circa 0.035 in Co 50Pt50alloys8 to slightly smaller or substantially larger values in multilay- ers made of chemically pure layers9. Direct measurements by conventional ferromagnetic resonance (FMR) are scarce as the high anisotropy of the material pushes the FMR fre- quencies far above610 GHz and results in a correlatively low permeability that challenges the sensitivity of commer- cial FMR instruments10. As a result most of the measure- ments of the damping of Co/Pt systems were made by all- optical techniques11,12in small intervals of applied fields. Un- fortunately this technique requires the static magnetization to be tilted away from the out-of-plane axis and this tilt ren- ders difficult the estimation of the contribution of the ma- terial disorder to the observed FMR linewidth using the es- tablished protocols13; this is problematic since in Co-Pt sys- tems there contributions of inhomogeneity line broadening and two-magnon scattering by the structural disorder (rough- ness, interdiffusion, granularity,...) are often large14,15. It is noticeable that past reports on the damping of Co/Pt systems concluded that it ought to be remeasured in sam- ples with atomically flat interfaces11. Besides, this measure- ment should be done in out-of-plane applied field since thiseases the separation of the Gilbert damping contribution to the linewidth from the contribution of structural disorder16. In this paper, we measure the damping parameter of [Co(5 Å)/Pt(3 Å)]6multilayers whose growth was optimized to maximize perpendicular anisotropy anisotropy. The sputter- deposition is performed at an extremely low17Argon pressure in remote plasma conditions which enables very abrupt inter- faces that are essentially free of intermixing. We show that in contrast to common thinking, the Gilbert damping parameter of Co/Pt multilayers can be low; its effective value is 0.021 but it still likely16includes contributions from spin-pumping that our protocol can unfortunately not suppress. II. EXPERIMENTAL Our objective is to report the high frequency properties of Co/Pt multilayers that were optimized for high anisotropy. The multilayer is grown by sputter-deposition on a Ru (50 Å) buffer and capped with a Ru(70 Å)/Ta(70 Å)/Ru(100 Å)/Ta(10 Å, cap) sequence (bottom to top order). The Ru buffer was chosen because it does not mix with Co-based multilayers even under tough annealing conditions18. The stacks were deposited by physical vapor deposition in a Canon-Anelva EC7800 300 mm system on oxidized silicon substrates at room temperature. The Argon plasma pressure is kept at 0.02 Pa, i.e. substantially lower than the usual conditions of 0.1-0.5 Pa used in typical deposition machines17. As this multilayer is meant to be the reference layer of bottom-pinned magnetic tunnel junctions, in some samples (fig. 1) the non-magnetic cap is replaced the following sequence: Ta cap / Fe 60Co20B20 / MgO / Fe 60Co20B20/ Ta / [Co(5 Å)/Pt(3 Å)] 4/ Ru sim- ilar to as in ref. 19 and 20 to form a bottom-pinned mag- netic tunnel junction with properties designed for spin-torque applications21. All samples were annealed at 300C for 30 minutes in an out-of-plane field of 1 T.arXiv:1807.04977v1 [cond-mat.mtrl-sci] 13 Jul 20182 Ru[Co5Å/Pt3Å]×6 RuMgOFeCoBFeCoBTaRuTa[Co5Å/Pt3Å]×4Co5Å(a)(b)(c) FIG. 1. (Color online). Structure and anisotropy of a Co-Pt multi- layer. (a) Transmission Electron Micrograph of a magnetic tunnel junction that embodies our Co/Pt as hard multilayer at the bottom of the reference synthetic antiferromagnet, similar to that of ref. 21. (b) Easy axis and (c) hard axis hysteresis loops of the hard multilayer when covered with Ru(70 Å)/Ta(70 Å)/Ru(100 Å)/Ta(10 Å, cap) III. STRUCTURE X-ray reflectivity scans (not shown) indicate Bragg reflex- ions at 2= 11 , 22.2 and 33.6 deg., consistent with the mul- tilayer periodicity of 8 Å. Consistently, the Pt to Co inter- mixing is sufficiently low that well formed 3Å Pt spacers can be seen the Transmission Electron Micrograph after anneal- ing [Fig. 1(a)]. Almost no roughness is observed throughout the Co/Pt multilayer. We emphasize that this quality of inter- faces is almost equivalent to that obtained in Molecular Beam Epitaxy conditions22. Indeed Co and Pt are strongly miscible such that hyperthermal (high energy) deposition techniques like sputter deposition do not easily yield this low degree of intermixing, except when the deposition is conducted under sufficiently low plasma pressure in remote plasma conditions, i.e. when the substrate-to-target distance is large to avoid di- rect plasma exposure to the film being deposited. IV . ANISOTROPY The magnetic material properties were measured by vibrat- ing sample magnetometry (VSM) and Vector Network Ana- lyzer ferromagnetic resonance23in both easy (z) and hard axis (x) configurations. For VNA-FMR the sample is mechanically pressed on the surface of a 50 microns wide coplanar waveg- uide terminated by an open circuit; data analysis is conducted following the methods described in ref. 24. The VSM signal indicated a magnetization Ms= 8:5105kA/m if assuming a magnetic thickness of 48 Å, i.e. assuming that the [Co(5 Å)/Pt(3 Å)]6multilayer can be described as a single mate- rial. The loops indicate a perpendicular anisotropy with full remanence. The reversal starts at 46.8 mT and completes be- /s45/s50 /s45/s49 /s48 /s49 /s50/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48 /s72/s101/s102/s102 /s107/s49/s43/s72 /s107/s50 /s105/s110/s45/s112/s108/s97/s110/s101/s32 /s102/s105/s101/s108/s100/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41 /s48/s72/s32/s40/s84/s41/s72/s101/s102/s102 /s107/s49/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32 /s102/s105/s101/s108/s100FIG. 2. (Color online). FMR frequencies versus in-plane (cross sym- bols) or out-of-plane (square symbols) applied field. The bold lines are fits using Eq. 1 and 2, yielding 0(Hk1Ms) = 1:3200:005T and0Hk2= 0:1200:015T. fore 48 mT with a tail-free square hysteresis loop. Careful attempts to demagnetize the sample using an acperpendicular field failed to produce a multidomain state at remanence. This indicates that the lowest nucleation field in the whole sample is larger that the domain wall propagation field everywhere in the film. This low propagation field indicates qualitatively that the effective anisotropy field is very uniform. The hard axis loop indicates an in-plane saturation field of 1:30:1T in line with the expectations for such composition3. The round- ing of the hard axis loop near saturation and its slight hys- teretic remanence [Fig. 1(c)] impedes a more precise deduc- tion of the anisotropy fields from the sole hard axis loop. We shall instead use the ferromagnetic resonance data because magnetization eigenfrequencies constitute absolute measurements of the effective fields acting on the mag- netization. Fig. 2 gathers the measured FMR frequen- cies measured for in-plane and out-of-plane applied fields from -2.5 to 2.5 T. To analyze the microwave susceptibil- ity data, we assume an energy density that reads E= 1 20Hk1MSsin2+1 40Hk2MSsin4withthe (suppos- edly uniform) angle between the magnetization and the sam- ple normal. Our convention is that the first and second order magneto-crystalline anisotropy fields Hk1= 2K1=(0MS) andHk2= 4K2=(0MS)are positive when they favor per- pendicular magnetization, i.e. = 0. In that framework, the ferromagnetic resonance frequencies in out-of-plane and in-plane applied fields saturating the mag- netization as: !perp= 0(Hz+Hk1Ms) (1) and !in-plane = 0p Hx(HxHk1Hk2+Ms); (2) where 0=j j0is the gyromagnetic ratio. (For in-plane fieldsHxlower thanHx;sat=Hk1Hk2Msthe magne- tization is tilted. A straightforward energy minimization was3 used to yield magnetization tilt that was subsequently in- jected to a Smit and Beljers equation to yield the FMR fre- quency). The best fit to the experimental data is obtained for0(Hk1Ms) = 1:3200:005 T (corresponding to K1= 106J/m3) and0Hk2= 0:1200:015T. Note that the second order anisotropy is small but non negligible such that the effective anisotropy fields deduced from easy and axis axis measurements would differ by circa 10% if Hk2was dis- regarded. V . GILBERT DAMPING A. Models We now turn to the analysis of the FMR linewidth (Fig. 3). As common in FMR, the linewidth comprises an intrin- sic Gilbert damping part and an extrinsic additional contribu- tion linked to the lateral non uniformity of the local effective fieldsHk1Ms. This can be gathered in a characteristic field
H0measuring the disorder relevant for FMR. In out- of-plane field FMR experiments, the proportionality between effective fields and resonance frequencies (Eq. 1) allows to write simply
H0=1 0
!j!!0, and for the perpendicular magnetization we follow the usual convention16and write: 1 0
!perp= 2(Hz+Hk1Ms) +
H0 (3) or equivalently
!perp= 2!perp+ 0
H0. For in-plane magnetization, the intrinsic linewidth above the in-plane saturation field is 1 0
!Gilbert in-plane =(2HxHk1Hk2+Ms) (4) The resonance frequency (Eq. 2) is non linear with the ef- fective fields such that the non uniformity
H0of the local effective fields translates in a linewidth broadening through the term 1 0
!disorder in-plane =d!in-plane d(MsHk1)
H0 (5) where the derivative term ispHx 2pHxHk1Hk2+Ms. In case of finite disorder, this factor diverges at the spatially-averaged in-plane saturation field Hx;sat. B. Results For each applied field, the real and imaginary parts of the transverse permeability (f)were fitted with the one expected for the uniform precession mode25with three free param- eters: the FMR frequency !FMR=(2), the FMR linewidth
!=(2))and a scaling (sensitivity) factor common to both real and imaginary parts of (f)as illustrated in Fig. 3b.When plotting the symmetric lorentzian-shaped imagi- nary part of the transverse permeability versus the asymet- ric lorentzian-shaped real part of the permeability for fre- quencies ranging from dcto infinity, a circle of diameter Ms=[2(Hz+Hk1Ms)]should be obtained for a spatially uniform sample18. The finite disorder
H0distorts the exper- imental imaginary part of the permeability towards a larger and more gaussian shape. It can also damp and smoothen the positive and negative peaks of the real part of the permeabil- ity; when the applied field is such that the inhomogeneous broadening is larger than the intrinsic Gilbert linewidth, this results in a visible ellipticity of the polar plot of (f). In our experimental polar plot of (f)(Fig. 3a) the deviations from perfect circularity are hardly visible which indicates that the inhomogeneous broadening is not the dominant contribution to the sample FMR linewidth in out-of-plane field conditions. To confirm this point we have plotted in Fig. 3c the de- pendence of FMR linewidth with FMR frequency for out-of- plane applied fields. A linear fit yields = 0:0210:002and
H040mT. A substantial part of the measured linewidth thus still comes from the contribution of the lateral inhomo- geneity of the effective anisotropy field within the film. As a result, low field measurements of the FMR linewidth would be insufficient to disentangle the Gilbert contribution and the structural disorder contributions to the total FMR linewidth. The in-plane applied field FMR linewidth can in principle be used to confirm this estimate of the damping factor. Un- fortunately we experience a weak signal to noise ratio in in- plane field FMR experiments such that only a crude estimation of the linewidth was possible.Within the error bar, it is inde- pendent from the applied field from 1.7 to 2.5 T (not shown) which indicates that the disorder still substantially contributes to the linewidth even at our maximum achievable field. At 2.5 T the linewidth was1 2
!in-plane3:00:3GHz. This is consistent width the expectations of that would predict 2.2 GHz of intrinsic contribution (Eq. 4) and 0.4 GHz of intrinsic contribution (Eq. 5). VI. DISCUSSION We conclude that the damping of Co/Pt multilayers can be of the order of 0.02 even for multilayers with anisotropies among the strongest reported (see ref. 26 for a survey of the anisotropy of Co/Pt multilayers). Note that 0:021is still a higher bound, as we are unable to measure and subtract the spin-pumping contribution. Measuring the spin-pumping contribution would require to vary the cap and buffer layer thicknesses without affecting the multilayer structure which is difficult to achieve. Still, we can conclude that the damp- ing of Co/Pt multilayers lies in the same range as other high anisotropy multilayers like Co/Ni (ref. 18 and 27) and Co/Pd (ref. 16) systems. This conclusion is in stark contrast with the common thinking7that Co/Pt systems alway s have a large damping. This widespread opinion is based on the standard models of magneto-crystalline anisotropy28and damping29that pre- dicts that they both scale with the square of the spin-orbit4 /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 /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 /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 /s40/s105/s109/s97/s103/s105/s110/s97/s114/s121/s32/s112/s97/s114/s116/s41 /s73/s109/s97/s103/s105/s110/s97/s114/s121/s32/s48/s46/s48/s52 /s32/s50/s102/s32 /s102 /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 /s40/s99/s41/s40/s98/s41/s77/s97/s99/s114/s111/s115/s112/s105/s110/s32/s102/s105/s116/s32/s32 /s40/s97/s41 FIG. 3. (Color online). Gilbert damping of the Co/Pt multilayer. (a) Imaginary part versus real part of the permeability for a field of 0.45 T applied perpendicularly to the plane. The bold lines are theoretical macrospin permeability curves with linewidth parameters (i.e. effec- tive damping) of 0.04. (b) Same data but versus frequency. (c): FMR half linewidth versus FMR frequency. The bold line is a guide to the eye with a slope = 0:021and zero frequency intercept of 0.6 GHz. coupling, which is particularly large in the Pt atoms. We emphasize that this expectation of large damping is not sys- tematically verified: in studies that make a thorough anal- ysis of the effects of structural disorder, no correlation wasfound between anisotropy and damping in comparable mate- rial systems11,16. Rather, a large correlation was found be- tweenHk1and
H0, indicating that when the anisotropy is strong, any local inhomogeneity thereof has a large impact on the FMR linewidth. Owing to the difficulty of achiev- ing well-defined Co/Pt interfaces, we believe that past con- clusions on the large damping of Co/Pt systems were based on systems likely to present some intermixing at the inter- face; indeed the presence of impurities with large spin-orbit coupling considerably degrades (increases) the damping of a magnetic material30and synchonously degrades (decreases) the magneto-crystalline anisotropy31. VII. CONCLUSION In summary, we have studied high anisotropy [Co(5 Å)/Pt(3 Å)]6multilayers grown by low pressure remote plasma sputter deposition. The deposition conditions were tuned to achieve abrupt interfaces with little intermixing. Broad- band ferromagnetic resonance was used to measure the first and second order uniaxial anisotropy fields. With the mag- netization measured by vibrating sample magnetometry, this yields an anisotropy energy of 1MJ/m3. The inhomogeneous broadening accounts for part of the ferromagnetic resonance linewidth, indicating some structural disorder leading to a equivalent 40 mT (or equivalently 600 MHz) of inhomogenity of the effective field in out-of-plane applied fields. This FMR- relevant inhomogeneity is comparable to the coercivity of 47 mT. Despite the large anisotropy a Gilbert damping parameter as low as 0.0210.002 is obtained. This unexpectedly rela- tively low damping factor indicates that the presence of the Pt heavy metal within the multilayer can in some condition not be detrimental to the damping. We interpret our results and literature values by analyzing the consequences of Pt/Co in- termixing: Pt impurities within a Cobalt layer reduce locally the interface anisotropy as they reduce the abruptness of the composition profile, but they also increase substantially the Gilbert damping. As a result, a large anisotropy together with a low damping can be obtained provided that intermixing is minimized at the Co/Pt interfaces. thibaut.devolder@u-psud.fr 1V . Mathet, T. Devolder, C. Chappert, J. Ferré, S. Lemerle, L. Bel- liard, and G. Guentherodt, Journal of Magnetism and Magnetic Materials 260, 295 (2003). 2S. Lemerle, J. Ferré, C. Chappert, V . Mathet, T. Giamarchi, and P. Le Doussal, Physical Review Letters 80, 849 (1998). 3C. Chappert, H. Bernas, J. Ferré, V . Kottler, J.-P. Jamet, Y . Chen, E. Cambril, T. Devolder, F. Rousseaux, V . Mathet, and H. Launois, Science 280, 1919 (1998). 4L. Belliard, J. Miltat, V . Kottler, V . Mathet, C. Chappert, and T. Valet, Journal of Applied Physics 81, 5315 (1997). 5K. Yakushiji, A. Sugihara, A. Fukushima, H. Kubota, and S. Yuasa, Applied Physics Letters 110, 092406 (2017).6N. Fujita, N. Inaba, F. Kirino, S. Igarashi, K. Koike, and H. Kato, Journal of Magnetism and Magnetic Materials Eighth Perpendic- ular Magnetic Recording Conference, 320, 3019 (2008). 7P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier, J. Ferré, V . Baltz, B. Rodmacq, B. Dieny, and R. L. Stamps, Physical Re- view Letters 99, 217208 (2007). 8V . Drchal, I. Turek, and J. Kudrnovský, Journal of Superconduc- tivity and Novel Magnetism 30, 1669 (2017). 9E. Barati and M. Cinal, Physical Review B 95, 134440 (2017). 10S. J. Yuan, L. Sun, H. Sang, J. Du, and S. M. Zhou, Physical Review B 68, 134443 (2003). 11S. Mizukami, E. P. Sajitha, D. Watanabe, F. Wu, T. Miyazaki, H. Naganuma, M. Oogane, and Y . Ando, Applied Physics Letters 96, 152502 (2010).5 12A. Barman, S. Wang, O. Hellwig, A. Berger, E. E. Fullerton, and H. Schmidt, Journal of Applied Physics 101, 09D102 (2007). 13S. Mizukami, Y . Ando, and T. Miyazaki, Physical Review B 66, 104413 (2002). 14N. Mo, J. Hohlfeld, M. ul Islam, C. S. Brown, E. Girt, P. Krivosik, W. Tong, A. Rebei, and C. E. Patton, Applied Physics Letters 92, 022506 (2008). 15A. J. Schellekens, L. Deen, D. Wang, J. T. Kohlhepp, H. J. M. Swagten, and B. Koopmans, Applied Physics Letters 102, 082405 (2013). 16J. M. Shaw, H. T. Nembach, and T. J. Silva, Physical Review B 85, 054412 (2012). 17J. Musil, Vacuum 50, 363 (1998). 18E. Liu, J. Swerts, T. Devolder, S. Couet, S. Mertens, T. Lin, V . Spampinato, A. Franquet, T. Conard, S. Van Elshocht, A. Furnemont, J. De Boeck, and G. Kar, Journal of Applied Physics 121, 043905 (2017). 19J. Swerts, S. Mertens, T. Lin, S. Couet, Y . Tomczak, K. Sankaran, G. Pourtois, W. Kim, J. Meersschaut, L. Souriau, D. Radisic, S. V . Elshocht, G. Kar, and A. Furnemont, Applied Physics Letters 106, 262407 (2015). 20T. Devolder, S. Couet, J. Swerts, and A. Furnemont, Applied Physics Letters 108, 172409 (2016).21T. Devolder, J.-V . Kim, F. Garcia-Sanchez, J. Swerts, W. Kim, S. Couet, G. Kar, and A. Furnemont, Physical Review B 93, 024420 (2016). 22D. Weller, L. Folks, M. Best, E. E. Fullerton, B. D. Terris, G. J. Kusinski, K. M. Krishnan, and G. Thomas, Journal of Applied Physics 89, 7525 (2001). 23C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P. P. Freitas, Journal of Applied Physics 101, 074505 (2007). 24C. Bilzer, T. Devolder, P. Crozat, and C. Chappert, IEEE Trans- actions on Magnetics 44, 3265 (2008). 25T. Devolder, Physical Review B 96, 104413 (2017). 26V . W. Guo, B. Lu, X. Wu, G. Ju, B. Valcu, and D. Weller, Journal of Applied Physics 99, 08E918 (2006). 27J.-M. L. Beaujour, W. Chen, K. Krycka, C.-C. Kao, J. Z. Sun, and A. D. Kent, The European Physical Journal B 59, 475 (2007). 28P. Bruno, Physical Review B 39, 865 (1989). 29V . Kambersky, Physical Review B 76(2007), 10.1103/Phys- RevB.76.134416. 30J. O. Rantscher, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F. Egelhoff, B. B. Maranville, D. Pulugurtha, A. P. Chen, and L. M. Connors, Journal of Applied Physics 101, 033911 (2007). 31T. Devolder, Physical Review B 62, 5794 (2000).

2018-07-13

Using broadband ferromagnetic resonance, we measure the damping parameter of [Co(5 \r{A})/Pt(3 \r{A})]${\times 6}$ multilayers whose growth was optimized to maximize the perpendicular anisotropy. Structural characterizations indicate abrupt interfaces essentially free of intermixing despite the miscible character of Co and Pt. Gilbert damping parameters as low as 0.021 can be obtained despite a magneto-crystalline anisotropy as large as $10^6~\textrm{J/m}^3$. The inhomogeneous broadening accounts for part of the ferromagnetic resonance linewidth, indicating some structural disorder leading to a equivalent 20 mT of inhomogenity of the effective field. The unexpectedly relatively low damping factor indicates that the presence of the Pt heavy metal within the multilayer may not be detrimental to the damping provided that intermixing is avoided at the Co/Pt interfaces.

Gilbert damping of high anisotropy Co/Pt multilayers

1807.04977v1

Lyapunov functions for linear damped wave equations in one-dimensional space with dynamic boundary conditions. Yacine Chitoura, Hoai-Minh Nguyenb, Christophe Romanc aLaboratoire des signaux et syst` emes, Universit´ e Paris Saclay, Centralesupelec CNRS, Gif-sur-Yvette, France. bLaboratoire Jacques Louis Lions, Sorbonne Universit´ e, Paris, France. cLaboratoire informatique et syst` eme, Aix-Marseille Universit´ e, Marseille, France. Abstract This paper considers a one-dimensional wave equation on [0 ,1], with dynamic boundary conditions of second order at x= 0 andx= 1, also referred to as Wentzell/Ventzel boundary conditions in the literature. In additions the wave is subjected to constant disturbance in the domain and at the boundary. This model is inspired by a real experiment. By the means of a proportional integral control, the regulation with exponential converge rate is obtained when the damping coefficient is a nowhere-vanishing function of space. The analysis is based on the determination of appropriate Lyapunov functions and some further analysis on an associated error system. The latter is proven to be exponentially stable towards an attractor. Numerical simulations on the output regulation problem and additional results on related wave equations are also provided. Key words: one-dimensional wave equation, Wentzel boundary conditions, regulation, output feedback control. The wave equation is one of the classical partial differen- tial equations. The actual reason is that the wave equa- tion is the continuous pendant of Newton’s second law of motion, i.e., where momentum is equal to the sum of the forces. As a consequence, it is also linked with the Euler-Lagrange framework, and therefore with the prin- ciple of least action. For stationary systems, the energy is conserved and the action (or Lagrangian) is station- ary. Other physical phenomena are therefore associated with the wave equation such that electromagnetic law, and quantum phenomena with the Klein-Gordon equa- tion. In the control community, the wave equation has been mainly used for the modelization, estimation, and con- trol of mechanical vibration and deformation phenom- ena. The regulation and control problem applied on the one-dimensional wave equation with dynamic bound- ary condition has attracted the attention of many re- searchers in the control community: crane regulation [8], [10], [14], and [6], hanging cable immersed in water [5], Email addresses: yacine.chitour@l2s.centralesupelec.fr (Yacine Chitour), hoai-minh.nguyen@sorbonne-universite.fr (Hoai-Minh Nguyen), christophe.roman@lis-lab.fr (Christophe Roman).drilling torsional vibrations [38], [45] ,[1], [48], piezoelec- tric control [24], and flexible structure [18]. There are nowadays two main classes of issues : on the one hand, longitudinal variation with for example overhead crane and underwater cable, and, on the other hand, torsional variation with drilling string dynamics. The difference is on the control objective: one aims at controlling the position in the first case, and instead the velocity in the second case. The behavior of the wave equation is strongly related to its boundary conditions. In the case of classical bound- ary condition (i.e., Dirichlet, Neumann, Robin) that is- sue is well understood in the linear case and without high-order terms. Particular terms at one boundary can compensate for anti-damping terms at other boundaries and even in the domain, for example, see [41], [40] and [35]. Moreover, there are cases where even if the energy of the one-dimensional linear wave equation decreases along trajectories, it still does not decay exponentially [23, Section 4]. The wave equation under consideration is subject to two dynamic boundary conditions. This model results from an identification problem associated with a laboratory experiment [36]. Preprint submitted to Automatica 6 February 2024arXiv:2305.01969v2 [math.AP] 5 Feb 20241 Problem statement. The considered system is defined for t⩾0 and for x∈ (0,1), by vtt(t, x) = (a(x)vx)x(t, x)−q(x)vt(t, x) +f(x),(1a) vtt(t,1) =−β1vx(t,1)−νvt(t,1) +U(t) +fc,(1b) vtt(t,0) = µ1vx(t,0)−γ1vt(t,0) +fac, (1c) v(0,·) =v0, v t(0,·) =v1. (1d) Here U(t) is the control input and we assume that (h1) the function a: [0,1]→R∗ +is in W1,∞(0,1) and that there exist a,a >0 such that a≤a(·)≤aa.e. on [0,1]. This function is associated with the mass and elasticity of the wave and it is also linked with the velocity. (h2) The function q: [0,1]→R∗ +, describing the in- domain damping is in L∞(0,1) and satisfies q≤ q(·)≤qa.e. on [0 ,1] for some q,q >0. (h3) The constants β1,γ1, µ1are positive real numbers, andνis real. (h4) The source terms f(·) is in L∞(0,1), and the real constants fc, facare unknown and therefore they cannot be used in the computation of the control lawU(t). The regularity of a(·) stated in ( h1) follows by classi- cal arguments. In detail, for the computation, we need a(·)vx(t,·) to be in H1(0,1). To be more precise every- thing will be the same as in the constant parameters case if a(·)vx(t,·) and vx(t,·) have the same regular- ity. To get strong solutions for (1), one needs to have that v∈H2(0,1). Next, it can be easily shown that ifa∈W1,∞and, v∈H2(0,1) then a(·)vx(t,·) is in H1(0,1). Note that this is just a sufficient condition for the regularity. We refer the reader to [43, Chapter 21] for more details about the regularity of a. In the sequel, we also need qvt(t,·)2need to be integrable, this means q∈L∞(0,1). For fwe actually only need it to be inte- grable, it holds nonetheless L∞(0,1)⊂L1(0,1). The objective of the paper is to regulate vt(t,·) to the constant reference value vref 1, by means of a proportional integral (PI) control law using the measurement of the velocity collocated with the actuation, vt(1, t), in other words, the control U(t) can take the form U(t) :=−k(vt(t,1)−vref 1)−kiZt 0(vt(s,1)−vref 1)ds,(2) where the constants k, kihave to be chosen. This can be equivalently written as U(t) =−k(vt(t,1)−vref 1)−kiηv(t), (3a) ˙ηv(t) =vt(t,1)−vref 1, η v(0) = 0 . (3b)In the literature boundary conditions of the type (1b)- (1c) can be recast as Wentzell’s boundary conditions [15]. It involves a modification of the usual state space which in our case requires the addition of two finite- dimensional state variables, in a similar way as in [39], [25], [10], [14] and [6]. When the wave equation is more than a one-dimensional, the reader is referred to [15] and [4] and references therein. This type of control problem lies in robust output reg- ulation. There has been an effort to extend the result and method from linear finite dimensional systems, to infinite dimensional systems. We refer the reader to [29], [27], [28], more recently [47] and reference within all of them. These papers establish general results for exam- ple [47] deals with non-linear systems. However, they are mostly based on either passivity, strong monoticity or exponential decay properties. These properties often remain to be proven as it is the case of the present pa- per. In [44], the authors establish general result on the PI control of infinite dimensional systems with the as- sumption beforehand on the exponential stability of the zero input system. The impact of the in-domain damping q(x) can be an issue for the decay rate, as we can have some overdamp- ing phenomenon. Intuitively, the damping should help the decay rate of the system. But as one can see in [17] where a semi-linear wave equation is considered, the de- cay rate of the non-damped system is finite time, the addition of the damping degrade this performance to an exponential decay rate. Note that for the present case the in-domain damping is mandatory for the proof. There are specific configurations of (1) that can be solved using more intricate and general control law designs, especially those tailored to address partial differential equations coupled with ordinary differential equations at the boundary. If disturbances are not considered, [11] and its extension [12] can be employed. Additionally, assuming that a(·) is constant allows the use of [37], [50], or [49]. These five papers primarily employ an infinite- dimensional backstepping approach, a development closely associated with the influential work of Miroslav Krstic [21]. Given that the wave equation can be ex- pressed as a coupled heterodirectional hyperbolic par- tial differential equation (PDE), the main strategy in the aforementioned papers involves using backstepping transformations to decouple or cascade the PDE. This transforms the closed-loop system into a target sys- tem, the stability of which is easier to analyze. Notably, the uniqueness of the present paper lies in achieving exponential stability without the need for decoupling, thereby establishing new potential target systems for backstepping based design. The closest approach associated with the present paper 2is [45] where the velocity regulation with a PI is consid- ered. However, the controlled boundary condition con- sidered in [45] is not a second order dynamic one, and thus is different from the one considered in this paper. Nevertheless, the boundary condition considered in [45] implies the exponential stability even with small viscous anti-damping at the boundary opposite to the actuation. In the case under consideration, only viscous damping at the opposite boundary is considered, and exponen- tial stability is achieved. In [6] the wave equation is sub- ject to two dynamic boundary conditions. The authors establish asymptotic stability for the position stabiliza- tion and that the decay rate is not exponential, and no viscous terms are considered for the zero input system. In [25], for the same model (as [6]) the exponential sta- bility toward the origin is obtained but the control law needs the knowledge of vxt(1, t). This can be related to the finite dimensional backstepping done in [14]. Studies have been conducted concerning the potential absence of exponential stabilization for wave-like equations, as evi- denced by works such as [26], [31], and references therein. In a broader context, investigations into this issue ex- tend to more general setups, as seen in [16], [46], and related references. PI controllers have been successfully and recently used in order to regulate linear and non-linear PDE, see [7], [22]. An identification procedure has been presented in [36] for the system (1) without source terms on exper- imental data. This means that the considered problem can be associated with an experimental setup. A first study has been made on this system in [33] using clas- sical form a Lyapunov functional but it failed to prove the exponential stability. Only asymptotic stability was established, by using the LaSalle invariance principle. This paper provides a new term in the Lyapunov func- tional and an associated methodology, for the present setup. The proof of the exponential stability is given in Section 3. In Section 4, this proof is compared with exist- ing results. Next the proof of the robustness of the con- trolled system is given in Section 5. Then in Section 6 we study, using the same approach, simpler cases where one boundary condition is a Dirichlet one and this al- lows us to establish the exponential stability of the zero input system in the undisturbed case. The last part of the paper deals with numerical simulations. The numer- ical scheme is not derived from the usual approxima- tion of space and time derivatives. We used the fact that the wave equation can be derived from the Lagrangian and the least action principle to approximate the sys- tem space energy by a finite dimensional continuous time Euler-Lagrange equation. The finite dimensional contin- uous time system is then numerically solved by using symplectic integrators. This suggested numerical scheme is new up to the authors’ knowledge and provides an interesting alternative compares to more standard dis- cretization schemes.Notations: IfIis an interval of real numbers, L2(I;R) denotes (the class of equivalence of) square-integrable functions from ItoR. Moreover L2([0,1];R) is abusively denoted L2(0,1). Furthermore Hndenotes the Sobolev space Wn,2, i.e., u∈H1⇔u∈L2, u′∈L2, (4) in which u′denotes the derivative of u. 2 Main result To achieve our objective, we perform a change of variable in order to obtain an error variable u(·,·) and to prove exponential decay of its partial derivatives. The error variable u(·,·) is defined as follows, for every (x, t)∈[0,1]×[0,∞) u(t, x) :=v(t, x)−tvref 1 +Zx 01 a(s)Zs 0[−vref 1q(χ) +f(χ)]dχds +a(0) µ1[−γ1vref 1+fac]Zx 01 a(s)ds, (5) η2(t) :=ηv(t)−β1 kia(1)Z1 0[−vref 1q(s) +f(s)]ds −β1a(0) kiµ1a(1)[−γ1vref 1+fac] +νvref 1−fc ki.(6) Note that, for every ( x, t)∈[0,1]×[0,∞), u(t, x) =v(t, x)−tvref 1+F(x), (7) ut(t, x) =vt(t, x)−vref 1, (8) where we have gathered all the uncertainties in the func- tionF(·) and it is immediate to deduce from (8) that proving exponential decay of ut(in an appropriate sense) is equivalent to prove it for vt−vref 1and hence to achieve the desired control objective. From now, we will therefore focus on the error variable u(·,·). Direct computations yield that it is the solution of the following system: utt−(a(x)ux)x=−q(x)ut, (t, x)∈R+×(0,1), (9a) ut(t,1) = η1(t) (9b) ut(t,0) = ξ1(t) (9c) ˙η1(t) =−α1η1(t)−α2η2(t)−β1ux(t,1),(9d) ˙η2(t) =η1(t), (9e) ˙ξ1(t) =−γ1ξ1(t) +µ1ux(t,0), (9f) u(0,·) =u0, u t(0,·) =u1on (0 ,1), (9g) η1(0) = η0, η 2(0) = η2,0ξ1(0) = ξ0. (9h) 3where α1:=k+νandα2:=ki, and kis chosen such thatα1is positive. Consider the following Hilbert spaces Xw: =H1((0,1);R)×L2((0,1);R)×R3,(10) Xs: =H2((0,1);R)×H1((0,1);R)×R3.(11) The wave equation is associated with the following ab- stract problem ˙X(t) +AX(t) = 0 , (12a) X(0) =X0∈Dom(A)⊂Xs⊂Xw, (12b) in which ∀z∈Dom(A),Az:= −z2 −(az′ 1)′+qz2 α1z3+α2z4+β1z′ 1(1) −z3 γ1z5−µ1z′ 1(0) ,(13) and Dom(A) :={z∈Xs;z2(1) = z3, z2(0) = z5}.(14) Our well-posed result goes as follows. Theorem 1 Considering assumption (h1)and(h2), the abstract problem (12) is well-posed. In order words for any initial data X0∈Dom(A), there exists a unique solution to the abstract problem (12), such that for any t≥0,X(t)∈Dom(A)⊂Xsand X ∈C0([0,∞);Dom(A))∩C1([0,∞);Xw),(15) Xwis the state space of weak solutions and the Hilbert space considered and is defined in (10).Xsis the state space of strong solutions and is defined in (11). In addition, for all initial data X0∈Xw, there exists a weak solution X(t)∈Xwto the abstract problem (12) given by X(t) =S(t)X0, (16) in which Sis the C0-semigroup generated by the un- bounded operator A. Moreover, it holds X ∈C0([0,∞);Xw). (17) The proof is based on finding a transformation such that the abstract problem is associated with a linear maxi- mal monotone operator. Then the conclusion is drawnby using the Hille-Yosida theorem. The part on weak solutions holds true from the fact that Dom( A) is dense inXw, and therefore S(t) defined a strongly continu- ous map from XwtoXw. Details are provided in Ap- pendix A. The state is X(t) := [ u(t,·), ut(t,·), η1(t), η2(t), ξ1(t)]∈Dom(A)⊂Xs.(18) We define the energy Euof a solution of (9) as ∀t≥0 Eu(t) :=1 2Z1 0(ut(t, x)2+a(x)ux(t, x)2)dx. (19) Note that this energy is invariant by translations with constants, i.e., Eu=Evifu−vis a constant func- tion. Moreover, the absolutely continuous function u(·,1)−η2(·) is constant along a trajectory of (9) and equal to u∗where u∗:=u0(1)−η2(0). (20) Our objective is to establish the exponential stability of the trajectory with respect to the following attractor S:={z∈Xw, z1(·) =d, d∈R, z2(·) = 0 , z3= 0, z4= 0, z5= 0}. (21) This attractor is the kernel of the following functional Γ(X(t)) :=Z1 0[u2 t(t, x) +u2 x(t, x)]dx +η2 1(t) +η2 2(t) +ξ2 1(t), (22) indeed it holds Γ(z) = 0⇔z∈S. (23) We establish the following result. Theorem 2 Consider the 1D wave equation (9)with the assumptions (h1),(h2),(h3), and with α2, α1>0. Then, there exist a positive constant ρ, and a positive constant Msuch that, for every weak solution X, it holds, Γ(X(t))⩽MΓ(X(0))e−ρt. (24) and the system is exponentially stable towards the attrac- torS. In addition it holds that max x∈[0,1]|u(t, x)−u∗|tends exponentially to zero as ttends to infinity, with a decay rate larger than or equal to ρ. 4Theorem 3 Under assumption (h1)-(h2), and for any ki=α2>0andβ1, µ1>0, the conclusion on Theorem 2 still holds if α1>−β1κ2 2a(1)+κ1κ2, (25) γ1>−µ1κ2 2a(0)+κ1κ2, (26) where κ1:=1 a 3a+∥ax∥L∞ 2+q 2
, (27) κ2:=2 q(1 +q 2+κ1). (28) Remark 1 The link between α1andνis defined right below (9). This theorem means in particular that (24) holds in a robust way and the regulation can even admit small anti-damping at the uncontrolled boundary for cer- tain value of µ1,a(·), and q(·). 3 Proof of Theorem 2 This proof follows a standard strategy: the result is first established for strong solutions by the determination of Lyapunov functions verifying an appropriate differential inequality, and then it is extended to weak solutions by a classical density argument. Hence, in the sequel, solu- tions of (9) are all assumed to be strong. We start with the time derivative of Eualong a strong solution. It holds for t≥0 ˙Eu=−Z1 0qu2 tdx+a(1)η1(t)ux(t,1) −a(0)ξ1(t)ux(t,0). (29) One also has, for t≥0, after using (9d) and (9e) a(1)η1(t)ux(t,1) =−a(1) β1η1(t) ˙η1(t) +α1η1(t) +α2η2(t) =−d dta(1) 2β1(η2 1(t) +α2η2 2(t)) −a(1)α1 β1η2 1(t). (30) Similarly, one also has, for t≥0, after using (9f) −a(0)ξ1(t)ux(t,0) =−a(0) µ1ξ1(t) ˙ξ1(t) +γ1ξ1(t) =−d dta(0) 2µ1ξ2 1(t) −a(0)γ1 µ1ξ2 1(t).(31)Define for t≥0 F(X(t)) := Eu(t) +a(1) 2β1η2 1(t) +a(0) 2µ1ξ2 1(t).(32) Then, by gathering (29), (30) and (31), one deduces that, fort≥0, d dt F+a(1)α2 2β1η2 2(t) =−Z1 0qu2 tdx −a(1)α1 β1η2 1(t)−a(0)γ1 µ1ξ2 1(t). (33) To conclude on the exponential stability we also need a negative term in u2 xandη2 2. We next consider an extra term which will be added in the candidate Lyapunov function in the sequel. From (20) it holds that η2(t) =u(t,1)−u∗, t≥0. (34) Set ξ2(t) :=u(t,0)−u∗, t≥0. (35) One has, for t≥0, that d dtZ1 0(u−u∗)utdx =Z1 0u2 t+Z1 0(u−u∗)utt =Z1 0u2 t+Z1 0(u−u∗)(aux)x−Z1 0q(u−u∗)ut =Z1 0u2 t−Z1 0au2 x−d dtZ1 0q 2(u−u∗)2dx +a(1)η2ux(t,1)−a(0)ξ2ux(t,0). (36) Using (9d) and (9f), one deduces after computations sim- ilar to those performed to get (30) and (31), that, for t≥0, η2ux(t,1) =−α2η2 2(t) +η2 1(t) β1 −d dtα1 2η2 2(t) +η1(t)η2(t) β1 , (37) −ξ2ux(t,0) =ξ2 1(t) µ1−d dtγ1 2ξ2 2(t) +ξ2(t)ξ1(t) µ1 .(38) We next define for t≥0 W(X(t)) =Z1 0(u−u∗)utdx+Z1 0q 2(u−u∗)2dx +a(1) β1α1 2η2 2(t) +η2(t)η1(t) +a(0) µ1γ1 2ξ2 2(t) +ξ2(t)ξ1(t) . (39) 5Gathering (36), (37) and (38), it holds for t≥0 ˙W=−Z1 0au2 x−a(1)α2 β1η2 2(t) +Z1 0u2 t +a(1) β1η2 1(t) +a(0) µ1ξ2 1(t). (40) We finally define the candidate Lyapunov function V used for proving Theorem 2, which is positive definite for some constant ℓsuch thatp2q> ℓ > 0 by V(X(t)) =F(X(t)) +a(1)α2 2β1η2 2(t) +ℓW(X(t)),≥0. (41) Putting together (32) and (39), it holds for t≥0, V(X(t)) =Eu(t) +ℓZ1 0 (u−u∗)ut+q 2(u−u∗)2 dx +a(1) 2β1 η2 1+α2η2 2+ℓ(2η2η1+α1η2 2) +a(0) 2µ1 ξ2 1+ℓ(2ξ2ξ1+γ1ξ2 2) . (42) and similarly, putting together (33) and (40), it holds fort≥0, ˙V=−Z1 0(q 2−ℓ)u2 tdx−ℓZ1 0au2 xdx −a(1) β1 (α1−ℓ)η2 1+α2ℓη2 2 −a(0)(γ1−ℓ) µ1ξ2 1. (43) The purpose of Vdefined in (42) compared with Fis to make negative terms in u2 xandη2 2appear. Next we compare the functional Vto the functional Γ defined in (22). Proposition 1 With the notations above, and Γdefined in(22), there exist ℓ > 0and two positive constants c, C, ρ > 0such that for every strong solution X(t)of (9), one gets, for t≥0, cΓ(X(t))≤V(X(t))≤CΓ(X(t)), (44) ˙V(X(t))≤ −CρΓ(X(t)). (45) Remark 2 Using α1andα2as tuning parameters one can show that a necessary condition for ˙V(X(t))⩽−CρΓ(X(t)). (46)is that Cρ < min{aq 4, aq 2q,aa(0)γ1 aµ1+a(0)}. (47) This upper bound is deduced from the next inequalities extracted from (43)and the condition for Vto be definite positive. q 2q> ℓ (48) q 2−ℓ > Cρ (49) ℓa > Cρ (50) a(0)(γ1−ℓ) µ1> Cρ (51) Moreover as candCdoes not depend on q. It holds for the decay rate ρ ρ−→ q→00. (52) The suggested approach allows us only to conclude for stability when q= 0, and in this case we can stop at (33). Nevertheless following [33] or [6] we could use LaSalle’s invariance principle to establish asymptotic stability. If in addition α2= 0, in the case of no integrator the system falls as a one-dimensional particular case of [4, Theorem 1.2], and therefore the decay rate is at least logarithmic. PROOF. Using (20) and (32) one can observe that for every t≥0 and x∈[0,1] it holds |u(t, x)−u∗|2≤2|u(t, x)−u(t,1)|2+ 2η2 2(t) ≤2Z1 0u2 x(t, x)dx+ 2η2 2(t) ≤4 aEu(t) + 2η2 2(t), (53) As an immediate consequence, one gets that, for t≥0, Z1 0(u−u∗)2dx≤4 aEu(t) + 2η2 2(t), (54) ξ2 2(t)≤4 aEu(t) + 2η2 2(t). (55) The proof of (44) relies now on the combination of (42), (54) and (55), several completions of squares and the Cauchy-Schwartz inequality. As for the argument of (45), it is obtained similarly by using (43), (54) and (55), Relying on Proposition 1, we complete the proof of The- orem 2. 6From (44) and (45), it follows that ˙V≤ −ρVhence yielding exponential decrease of Vat the rate ρand the similar conclusion holds for Γ, thanks to (45). All items of Theorem 2 are proven after using (53). 4 Discussion on the proof of the Theorem 2 There exist cases where the linear one-dimensional wave does not decay exponentially. For example, the solution uof the system utt(t, x) =uxx(t, x),∈R+×(0,1), (56a) u(t,0) = 0 , (56b) utt(t,1) =−ux(t,1)−ut(t,1), (56c) does not decrease exponentially towards the origin, see [23, Section 4]. It follows a t−1sharp decay rate. The addition/suppression of one term can make the decay rate drastically different, for example utt(t, x) = (aux)x(t, x),∈R+×(0,1),(57a) utt(t,0) = ux(t,0), (57b) utt(t,1) =−ux(t,1)−ut(t,1)−u(t,1) utt(t,1) =−uxt(t,1), (57c) is exponentially stable [25], whereas utt(t, x) = (aux)x(t, x),∈R+×(0,1),(58a) utt(t,0) = ux(t,0), (58b) utt(t,1) =−ux(t,1)−ut(t,1)−u(t,1),(58c) is not exponentially stable, see [6]. However the solution of (57) need to be more regular, see [25]. The energy of the following two systems utt(t, x) =uxx(t, x),∈R+×(0,1), (59a) ux(t,0) = ut(t,0), (59b) ux(t,1) =−ut(t,1), (59c) and utt(t, x) =uxx(t, x),∈R+×(0,1), (60a) ux(t,0) = ut(t,0), (60b) utt(t,1) =−ux(t,1)−ut(t,1), (60c) are exponentially decreasing [34]. Typically, for both pre- vious cases, the exponential decrease and stability can be obtained via Energy/Lyapunov approach using cross terms in the following form. Z1 0(1 +x)utuxdx, (61)which can make negative term as u2 xandu2 tappear for the Energy/Lyapunov functional derivative. This pervi- ous term implies boundary terms in the following form  u2 x+u2 t1 x=0, (62) in the case of (59b)-(59c) or (60b)-(60c) we can manage to handle this term. However, this is problematic when considering both boundary conditions as (9d) and (9f). Indeed, even when α2= 0, we do not arrive to cope with the term u2 xboth in 0 and 1. This incapacity to handle the term u2 xwith both dynamic boundary conditions is properly shown in [33] with more general form of uxut cross terms, and considering a large family of reforma- tion as hyperbolic PDE for example. In particular the term (62) can be also taken care of if we have damped position terms ( u) on the domain. Indeed in this case, this term enable us to use cross terms like Z1 0uut, (63) The exponential stability of the linear wave equation at the origin with both dynamic boundary condition and damped in velocity and position everywhere is estab- lished in [32, Chapter 9]. We stress that the paper deals with velocity regulation which has been transformed to velocity exponential stability. The term (63) is close to the one we suggest Z1 0(u−u∗)ut, (64) This mostly corresponds to the beforehand knowledge of the limit value of ufor the system. This can be made because the integrator part of the system captures the distance between the state and the attractor. In our case this term can be added because qis strictly positive, see (36). 5 Proof on Theorem 3 We start from the proof of Theorem 2, in (43), then we compute the derivative of the following cross term, using integration by parts d dtZ1 0(1−2x)uxutdx =−a(1)u2 x(t,1) +a(0)u2 x(t,0) 2 +Z1 0(2a+(a(1−2x))′ 2)u2 x−η2 1(t) +ξ2 1(t) 2dx +Z1 0u2 tdx−Z1 0(1−2x)quxutdx. (65) 7The above cross term can be used to make negative terms inη2 1andξ2 1appear at the cost of positive terms in u2 t andu2 x. Consider that |ℓ2|<√athen Gu=Vu+ℓ2Z1 0(1−2x)uxutdx, (66) is positive. Gathering (43) and (65), and using the Young’s inequal- ity, the derivative of Gualong the trajectory is ˙Gu≤ −Z1 0(q 2−ℓ−ℓ2−ℓ2q 2)u2 tdx −Z1 0(ℓa−ℓ2(2a+(a(1−2x))′ 2)−ℓ2q 2)u2 xdx −a(1) β1(α1−ℓ) +ℓ2 2 η2 1−(a(0)(γ1−ℓ) µ1+ℓ2 2)ξ2 1 −a(1) β1α2ℓη2 2 (67) The exponential stability still holds if the following in- equalities hold q 2−ℓ−ℓ2(1 +q 2)>0, (68) ℓa−ℓ2(3a+a′ 2+q 2)>0, (69) 2a(1)(α1−ℓ) +β1ℓ2>0, (70) 2a(0)(γ1−ℓ) +µ1ℓ2>0. (71) A sufficient condition for the four previous inequalities to hold is ℓ2< κ2, (72) ℓ > κ 1κ2, (73) 2a(1)(α1−κ1κ2) +β1κ2>0, (74) 2a(0)(γ1−κ1κ2) +µ1κ2>0. (75) where κ1andκ2are defined in (27)-(28). This concludes the proof. 6 Exponential stability for the zero input sys- tem with no disturbance. In the following we investigate and establish results on associated problems. We first start with a wave equa- tion subject to a Dirichlet’s boundary conditions and a 2nd order dynamic boundary condition. The second sys- tem we add an integral action to the dynamics boundary condition. The third and last system consist of a wave equation with both 2nd order dynamics boundary con- ditions, and correspond to the zero input system with no disturbance.Proposition 2 Consider the following 1D wave equa- tion utt−(aux)x=−qut,(t, x)∈R+×(0,1),(76a) ut(t,1) = η1(t), (76b) ˙η1(t) =−α1η1(t)−β1ux(t,1), (76c) u(t,0) = 0 , t≥0, (76d) u(0,·) =u0, u t(0,·) =u1,on(0,1), (76e) η1(0) = η0. (76f) where a(·),q(·)are respecting (h1)-(h2), and with α1and β1are strictly positive. The state of this system is X2(t) =[u(t,·), ut(t,·), η1(t)]∈Dom(A2), (77) where A2is the unbounded operator associated with (76). The domain is defined as Dom(A2) ={z∈X2,s, z1(0) = 0 , z2(1) = z3},(78) where X2,sis the space of strong solutions, and X2,wis the space of weak solutions defined as X2,s=H2×H1×R, (79) X2,w=H1×L2×R. (80) Finally, consider Γ2(X2(t)) =Z1 0(u2 t(t, x) +u2 x(t, x))dx +η2 1(t). (81) Then, there exist a positive constant ρand a positive constant Msuch that for every weak solution X2, it holds Γ2(X2(t))⩽MΓ2(X2(0))e−ρt. (82) And the system is exponentially stable towards the origin ofX2,w. In addition, it holds that max x∈[0,1]|u(t, x)|tends expo- nentially to zero as ttends to infinity, with a decay rate larger than or equal to ρ. The following system is when we consider an integral part at the dynamic boundary for (76). Proposition 3 Consider the following 1D wave equa- 8tion, utt−(aux)x=−qut,(t, x)∈R+×(0,1),(83a) ut(t,1) = η1(t), (83b) ˙η1(t) =−α1η1(t)−α2η2(t)−β1ux(t,1), (83c) ˙η2(t) =η1(t), (83d) u(t,0) = 0 , t≥0, (83e) u(0,·) =u0, u t(0,·) =u1,on(0,1), (83f) η1(0) = η0, η 2(0) = η2,0. (83g) where a(·),q(·)are respecting (h1)-(h2), and with α1,α2 andβ1are strictly positive. Forx∈[0,1], define v(x) :=C2Zx 0ds a(s), (84) C2:=a(1)α2 a(1)α2R1 0ds a(s)+β1u∗(1). (85) The state of this system is X3(t) = [u(t,·), ut(t,·), η1(t), η2(t)]∈Dom(A3), (86) where A3is the unbounded operator associated with (83). The domain is defined as Dom(A3) ={z∈X3,s, z1(0) = 0 , z2(1) = z3},(87) where X3,sis the space of strong solutions, and X3,wis the space of weak solutions defined as X3,s=H2×H1×R2, (88) X3,w=H1×L2×R2. (89) Finally, consider Γ3(X2(t)) =Z1 0((u(t, x)−v(x))2+u2 t(t, x) +u2 x(t, x))dx +η2 1(t) + (η2(t)−β1C2 a(1)α2)2.(90) Then, there exists a positive constant ρand a positive constant Msuch that for every weak solution X2, it holds Γ3(X3(t))⩽MΓ3(X3(0))e−ρt. (91) And the system is exponentially stable towards the at- tractor defined as ker (Γ3(·)). In additions, it holds that max x∈[0,1]|u(t, x)−v(x)|tends exponentially to zero as ttends to infinity, with a decay rate larger than or equal to ρ.Now we consider the case where α2= 0 in (9). This system has been studied in a more general and multidi- mensional setup in [4], the author establishes with lessen hypothesis logarithmic decay rates. Proposition 4 Consider the following 1D wave equa- tion, utt−(aux)x=−qut,(t, x)∈R+×(0,1),(92a) ut(t,1) = η1(t), (92b) ut(t,0) = ξ1(t), (92c) ˙η1(t) =−α1η1(t)−β1ux(t,1), (92d) ˙ξ1(t) =−γ1ξ1(t) +µ1ux(t,0), (92e) u(0,·) =u0, u t(0,·) =u1on(0,1), (92f) η1(0) = η0, ξ 1(0) = ξ0. (92g) where a(·),q(·)are respecting (h1)-(h2), and with α1, β1, γ1andµ1are positive. The state of this system is X4(t) =[u(t,·), ut(t,·), η1(t), ξ1(t)]∈Dom(A4),(93) where A4is the unbounded operator associated with (92). The domain is defined as Dom(A4) ={z∈X3,s; z2(0) = z4, z2(1) = z3}, (94) where X3,sis the space of strong solutions, and X3,w is the space of weak solutions, both defined in (88)-(89) Finally, consider Γ4(X4(t)) =Z1 0(u2 t(t, x) +u2 x(t, x))dx +η2 1(t) +ξ2 1(t). (95) Then, there exists a positive constant ρand a positive constant Msuch that, for every weak solution X4, it holds Γ4(X4(t))⩽MΓ4(X4(0))e−ρt. (96) And the system is exponentially stable towards the at- tractor S4defined by S4={z∈X3,w, z1(·) =d, d∈R, z2(·) = 0 , z3= 0, z4= 0}. (97) which is the kernel of Γ4(·). In addition, there exists u∗so that max x∈[0,1]|u(t, x)−u∗| tends exponentially to zero as ttends to infinity. PROOF. We start by proving Proposition 2. As before, the ar- gument is based on an appropriate Lyapunov function 9¯Vu=¯Fu+ℓ¯Wuwhere ℓis a positive constant to be cho- sen and ¯Fu(t) :=1 2Z1 0(u2 t+au2 x)dx+a(1) 2β1η2 1, (98) ¯Wu(t) :=Z1 0uutdx+1 2Z1 0qu2dx +a(1) β1η1u(t,1)+a(1)α1 2β1u(t,1)2. (99) One gets, using integration by parts, (76c), and (76d) ˙¯Fu(t) :=−Z1 0(qu2 t)dx−a(1)α1 β1η2 1, (100) ˙¯Wu(t) :=Z1 0u2 tdx−Z1 0au2 xdx+a(1) β1η2 1.(101) Therefore ˙¯Vu=−Z1 0(q−ℓ)u2 tdx−ℓZ1 0au2 xdx −a(1) β1(α1−ℓ)η2 1. (102) The conclusion follows by taking ℓ >0 small enough and noting that, thanks to the Dirichlet boundary condition (83e), for every t≥0 and x∈[0,1] |u(t, x)|2=|u(t, x)−u(t,0)|2 ≤Z1 0u2 x(t, x)dx≤2Eu(t)≤2¯Fu(t).(103) One proceeds by establishing an analog to Proposition 1 where Γ and Vare replaced by Γ 2and¯Vuin order first to obtain that˙¯Vu≤ −ρ¯Vufor some positive constant ρ independent of the state and finally to conclude as in the final part of the argument of Theorem 2. We next turn to the proof of Proposition 3. Using the notations of the proposition, we set w(t, x) : = u(t, x)−v(x), t≥0, x∈[0,1], ¯η2(t) : = η2(t) +β1C2 a(1)α2, t≥0. (104) It is a matter of elementary computations to check that wis the solution of (83) with different and corresponding initial conditions with the Dirichlet boundary condition atx= 0 (since v(0) = 0) and the boundary condition given by ˙η1(t) =−α1η1(t)−α2¯η2(t)−β1wx(t,1), (105) ˙¯η2(t) =η1(t). (106)It holds w∗(1) := w(t,1)−¯η2(t) =u∗(1)−v(1)−β1C2 a(1)α2 =u∗(1)−C2(Z1 0ds a(s)+β1C2 a(1)α2) = 0 .(107) We have essentially reduced the problem to only deal with solutions of (92) with the Dirichlet boundary con- dition at x= 0, with the additional constraint that w∗(1) = 0. In that case, we consider the candidate Lya- punov function ˜Vw=˜Fw+ℓ˜Wwwhere ℓis a positive constant to be chosen and ˜Fw(t) : =1 2Z1 0(w2 t+aw2 x)dx +a(1) 2β1(η2 1+α2¯η2 2), (108) ˜Ww(t) :=Z1 0wwtdx+1 2Z1 0qw2dx +a(1) β1α1 2¯η2 2+η1¯η2 . (109) One gets ˙˜Vu=−Z1 0(q−ℓ)w2 tdx−ℓZ1 0aw2 xdx −a(1) β1(α1−ℓ)η2 1−ℓa(1)α2 β1¯η2 2, (110) where we have repeatedly used the equality w(t,1) = ¯η2(t). By following what has been done previously, the conclusion follows. We finally prove Proposition 4. As before the argument is based on an appropriate Lyapunov function ¯Vudefined later. We first consider Fgiven in (32) and note that for t≥0 it holds ˙F=−Z1 0qu2 tdx−a(1)α1 β1η2 1(t)−a(0)γ1 µ1ξ2 1(t).(111) We next compute along solutions of (92) the following time derivative d dtZ1 0 u(t, x)−u(t,1) ut(t, x)dx = +Z1 0u2 tdx−Z1 0q u(t, x)−u(t,1)
ut(t, x)dx + u(t,1)−u(t,0) a(0)ux(t,0) −Z1 0au2 xdx−η1Z1 0ut(t, x)dx. (112) 10In the above equation, we use (92e) to get rid of ux(t,0) and, to obtain for t≥0 that  u(t,1)−u(t,0) ux(t,0) = + u(t,1)−u(t,0)˙ξ1+γ1ξ1 µ1 =d dt u(t,1)−u(t,0)ξ1 µ1 −(η1−ξ1)ξ1 µ1+γ1ξ1 µ1 u(t,1)−u(t,0) . (113) Setting for t≥0 Gu(t) : =Z1 0 u(t, x)−u(t,1) ut(t, x)dx −a(0)ξ1 µ1 u(t,1)−u(t,0) , (114) we deduce from the above that along with solutions of (92) that ˙Gu=Z1 0u2 tdx−Z1 0au2 xdx−a(0)ξ1 µ1(η1−ξ1) +a(0)γ1ξ1 µ1 u(t,1)−u(t,0) −η1Z1 0ut(t, x)dx −Z1 0q u(t, x)−u(t,1) utdx. (115) We finally recall that there exists a positive constant Ca(independent of the solutions of (92)) such that, for t≥0, Z1 0|ut|dx+ max x∈[0,1]|u(t, x)−u(t,1)| ≤Z1 0(|ut|+|ux|)dx ≤CaE1/2 u(t). (116) We now choose ¯Vu=F+ℓGuforℓ >0 small enough. Us- ing repeatedly the Cauchy-Schwarz inequality, and (116) in (111) and (115), one gets for εandℓsmall enough that (44) and (45) hold true, from which one deduces Item( i) of Proposition 4. Finally, to get Item( ii) of Proposition 4, first notice that u(t,1) admits a limit u∗asttends to infinity since, for every t, t′>0 it holds u(t,1)−u(t′,1) =Rt t′η1andη1 decreases to zero exponentially. The conclusion follows now by using (116). Remark 3 In the proofs of all our results, one could use the function Gu(especially the integral term) to obtainthe exponential decrease of Euand some of the compo- nents of the Wentzell’s boundary conditions. However, this does not allow one to determine the limit u∗for the solution uin terms of initial conditions. In particular, we are not able to characterize u∗in Proposition 4. Note also that u(t, x)−u(t,1) =−Z1 xux(t, s)ds. (117) This can be related with the means of uxand therefore we have extended our Lyapunov function with a space moving evaluation of the mean of the force/torque. Indeed uxis associated with the torque or the force in mechanical setup. 7 Numerical schemes and simulations. There exist several ways to compute numerical approxi- mation of the solution of evolution problems associated with partial differential equation, [42]. In the case un- der consideration, spectral methods lead to an estima- tion of the base function at each time step due to the dynamics boundary condition. This requires an impor- tant computing power. As we have only one dimension in space finite-element methods reduce to finite differ- ence methods with (possibly unequal) spacial step. Fi- nite different methods can be delicate to design in or- der to ensure at the same time numerical stability and good approximation. Note that there also exist specific schemes based on Riemann invariants [2]. These last schemes have good numerical property, but their exten- sion to dynamic boundary conditions is not obvious. In this paper, we suggest a new approach, which pro- vides numerical scheme stability and therefore achieves structural stability. It is based on the discretization of the Lagrangian associated with the wave equation. This approach leads to a special finite difference scheme. As previously said the wave equation in its stationary form can be associated with a Lagrangian. For the case un- der consideration (1), (in the stationary case where ν= U(t) =fc=γ1=fac= 0), this Lagrangian is given by L(v(t,·)) =Z1 01 2(v2 t(t, x)−a(x)v2 x(t, x))dx +1 2(a(1) β1vt(t,1)2+a(0) µ1vt(t,0)2).(118) Following the strategy in [20] and the least action princi- ple, the dynamics of the system is associated with a sta- tionary action. The action for any time interval is given as I(v) =Ztf tiL(v(t,·))dt. (119) 11A stationary action means that the first variation is equal to zero δI(v, δv) = 0 , (120) where the first variation is defined as δI(v, δv) =δI(v+δv)−δI(v) +O(∥δv∥2).(121) Computation gives the following stationary system vtt−(avx)x= 0,(x, t) inR+×(0,1),(122a) vtt(t,1) =−β1vx(t,1), (122b) vtt(t,0) = µ1vx(t,0). (122c) This is the stationary part of (1), as usual the less ac- tion principle, the dissipation and the input are added afterward to obtain exactly (1). Now consider a discrete version of (118) Ld(vd(t)[·]) =1 2N−1X i=1[ ˙vd(t)[i]2 −ai−1 12(vd(t)[i]−vd(t)[i−1] dxi)2 −ai 3(vd(t)[i+ 1]−vd(t)[i−1] dxi+dxi+1)2 −ai+1 12(vd(t)[i+ 1]−vd(t)[i] dxi+1)2]dxi +1 2aN β1˙vd(t)[N]2+1 2a0 µ1˙vd(t)[0]2.(123) The integral part in v2 xhas approximated using Simp- son’s 1 /3 rule. The derivation of the Euler-Lagrange equation can then be done by a symbolic numerical com- putation. This gives an autonomous stationary linear fi- nite dimensional system: E¨vd(t) =Avd(t), (124) with σ(A)∈iR. It holds E= diagh 1 β1dx1. . . dx N−11 µ1i . (125) Then we add dissipation with a positive symmetric ma- trixR, source term (disturbance and action) and obser- vation, E¨vd(t) =Avd(t)−R˙vd(t) +BU(t) +fet,(126a) y(t) =C˙vd(t). (126b) with fT et=h fcf1f2. . . f aci (127)which represents the disturbance, and with R= diagh ν q1. . . q N−1γ1i . (128) The control U(t) is computed through ˙ηv(t) =y(t)−yref, (129a) U(t) =−kiηv(t)−kp(y(t)−yref). (129b) As the main idea of this discretization scheme is to have a good approximation of the energy, we suggest going on with this idea using symplectic integrator scheme, see [13] and references within. These methods, like the Crank-Nicolson method have the property preserve the energy as time evolves. It is known that for a system which has an eigenvalue in iRexplicit schemes are un- stable, and implicit schemes are exponentially stable see [13]. As our system has structurally the zero eigenvalue, symplectic numerical discretization schemes tend to give better behaviors approximation. The idea of a symplectic scheme is to combine an implicit scheme together with an explicit one. This leads to vd[k+ 1] = vd[k] + ∆ t˙vd[k+ 1], (130) E˙vd[k+ 1] = E˙vd[k] + ∆ t Avd[k]−∆t R˙vd[k+ 1] + ∆t BU[k] + ∆ tf. (131) The second line is implicit, but Rin our case is a diagonal matrix and so the associated inverse matrix is easily computed ˙vd[k+ 1] =(1 + ∆ tE−1R)−1( ˙vd[k] + ∆ t E−1Avd[k] + ∆t E−1BU[k] + ∆ tE−1f). (132) There are several key points to note in this last equation. First, the term (1 + E−1∆tR)−1correspond to a con- traction map in the case where Ris positive, and there- fore is associated with dissipation terms. Second, in the case where Rrepresent anti-dissipation term, there exist discretized steps∆t dxwhere the numerical shame is unde- fined. Third, where R= 0, these equations are two-step explicit ones. The value selected for the numerical simu- lation for the output regulation problem is summarized in Table 1. The Figure 1 illustrates the behavior of the output reg- ulation problem, we observe that boundary velocities of the system goes exponentially towards the constant ref- erence. In Figure 2 the time response of the regulation problem objectives are depicted. The in-domain velocity converges in L2norm towards the reference. The con- trol law associated with these time responses are given in Figure 3. It is not clear how to select the control gain to provide rapidity and robustness. Getting an urge in- tegrator gain in order to have the control go faster to- wards its steady state may cause some heavy oscillation. However, as proven the exponential stability still holds. 12Symbol value Symbol value N 199 fc −1 a(x) sin(2 x) + 2 fac 1 q(x) .01 + .1x2kp 10 f(x) sin(2πx) ki=α2 20 β1 20 vref 1 5 µ1 20 vd[·] 0 ν 1 ˙vd[0 :N] 0 γ1 1 ∆t 0.001 Table 1 Parameter values for the simulation. The time response of the wave equation velocity is drawn as a surface in a 3d perspective in Figure 4. There is first some important oscillation, with traveling wave go- ing back and forth from the boundary, then the oscilla- tion rapidly goes smaller, and finally the velocity goes smoothly towards the reference. The time response of the position is given in Figure 5. The impact of the con- stant disturbance are more visible in this graph. The os- cillations observed in Figure 5 are mainly due to the dis- turbance which needs a particular distribution of the po- sition along the space. Once this particular distribution is obtained, the constant disturbance is compensated by the integrator. The last figure, Figure 6 depicts vx(t,·) it allows to observe the effect of the disturbance and the in-domain damping. The smooth convergence of the ve- locity can be compared with the behavior of vx(t,·). 0 2 4 6 8 10 12 14 time t0.02.55.07.510.0velocityvt(0, t) vt(1, t) Fig. 1. The boundary velocities times responses. 8 Conclusion This paper presents the first systematic Lypunov anal- ysis for a 1-dimensional damped wave equation subject to various dynamic (or Wentzell) boundary conditions, in the case where the damping is everywhere active. As a particular case, we also provide a regulation law for a wave equation (with dynamic boundary conditions) by the means of a PI control. The control law achieved ex- ponential decay rate towards the constant reference, and 0.0 2.5 5.0 7.5 10.0 12.5 15.0 time t−5.0−2.50.02.55.0objectivevt(0, t)−vref t vt(1, t)−vref t∫x 0(vt(x, t)−vref t)dxFig. 2. The objectives times responses. 0 2 4 6 8 10 12 14 time t20406080100U(t) Fig. 3. The control law time response. Fig. 4. The distributed velocity ˙ v(t, x) time response. the rejection of constant disturbance. The possible re- jection of the disturbance by the integral action can be explained by the interne model principle. 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Finally, we conclude using the following theorem. 15Theorem 4 (Hille-Yosida [3, Theorem 7.4 ]) Let Abe a maximal operator on the Hilbert space Hthen for every X0∈ D A
there exists a unique solution Xto the following abstract problem. dX dt(t) +AX(t) = 0 , (A.1a) X(0) = X0. (A.1b) with X∈C1([0,∞);H)∩C([0,∞);D A
). (A.2) Now consider the following operator ∀z∈ D(G), Gz = −z2 −(az′ 1)′+z2+z1 β1z′ 1(1) 0 −µ1z′ 1(0) , (A.3) and the following matrix B= 0 0 0 0 0 1−q+ 1 0 0 0 0 0 −α1−α20 0 0 1 0 0 0 0 0 0 −γ1 . (A.4) The domain of Gis equal to the domain of A. One gets A=G+B. (A.5) Gis a monotone part, this is established in the following lemma and Bis a bounded operator. Lemma 5 The unbounded linear operator Gdefined in (A.3) is a maximal monotone operator on Xwdefined in (11). PROOF. Considering the following scalar product onXw ⟨z, q⟩=Z1 0(z1ν+z2q2+az′ 1ν′)dx+ a(1) β1z3q3+z4q4+a(0) µ1z5q5, (A.6) ⟨z, Gz⟩=Z1 0[−z1z2+z2(−(az′ 1)′+z2+z1) −a(x)z′ 1z′ 2]dx+a(1)z3z′ 1(1) −a(0)z5z′ 1(0), (A.7) using integration by parts and the fact that z∈ D(A), one obtains ⟨z, Gz⟩=Z1 0z2 2dx⩾0. (A.8) Thus the operator Gis monotone (see [3, Chapter 7 on Page 181]) on the Hilbert Xw. In addition, if we establish that R(I+G) =Xw, (A.9) then the operator Gis maximal monotone (see [3, Chap- ter 7 on Page 181]), Rstands for the range of the oper- ator. Let y∈Xw, we have to solve z∈ D(A), z +Gz=y, (A.10) which means that z1−z2=y1, (A.11) z2−(az′ 1)′+z2+z1=y2, (A.12) z3+βz′ 1(1) = y3, (A.13) z4+ 0 = y4, (A.14) z5−µ1z′ 1(0) = y5, (A.15) using the fact that z∈ D(A) one gets 3z1−(az′ 1)′= 2y1+y2, (A.16) β1z′ 1(1) + z1(1) = ( y3+y1(1)), (A.17) −µ1z′ 1(0) + z1(0) = ( y5+y1(0)). (A.18) This is a classical stationary problem (e.g., see [3]) with Robin’s boundaries conditions, using standard result (as done in [3, Example 6, On Page 226] ) one gets that as 2y1+y2∈L2(0,1), (A.16)-(A.18) has a unique solution z1∈H2(0,1). Now one can check that the element z= (z1, z2, z3, z4, z5) with z1is the solution to (A.16)-(A.18) ,(A.19a) z2=z1−y1, (A.19b) z3=y3−a(1)z′ 1(1), (A.19c) z4=y4z4=y5+a(0)z′ 1(0), (A.19d) satisfies (A.11)-(A.15). Moreover using (A.16)-(A.18) on (A.19) one gets that zsatisfying (A.19) is in D(A). 16Now, we are ready to state the proof of the well posedness of (12). Note that the fact that Gis maximal monotone implies that D(A) is dense in Xw(i.e.,D(A) =Xw). Using the bijective change of variable ze(t) =z(t)eBt, (A.20) zis the solution to (12) is equivalent to, ze∈ D A
is the solution to d dtze(t) +Gze(t) = 0 , (A.21a) ze(0) = z0, (A.21b) where Bis defined in (A.4) and Gis defined in (A.3). From Lemma 5, using Theorem 4 on (A.21), and the change of variable (A.20), one establishes (i). Using ar- gument of density of D(A) inXw, and C0-semigroup theory one obtains the regularity of weak solutions. Note that we refer the reader to [19], [30] for the notion weak solutions. Moreover part of the proof are inspired from [6] and [9] which in turn originates from [39]. B Additional materials This section pertains to additional materials that are not included in the accepted version of the paper and includes links to online resources. The first line of Ais h −a0 6dx1−2a1dx1 3(dx1+dx2)2,a0 6dx1,2a1dx1 3(dx1+dx2)2, . . .i (B.1) The second line is  a0 6dx1,−a0 6dx1−a1 6dx2−a2dx1 6dx2 2−2a2dx2 3(dx2+dx3)2, a1 6dx2+a2dx1 6dx2 2,2a2dx2 3(dx2+dx3)2, . . . (B.2) Thei-line for i∈[3, N−2] for column i−2 ati+ 2 is  2ai−2dxi−2 3(dxi−2+dxi−1)2,ai−1dxi−2+ai−2dxi−1 6dx2 i−1, −2ai−2dxi−2 3(dxi−2+dxi−1)2−ai−1dxi−2+ai−2dxi−1 6dx2 i−1 −aidxi−1+ai−1dxi 6dx2 i−2aidxi 3(dxi+dxi+1)2, +aidxi−1+ai−1dxi 6dx2 i,2aidxi 3(dxi+dxi+1)2 (B.3)and zero elsewhere. The N−1 line  . . . ,2aN−2dxN−2 3(dxN−2+dxN−1)2,aN−2 6dxN−1+aN−1dxN−2 6dx2 N−1, −2aN−2dxN−2 3(dxN−2+dxN−1)2−aN−2 6dxN−1−aN−1dxN−2 6dx2 N−1 −aNdxN−1 6dx2 N,aNdxN−1 6dx2 N (B.4) TheNline  . . . ,2aN−1dxN−1 3(dxN−1+dxN)2aNdxN−1 6dx2 N, −2aN−1dxN−1 3(dxN−1+dxN)2−aNdxN−1 6dx2 N (B.5) The reader will find an online environ- ment for the numerical simulation at https://colab.research.google.com/drive/ 1m6uhaur3eySqQ6eyjKf6SXxHXxXhSsWd?usp=sharing and a git-hub depot of the numerical simulation at https://github.com/christoautom/wave_1d . 17

2023-05-03

We establish the exponential decay of the solutions of the damped wave equations in one-dimensional space where the damping coefficient is a nowhere-vanishing function of space. The considered PDE is associated with several dynamic boundary conditions, also referred to as Wentzell/Ventzel boundary conditions in the literature. The analysis is based on the determination of appropriate Lyapunov functions and some further analysis. This result is associated with a regulation problem inspired by a real experiment with a proportional-integral control. Some numerical simulations and additional results on closed wave equations are also provided.

Lyapunov functions for linear damped wave equations in one-dimensional space with dynamic boundary conditions

2305.01969v2

1 Large thermo -spin effects in Heusler alloy based spin -gapless semiconductor thin film s Amit Chanda1*, Deepika Rani2, Derick DeT ellem1, Noha Alzahrani1, Dario A. Arena1, Sarath Witanachchi1, Ratnamala Chatterjee2, Manh -Huong Phan1 and Hari haran Srikanth1* 1 Department of Physics, University of South Florida, Tampa FL 33620 2 Physics Department, Indian Institute of Technology Delhi, New Delhi - 110016 *Corresponding authors: achanda@usf.edu ; sharihar@usf.edu Keywords: Longitudinal spin Seebeck effect , Anomalous Nernst effect , Spin gapless semiconductor, Heusler alloy , Magnetic anisotropy, Gilbert damping Abstract Recently, H eusler alloys -based spin gapless semiconductors (SGSs) with high Curie temperature ( 𝑇𝐶) and sizeable spin polarization have emerged as potential candidates for tunable spintronic applications. We report comprehensive investigation of the temperature depe ndent ANE and intrinsic longitudinal spin Seebeck effect (LSSE) in CoFeCrGa thin films grown on MgO substrates. Our findings show the anomalous Nernst coefficient for the MgO/CoFeCrGa (95 nm) film is ≈ 1.86 μV.K−1 at room temperature which is nearly two orders of magnitude higher than that of the bulk polycrystalline sample of CoFeCrGa (≈ 0.018 μV.K−1) but comparable to that of the magnetic Weyl semimetal Co 2MnGa thin film (≈ 2−3 μV.K−1). Furthermore, the LSSE coefficient for our MgO/CoFeCrGa (95nm)/Pt(5nm) heterostructure is ≈20.5 nV.K−1.Ω−1 at room temperature which is twice larger than that of the half-metallic ferromagnet ic La0.7Sr0.3MnO 3 thin films (≈9 nV.K−1.Ω−1). We show that both ANE and LSSE coefficients follow identical temperature dependences and exhibit a 2 maximum at ≈225 K which is understood as the combined effects of inelastic magnon scatterings and reduced magnon population at low temperatures . Our analys es not only indicate that the extrinsic skew scattering is the dominating mechanism for ANE in these films but also provide critical insight s into the functional form of the observed temperature dependent LSSE at low temperatures . Furthermore, by employing radio frequency transverse susceptibility and broadband ferromagnetic resonance in combination with the LSSE measurements, we establish a correlation among the observed LSSE signal, magnetic anisotropy and Gilbert damping of the CoFeCrGa thin films , which will be beneficial for fabricating tunable and highly efficient Heusler alloys based spincaloritronic nanodevices. 3 1. INTRODUCTION The p ast few years have witnessed extensive resear ch efforts in the field of spin caloritronics for the development of highly efficient next -generation spin -based electronic devices by combining the versatile advantages of spintronics and thermoelectricity , with the aim of finding novel avenues for waste heat recovery and thermoelectric energy conversion1,2. Fundamental knowledge of the interplay between heat, charge, and spin degrees of freedom not only allowed us to understand how thermal gradients can be utilized to manipulate and control the flow of spin angular momenta inside a material a t nanoscale , but also helped the scientific community to explore various intriguing thermo -spin transport phenomena, such as the anomalous Nernst effect (ANE)3, spin Nernst effect4, spin Seebeck effect5,6, spin Peltier effect7 and so on . The ANE refers to the generation of a transverse thermoele ctric voltage in a magnetic conductor/semiconductor by the application of a thermal gradient and an external magnetic field8,9. The ANE has been observed in a large range of magnetic materials, from half-metallic ferromagnets such as hole -doped manganites10, cobaltite s11–13, spin gapless semiconductors14 to ferrimagnets such as iron oxide15, Mn-based nitride16, as well as unconventional magnetic systems with topologically non -trivial phases such as topological full Heusler ferroma gnets3,17– 19, ferro magnetic Weyl semimetal s20,21, two -dimensional topological van der Waals ferromagnets22,23, chiral8 and canted24 topological antiferromagnet s etc. In a topological magnetic material , charge carriers moving through a periodic potential with strong spin -orbit coupling (SOC) acquire an additional anomalous velocity perpendicular to their original trajectory due to the non -zero Berry curvature at the Fermi level25. This anomalous velocity causes a real space spin selective deflection of the charge carriers and leads to a potentially large ANE response in these topological magnetic materia ls compared to conventional magnets 25. In addition to the aforementioned intrinsic origin, ANE can also originate from extrinsic 4 effects for example, asymmetric skew scattering of charge carriers as observed in Heusler ferromagnets14,26,27, hole -doped manganites10, cobaltite s11–13, spin gapless semiconductors14, iron oxide15 etc. On the other hand, the longitudinal spin Seebeck effect (LSSE) refers to the thermal generation of magnonic spin current in a ferromagnetic (FM) material by the concurrent applications of a temperature gradient and an external magnetic field across a FM/heavy met al (HM) bilayer structure and injection of th at spin current to the adjacent HM layer with strong SOC , which is then converted into electrically detectable charge current in the HM layer via the inverse spin Hall effect (ISHE)1,28–30. The bilayer structure consisting of the ferrimagnetic insulator Y 3Fe5O12 (YIG) and Pt is known as the benchmark system for generating pure spin current and hence , LSSE28,30 –34. Apart from YIG, other magnetic insulators for example, the compensated ferrimagnetic insulator Gd 3Fe5O12,35,36 insulating spinel ferrites CoFe 2O4, NiFe 2O437,38, noncollinear antiferromagnet ic insulator LuFeO 339 etc., have also emerged as promising spincaloritronic materials . Nevertheless, observation of LSSE is not only restricted to magnetic insulators , but it has also been observed in metallic5,40, half-metallic41–43 and semiconducting ferromagnet s44. Although ANE and LSSE are two distinct types of magnetothermoelectric phenomena, they share common origin for materials exhibiting extrinsic effects dominated ANE45. In both the cases, simultaneous application of thermal gradient and external magnetic field generates magnonic excitations . While in the case of ANE, the thermally generated magnons transfer spin angular momenta to the itinerant electrons of the FM via the electr on-magnon scattering and thereby dynamically spin polarizes them, in the case of LSSE, a spatial gradient of those thermally generated magnons leads to magnon accumulation close to the FM/HM interface and 5 pumps spin current to the HM layer45. Large magnon -induced ANE has been observed in MnBi single crystal45. However, observation of large ANE in a FM conductor does not necessarily indicate a promise for a large LSSE, and vice versa. Therefore, it would be technologically advantageous from the perspective of spincaloritronic device applications and thermal energy harnessing to search for a FM material that can simultaneously exhibit large LSSE and ANE. In recent years , Heusler alloys -based spin gapless semiconductors (SGSs) have emerged as promising magnetic materials for tunable spintronic applications as they not only combine the characteristics of both half -metallic ferromagnets and gapless semiconductors ,46 but also possess high Curie temperature ( 𝑇𝐶) and substantial spin polarization47–50. We have recently observed large ANE in the bulk sample of Heusler alloy based SGS : CoFeCrGa with 𝑇𝐶≈690 K, 14,50,51 which was the first experimental observation of ANE in the SGS family. Our fascinating observation motivated us to explore ANE a s well as LSSE in the CoFeCrGa thin films. Although SGS has been theoretically predicted to be a promising candidate for spintr onic applications52, there is no previous experimental study on the thermo -spin transport phenomena, especially LSSE in SGS thin films . In this paper , we report on the temperature dependent ANE and LSSE in the CoFeCrGa single layer and CoFeCrGa /Pt bilayer films with different CoFeCrGa film thicknesses. We found that both ANE and LSSE coefficients follow identical temperature dependences and exhibit a maximum at ≈225 K which is understood as the combined effects of inelastic magnon scatterings and reduced magnon population at low temperatures . Our analys es not only indicate that the extrinsic skew scattering is the dominating mechanism for ANE in these films but also provi de critical insight s into the functional form of the observed temperature dependent LSSE. Furthermore, we have established a correlation among the observed LSSE signal , magnetic anisotropy and Gilbert damping of the CoFeCrGa thin films which will be beneficial for fabricating tunable and efficient spincaloritronic device s. 6 2. EXPERIMENTAL SECTION The thin film s of CoFeCrGa w ere grown on single crystal MgO (001) substrates of surface area 5×5 mm2 using an excimer KrF pulsed laser deposition (PLD) system. The films were deposited at 500 °C and were further annealed in -situ at 500 °C for 30 min to further enhance the chemical order and crystallization . The film surface morphology was investigated by field emission gun – scanning electron microscop y (FEG -SEM) and atomic force microscopy (AFM), while the structural properties of the thin films were identified by x -ray diffraction (XRD) using monochromatic Cu Kα radiation. AFM and t emperature dependent magnetic force microscopy ( MFM ) measurements were performed on a Hitachi 5300E system. All measurements were done under high vacuum (P ≤ 10-6 Torr). MFM measurements utilized PPP-MFMR tips, which were magnetized out -of- plane with respect to the tip surface via a permanent magnet. Films were first magnetized to their saturation magnetization by being placed in a 1T static magnetic field, in -plane with the film surface. After that AC demagnetization of th e film was implement ed before initiating the MFM scans. After scans were performed, first a linear background was subtracted which comes from the film not being completely flat on the sample stage . After that, a parabolic background was subtracted, which a rises from the nonlinear motion of the piezoelectric crystal that drives the x-y translation. Phase standard deviation was determined by fitting a Gaussian to the image phase distribution and extracting the standard deviation from the fit parameters. The DC magnetic measurements on the samples at temperatures between 100 K and 300 K were performed using a vibrating sample magnetometer (VSM) attached to a physical property measurement system (PPMS), Quantum Design. A linear background originating from the diamagnetic MgO substrate was thereby subtracted. Due to a trapped remanent field 7 inside the superconducting coils, the measured magnetic field was corrected using a paramagnetic reference sample. The longitudinal electrical resistivity, longitudinal Seebeck coefficient, and thermal conductivity of the bulk samples were simultaneously measured with the thermal transport option (TTO) of the PPMS. The electrical resistivity and Hall measurements on th e thin film samples were performed using the DC resistivity option of the PPMS by employing a standard four point measurement technique with sourcing currents of 500 A and 1 mA, respectively. The temperature dependence of the effective magnetic anisotropy fields of the MgO/CoFeCrGa films were measured by using a radio frequency (RF) transverse susceptibility (TS) measurement technique that exploits a self -resonant tunnel diode oscillator (TDO) circuit with a resonance frequency of ≈12 MHz53,54. The PPMS was used as a platform to sweep the external DC magnetic field and temperature. During the TS measurement, the MgO/CoFeCrGa thin film samples were firmly placed inside an inductor coil (L), which is a component of an LC resonator circuit. The coil containing the sample was positioned at the ba se of the PPMS sample chamber through a multifunctional PPMS probe in such a way that the axial RF magnetic field generated inside the coil stay ed parallel to the film surface, but perpendicular to the DC magnetic field generated by the superconducting mag net of the PPMS. In presence of both the RF and DC magnetic fields, the dynamic transverse susceptibility of the sample changes which eventually changes the resonance frequency of the LC circuit53. From the magnetic field dependence of the shift in the resonance frequency recorded by an Agilent frequency counter, we obtained the fie ld dependent transverse susceptibility. 8 The ANE and LSSE measurements were performed using a custom -designed setup assembled on a universal PPMS sample puck , as shown in our previous reports14,36. For both the measurements , the thin film samples were sandwiched between two copper plates. A single layer of thin Kapton tape was thermally anchored to the bare surfaces of the top (cold) and bottom (hot) copper plates . Cryogenic Apiezon N -grease was used to create good thermal connectivity between the thin film surface and that of the Kapton tape s. A resistive heater (PT - 100 RTD sensor) and a calibrated Si -diode thermometer (DT-621-HR silicon diode sensor) were attached to each of th ose copper plates . The temperature s of both these copper plates were monitored and controlled individually by employing two distinct separate temperature controllers (Scientific Instruments Model no. 9700). The top copper plate was thermally linked to the base of the PPMS universal puck us ing a pair of molybdenum screws and a 4 mm thick Teflon block was thermally sandwiched between the universal PPMS puck base and the bottom copper plate to maintain a temperature difference of ~ 10 K between the hot copper plate and the PPMS universal puck base. The Ohmic contacts for the ANE and LSSE voltage measurements were made by using a pair of thin gold wires of 25 µm diameter to the Pt layer by high quality conducting silver paint (SPI Supplies). In presence of an applied temperature gradient along the z-direction , and an i n-plane external DC magnetic field applied along the x- direction, the transverse thermoelectric voltage generated along the y-direction across the Pt layer due to the ISHE ( 𝑉𝐼𝑆𝐻𝐸) and across the CoFeCrGa film itself due to the ANE was recorded with a Keithley 2182a nanovoltmeter. Broadband ferromagnetic resonance (FMR) measurements were performed using a broadband FMR spectrometer (NanOscTM Phase -FMR, Quantum Design Inc., USA) integrated to the Dynacool PPMS55. 9 3. RESULTS AND DISCUSSION 3.1. Structural and morphological properties Figure 1 (a) shows the X -ray 2- (out of plane) diffraction pattern for CoFeCrGa (95nm) film grown on MgO (001) substrate. In addition to the peaks corresponding to the MgO substrate, there are additional (002) and (004) diffraction peaks from the film, indicating the growth in the (001) orientation. The formation of B2 CoFeCrGa structure is confirmed by t he presence of (002) peak . To find the CoFeCrGa (220) peak intensity , a 2- scan was performed with = 45 as shown in Fig. 1 (b). The lattice parameter as estimated by applying the Bragg equation to the (0 22) peak, was found to be 5.76 Å. Figure 1. (a) XRD of MgO/ CoFeCrGa (95nm) film: ω–2θ (ou t-of-plane) scan . (b) The 2θ –θ scan of the (022) plane. (c) Phi -scan of the (022) plane. (d) 𝜙 -scan of the (111) plane . (e) FEG - SEM , (f) Cross sectional SEM image and (g) AFM image for the MgO/ CoFeCrGa (95nm) film. 10 To further confirm the epitaxial growth of the CoFeCrGa (95nm) film, 𝜙-scan was performed for the (220) and (111) planes by tilting the sample, i.e., = 45 for the (220) plane and = 54.7 for the (111) plane ( Fig. 1 (c), and Fig. 1 (d)). The 𝜙 - scans of both (220) and (111) plane show a four -fold symmetry, as four well defined peaks periodically separated from each other by 90 were observed. The presence of both (111) and (200) peaks rule out the possibility of complete A2 or B2 disorder, however partia l disorder can still be present . The chemical composition was interpreted by the scanning electron microscopy energy -dispersive spectroscopy (SEM -EDS) measurements an d was found to be Co 1.05Fe1.05Cr0.9Ga0.99, which is very close to the ideal stoichiometric composition expected for an equiatomic quaternary Heusler alloy. The surface morphology of the film obtained from the F EG-SEM image is shown in Fig. 1 (e), which indicates that the film is homogenous , which was further confirmed by AFM measurements as shown in Fig. 1 (g). A low root -mean -square (RMS) roughness of ≈ 2.5nm is achieved for the CoFeCrGa film , as noticeable in the AFM image shown in Fig. 1 (g). The cross -section SEM imag e of the film is shown in Fig. 1 (f), which indicates that the film thickness (~95± 5 nm). In Fig. 2 , we show the temperature dependent magnetic force microscopy ( MFM ) images recorded on the MgO/ CoFeCrGa (95nm) film. The MFM image at 300 K ( see Fig. 2(a)) shows a bright /dark contrast with highly irregular shape d features indicating cloudlike domain - clusters56. Note that in MFM, the domain -image contrast is determined by the magnetic force - gradient (𝑑𝐹 𝑑𝑧) between the sample and the MFM tip (magnetized ⊥ to the film -surface), which is proportional to the perpendicular component of the stray field of the film57,58. For our film, due to low bright/dark c ontrast patterns of the MFM images in the T-range: 160K ≤𝑇≤ 300 K (Fig. 2(a)-(e)), the domain boundaries are not as well -defined as observed in films with strong PMA59. 11 Figure 2. Magnetic force microscopy (MFM) images of MgO/ CoFeCrGa (95nm) film measured at (a) T = 300 K, (b) T = 250 K, (c) T = 200 K, (d) T = 180 K, and (e) T = 160 K while cooling the sample after applying an IP magnetic field (higher than the IP saturation field) and then AC demagnetization of the sample at 300 K. (f) The RMS value of the phase shift, Δ𝜙𝑅𝑀𝑆 as a function of temperature for the MgO/ CoFeCrGa (95nm) film extracted from the MFM images. (g) The average domain width as a function of temperature obtained from the MFM images. A steep increase in the root mean square ( RMS ) value of the phase shift,57 Δ𝜙𝑅𝑀𝑆≈ 𝑄 𝐾[𝑑𝐹 𝑑𝑧] (Q = quality factor and 𝐾 = spring constant of the tip; hence, Δ𝜙𝑅𝑀𝑆 ∝ average domain contrast58) has also been observed below 300 K (see Fig. 2(f)). However, Δ𝜙𝑅𝑀𝑆 decreases slightly below 180 K. Averag e domain widths were determined by calculating the 2D autocorrelation across the MFM image s, then determining the full -width half -max (FWHM) of arbitrary lines through the 2D autocorrelation spectra. As shown in Fig. 2(g), the average domain width also increases with decreasing temperature followed by a slight decrease below 170 K. 12 3.2. Magnetic and electrical t ransport properties Previous studies on bulk CoFeCrGa50,51 as well as MgO/CoFeCrGa thin films60 reveal that the ferromagnetic transition temperature of this sample is very high (at least ≥500 K). The main panel of Fig. 3(a) shows the magnetic field dependence of magnetization, 𝑀(𝐻) of our MgO/CoFeCrGa film measured at selected temperature s in the range: 125 K ≤𝑇 ≤300 K in presence of an in -plane sweeping magnetic field . The 𝑀(𝐻) loops exhibit very small coercivity throughout the measured temperature range. Figure 3. (a) Main panel: magnetic field dependence of magnetization, 𝑀(𝐻) of our MgO/CoFeCrGa (95nm) film measured at selected temperatures in the range: 125 K ≤𝑇 ≤ 300 K in presence of an in -plane sweeping magnetic field , inset: temperature dependence of the saturation magnetizati on, MS. (b) T emperature dependence of magnetization, 𝑀(𝑇) measured in zero -field-cooled warming (ZFCW) and field -cooled -warming (FCW) protocols in presence of an external magnetic field : 𝜇0𝐻=0.1 T. (c) N ormalized 𝑀(𝐻) hysteresis loops at 𝑇=300 K for the in -plane (IP) and out -of-plane (OOP) configurations . (d) Main panel: 13 temperature dependence of longitudinal resistivity, 𝜌𝑥𝑥(𝑇) for the MgO/CoFeCrGa film in the temperature range: 10 K ≤𝑇 ≤300 K, inset shows corresponding temperature dependence of electrical conductivity, 𝜎𝑥𝑥(𝑇). (e) The bipolar field scan s (+𝐻𝐷𝐶𝑚𝑎𝑥−𝐻𝐷𝐶𝑚𝑎𝑥 +𝐻𝐷𝐶𝑚𝑎𝑥) of ∆𝜒𝑇 𝜒𝑇(𝐻𝐷𝐶) for MgO/CoFeCrGa (95nm) film measured at T = 20 K for both IP ( HDC is parallel to the film surface) and OOP ( HDC is perpendicular to the film surface) configurations. (f) Temperature variations of the effective anisotropy fields: 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃for our MgO/CoFeCrGa (95nm) film. As shown in the inset of Fig. 3(a), the saturation magnetization, 𝑀𝑆 increases almost linearly with decreasing temperature , which is in agreement with the temperature dependent Δ𝜙𝑅𝑀𝑆 obtained from the MFM images58. In Fig. 3(b), we show the temperature dependence of magnetization, 𝑀(𝑇) measured in zero -field-cooled warming (ZFCW) and field -cooled - warming (FCW) protocols in presence of an external magnetic field : 𝜇0𝐻=0.1 T. It is evident that both ZFCW and FCW 𝑀(𝑇) increases with decreasing temperature down to 10 K below which it shows a slight up -turn. Furthermore, the ZFCW and FCW 𝑀(𝑇) curves do not exhibit any considerable bifurcation at low temperatures which is indicative of the absence of any glassy magnetic ground state. Fig. 3(c) shows the normalized 𝑀(𝐻) hysteresis loops at 𝑇= 300 K for the in -plane (IP) and out -of-plane (OOP) configurations confirming the soft ferromagnetic nature of the film along the I P direction, which is consistent with a recent report on this system60. The main panel of Fig. 3(d) demonstrates the T-dependence of longitudinal resistivity, 𝜌𝑥𝑥(𝑇) for the MgO/CoFeCrGa film in the temperature range: 10 K ≤𝑇 ≤300 K. It is obvious that 𝜌𝑥𝑥(𝑇) exhibits semiconducting -like resistivity (𝜕𝜌𝑥𝑥 𝜕𝑇>0) throughout the temperature range . The inset of Fig. 3(d) shows the T-dependence of electrical conductivity , 𝜎𝑥𝑥(𝑇) for the MgO/CoFeCrGa film. Note that the values of both 𝜌𝑥𝑥(𝑇) and 𝜎𝑥𝑥(𝑇) for our MgO/CoFeCrGa film are quite close to those reported on the same film with 12 nm thickness60. 14 Furthermore, the linear temperature coefficient of the resistivity for our MgO/CoFeCrGa film was found to be ≈−1.37 ×10−10Ω m/K, which is of the same magnitude to that reported for different Heusler alloys -based spin gapless semiconductors ( SGSs), such as Mn2CoAl (−1.4 ×10−9Ω m/K),47 CoFeMnSi (−7 ×10−10Ω m/K),61 CoFeCrAl (−5 ×10−9Ω m/ K),62 and CoFeCrGa (−1.9 ×10−9Ω m/K)60 thin films . We have also performed radio frequency (RF) transverse susceptibility (TS) measurements on our MgO/CoFeCrGa film in the temperature range: 20 K ≤𝑇 ≤300 K to determine the temperature evolution of effective magnetic anisotropy. This technique can accurately determine the dynamical magnetic response of a magnetic material in presence of a DC magnetic field ( HDC) and a transverse RF magnetic field ( HRF) with small and fixed amplitude.63 When HDC is scanned from positive to negative saturations , the TS of a magnetic material with uniaxial anisotropy demonstrates well -defined peaks at the anisotropy fields, HDC = ± 𝐻𝐾.64 But for a magnetic material comprising of randomly dispersed magnetic easy axes, the TS shows broad maxima at the effective anisotropy fields, HDC = ±𝐻𝐾𝑒𝑓𝑓. Here, we show the TS spectra as percentage change of the measured transverse susceptibility as, ∆𝜒𝑇 𝜒𝑇(𝐻𝐷𝐶)= 𝜒𝑇(𝐻𝐷𝐶)−𝜒𝑇(𝐻𝐷𝐶𝑚𝑎𝑥) 𝜒𝑇(𝐻𝐷𝐶𝑚𝑎𝑥)×100% , where 𝜒𝑇(𝐻𝐷𝐶𝑚𝑎𝑥) is the value of 𝜒𝑇 at the maximum value of the applied DC magnetic field, 𝐻𝐷𝐶𝑚𝑎𝑥 which is chosen in such a way that 𝐻𝐷𝐶𝑚𝑎𝑥≫𝐻𝐷𝐶𝑠𝑎𝑡, where 𝐻𝐷𝐶𝑠𝑎𝑡 is the saturation magnetic field. Fig. 3(e) shows the bipolar field scan ( +𝐻𝐷𝐶𝑚𝑎𝑥− 𝐻𝐷𝐶𝑚𝑎𝑥 +𝐻𝐷𝐶𝑚𝑎𝑥) of ∆𝜒𝑇 𝜒𝑇(𝐻𝐷𝐶) for MgO/CoFeCrGa film measure d at T = 20 K for both IP ( HDC is parallel to the film surface) and OOP ( HDC is perpendicular to the film surface) configurations. For both the configurations, the TS shows maxima centering at 𝐻𝐷𝐶=±𝐻𝐾𝑒𝑓𝑓. Here, we define 𝐻𝐾𝑒𝑓𝑓= 𝐻𝐾𝐼𝑃 as the IP effective anisotropy field (for IP configuration) and 𝐻𝐾𝑒𝑓𝑓= 𝐻𝐾𝑂𝑂𝑃 as the OOP effective anisotropy field (for OOP configuration). We found that 15 |𝐻𝐾𝑂𝑂𝑃|> |𝐻𝐾𝐼𝑃| at all the temperatures indicating IP easy axis of this film in the te mperature range: 20 K ≤𝑇 ≤300 K. Furthermore, it is evident that the peaks at +𝐻𝐾𝐼𝑃(+𝐻𝐾𝑂𝑂𝑃) and −𝐻𝐾𝐼𝑃(−𝐻𝐾𝑂𝑂𝑃) are asymmetric with unequal peak heights which is indicative of significant anisotropy dispersion in our MgO/CoFeCrGa film for the both IP and OOP configurations. The temperature variations of 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃for our MgO/CoFeCrGa (95nm) film are shown in Fig. 3(f). Clearly, both 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃increase with decreasing temperature and 𝐻𝐾𝑂𝑂𝑃> 𝐻𝐾𝐼𝑃 throughout the measured temperature range. Interestingly, with decreasing temperature, 𝐻𝐾𝑂𝑂𝑃 increases more rapidly than 𝐻𝐾𝐼𝑃 which gives rise to large difference between 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃 at low temperatures. Additionally, both 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃 increases more rapidly below ≈ 200 K compared to the temperature range of 200 K ≤𝑇 ≤300 K. 3.3. Thermal spin transport properties : ANE and LSSE Next, we focus on the thermo -spin transport properties of our MgO/CoFeCrGa (95nm) film. We have performed anomalous Nernst effect ( ANE) and longitudinal spin Seebeck effect (LSSE) measurements on MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt(5nm) films, respectively. Figs. 4(a) and (b) demonstrate the schematic illustration s of our ANE and LSSE measurements . Both t he ANE and LSSE measurements on MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively were performed by sandwiching the film between two copper blocks and applying a temperature gradient (along the + z-direction) that creates a temperature difference, ∆𝑇 between th ose copper blocks in presence of an external DC magnetic field applied along the x-direction. The thermally generated Nernst and LSSE voltage s generated along the y-direction w ere recorded us ing a Keithley 2182a nanovoltmeter while scanning the DC magnetic field. According to the theory of thermally generated magnon - driven interfacial spin pumping mechanism , simultaneous application of a vertical ( z-axis) temperature gradient ( 𝛁𝑻⃗⃗⃗⃗⃗ ) and an external transverse dc magnetic field ( 𝝁𝟎𝑯⃗⃗⃗⃗⃗⃗⃗⃗ ) (x-axis) across 16 the MgO/CoFeCrGa (95nm)/Pt film gives rise to transverse spin current pumping from the CoFeCrGa layer into the Pt layer with the interfacial spin current density : 𝑱𝑺⃗⃗⃗ = 𝐺↑↓ 2𝜋𝛾ℏ 𝑀𝑆𝑉𝑎𝑘𝐵𝛁𝑻⃗⃗⃗⃗⃗ at the CoFeCrGa /Pt interface, where 𝐺↑↓, ℏ, 𝛾, 𝑀𝑆, 𝑉𝑎 and 𝑘𝐵 are the interfacial spin -mixing conductance, the reduced Planck’s constant (ℏ= ℎ 2𝜋), the gyromagnetic ratio, the saturation magnetization of CoFeCrGa , the magnon coherence volume and the Boltzmann constant, respectively42,65,66. The magnetic coherence volume is expressed as : 𝑉𝑎= 2 3𝜁(52⁄)(4𝜋𝐷 𝑘𝐵𝑇)3/2 ; where, 𝜁 is the Riemann Zeta function and 𝐷 is the spin -wave stif fness constant65,66. This transverse spin current, 𝑱𝑺⃗⃗⃗ is then converted into charge current, 𝑱𝑪⃗⃗⃗ = (2𝑒 ℏ)𝜃𝑆𝐻𝑃𝑡(𝑱𝑺⃗⃗⃗ × 𝝈𝑺⃗⃗⃗⃗⃗ ) along the y-axis via the inverse spin Hall effect (ISHE), where e, 𝜃𝑆𝐻𝑃𝑡, and 𝝈𝑺⃗⃗⃗⃗⃗ are the electron charge, the spin Hall angle of Pt, and the spin -polarization vector, respectively . The corresponding voltage along the y-axis can be expressed as,42,67,68 𝑉𝐿𝑆𝑆𝐸= 𝑅𝑦𝐿𝑦𝜆𝑃𝑡(2𝑒 ℏ)𝜃𝑆𝐻𝑃𝑡 𝐽𝑆tanh(𝑡𝑃𝑡 2𝜆𝑃𝑡), (1) where 𝑅𝑦,𝐿𝑦,𝜆𝑃𝑡,and 𝑡𝑃𝑡 are the electrical resistance between the voltage leads, the distance between the voltage leads, the spin diffusion length of Pt, and the thickness of Pt layer (= 5 nm ), respectively . Since CoFeCrGa is a spin -gapless semiconductor with soft ferromagnetic behavior,51,60 concomitant application of the temperature gradient (z-axis) and dc magnetic field (x-axis) also generates a spin -polarized current in the CoFeCrGa layer along the y-axis due to ANE ,69 which gives rise to an additional contribution ( 𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎𝐴𝑁𝐸) to the total voltage signal measured across the Pt layer in the MgO/CoFeCrGa (95nm)/Pt heterostructure. 17 Figure 4. (a) and (b) the schematic illustrations of our ANE and LSSE measurements , respectively . (c) and (d) show the magnetic field dependence of the ANE voltage, 𝑉𝐴𝑁𝐸(𝐻) and ISHE -induced in-plane voltage, 𝑉𝐼𝑆𝐻𝐸(𝐻) measured on the MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively for different values of the temperature difference between the hot ( 𝑇ℎ𝑜𝑡) and cold ( 𝑇𝑐𝑜𝑙𝑑) copper blocks, ∆𝑇= (𝑇ℎ𝑜𝑡−𝑇𝑐𝑜𝑙𝑑) in the range: +5 K≤ ∆𝑇 ≤+18 K at a fixed average sample temperature 𝑇= 𝑇ℎ𝑜𝑡+𝑇𝑐𝑜𝑙𝑑 2 = 295 K. (e) and (f) exhibit the ∆𝑇dependence of the background -corrected ANE voltage, 𝑉𝐴𝑁𝐸(Δ𝑇)= [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥, Δ𝑇)−𝑉𝐴𝑁𝐸( −𝜇0𝐻𝑚𝑎𝑥, Δ𝑇) 2] and the background -corrected (ANE+ LSSE ) voltage, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(Δ𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥, Δ𝑇)−𝑉𝐼𝑆𝐻𝐸( −𝜇0𝐻𝑚𝑎𝑥, Δ𝑇) 2], respectively . 18 In presence of a transverse temperature gradient (𝛁𝑻⃗⃗⃗⃗⃗ ), the electric field generated by ANE in a magnetic conductor/semiconductor with magnetization 𝑴⃗⃗⃗ can be expressed as,15 𝑬𝑨𝑵𝑬⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ∝ 𝑆𝐴𝑁𝐸(𝜇0𝑴⃗⃗⃗ ×𝛁𝑻⃗⃗⃗⃗⃗ ) (2) where, 𝑆𝐴𝑁𝐸 is the anomalous Nernst coefficient. Furthermore, an additional voltage contribution ( 𝑉𝑃𝑟𝑜𝑥𝐴𝑁𝐸) can appear due to the magnetic proximity effect (MPE) induced ANE in the non -magnetic Pt layer.69,70 Note that, onl y a few layers of Pt close to the CoFeCrGa (95nm)/Pt interface gets magnetized (proximitized) due to the MPE, whereas the remaining layers remain unmagnetized. Hence both 𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎𝐴𝑁𝐸 and 𝑉𝑃𝑟𝑜𝑥𝐴𝑁𝐸 are suppressed due to the inclusion of t he 5 nm thick Pt layer on the top of CoFeCrGa layer .41 Therefore, the resultant voltage measured across the Pt layer of our MgO/CoFeCrGa (95nm)/Pt heterostructure can be expressed as,71 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸= 𝑉𝐿𝑆𝑆𝐸+ 𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎 , 𝑆𝑢𝑝𝐴𝑁𝐸+ 𝑉𝑃𝑟𝑜𝑥, 𝑆𝑢𝑝𝐴𝑁𝐸; where , 𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎 , 𝑆𝑢𝑝𝐴𝑁𝐸 and 𝑉𝑃𝑟𝑜𝑥, 𝑆𝑢𝑝𝐴𝑁𝐸 account for the suppressed ANE voltages due to the CoFeCrGa layer and the MPE - induced ANE voltage in the Pt layer, respectively. Previous studies show that the contribution from the MPE -induced ANE in the Pt layer is negligibly small for bilayers consisting of magnetic semiconductors and Pt.41,69 Also, in our previous report ,71 we have shown that the MPE - induced LSSE contribution of the proximitized Pt layer is negligible as only a few layers of Pt close to the CoFeCrGa (95nm)/Pt interface are magnetized due to the MPE41. Therefore, the resultant voltage measured across the Pt layer of our MgO/CoFeCrGa (95nm)/Pt heterostructure can be expressed as: 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸= 𝑉𝐿𝑆𝑆𝐸+ 𝑉𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎 , 𝑆𝑢𝑝𝐴𝑁𝐸. Considering a parallel circuit configuration of CoFeCrGa and Pt layers, the suppressed ANE voltage (due to the CoFeCrGa layer) across the Pt layer of the MgO/CoFeCrGa (95nm)/Pt heterostructure can be expressed as,41,69 𝑉𝐶𝑜𝐹𝑒𝐶𝑅𝐺𝑎 , 𝑆𝑢𝑝𝐴𝑁𝐸= (𝐹 1+𝐹)𝑉𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎𝐴𝑁𝐸 (3) 19 where, 𝐹= 𝜌𝑃𝑡 𝜌CoFeCrGa∙ 𝑡CoFeCrGa 𝑡𝑃𝑡, 𝜌CoFeCrGa (𝜌𝑃𝑡) is the electrical resistivity of the CoFeCrGa (Pt) layer, and 𝑡CoFeCrGa (𝑡𝑃𝑡) is the thickness of the CoFeCrGa (Pt) layer, respectively. Therefore, the intrinsic LSSE voltage contribution can be disentangled from the ANE contribution using the expressio n,41,71 𝑉𝐿𝑆𝑆𝐸= 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸− (𝐹 1+𝐹)𝑉𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎𝐴𝑁𝐸 (4) Figs. 4(c) and (d) show the magnetic field dependence of the ANE voltage, 𝑉𝐴𝑁𝐸(𝐻) and ISHE - induced in-plane voltage, 𝑉𝐼𝑆𝐻𝐸(𝐻) measured on the MgO/CoFeCrGa (95nm) and MgO/ CoFeCrGa (95nm)/Pt films, respectively for different values of the temperature difference between the hot ( 𝑇ℎ𝑜𝑡) and cold ( 𝑇𝑐𝑜𝑙𝑑) copper blocks, ∆𝑇= (𝑇ℎ𝑜𝑡−𝑇𝑐𝑜𝑙𝑑) in the range: +5 K≤ ∆𝑇 ≤+18 K at a fixed average sample temperature 𝑇= 𝑇ℎ𝑜𝑡+𝑇𝑐𝑜𝑙𝑑 2 = 295 K. Clearly, both 𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) signals increase upon increasing Δ𝑇. Figs. 4(e) and (f) exhibit the ∆𝑇dependence of the background -corrected ANE voltage, 𝑉𝐴𝑁𝐸(Δ𝑇)= [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥, Δ𝑇)−𝑉𝐴𝑁𝐸( −𝜇0𝐻𝑚𝑎𝑥, Δ𝑇) 2] and the background -corrected (ANE+ LSSE ) voltage, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(Δ𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥, Δ𝑇)−𝑉𝐼𝑆𝐻𝐸( −𝜇0𝐻𝑚𝑎𝑥, Δ𝑇) 2], respectively, where 𝜇0𝐻𝑚𝑎𝑥 (𝜇0𝐻𝑚𝑎𝑥≫𝜇0𝐻𝑠𝑎𝑡) is the maximum value of the applied magnetic field strength and 𝜇0𝐻𝑠𝑎𝑡 = saturation magnetic field . Evidently , both 𝑉𝐴𝑁𝐸 and 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸 scale linearly with Δ𝑇 and |𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸|>|𝑉𝐴𝑁𝐸|, which confirm that the observed field dependen ces originate from the ANE and (ANE+LSSE), respectively41,71. 20 Figure 5. (a) and (b) 𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured at selected average sample temperatures in the temperature range: 125 K≤ ∆𝑇 ≤295 K for a fixed value of Δ𝑇 = +1 5 K on MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively . (c) The 𝑇-dependence of the background -corrected ANE voltage, 𝑉𝐴𝑁𝐸(𝑇)= [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥, 𝑇)−𝑉𝐴𝑁𝐸( −𝜇0𝐻𝑚𝑎𝑥, 𝑇) 2] and the background -corrected (ANE+ LSSE ) voltage, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥, 𝑇)−𝑉𝐼𝑆𝐻𝐸( −𝜇0𝐻𝑚𝑎𝑥, 𝑇) 2] measured on MgO/ CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively . (d) Right y -scale: the temperature dependence of the in trinsic LSSE voltage, 𝑉𝐿𝑆𝑆𝐸(𝑇) and the left y-scale: the temperature dependence of [𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)−𝑉𝐴𝑁𝐸(𝑇)]. In Figs. 5(a) and (b), we show 𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured at selected average sample temperatures in the temperature range: 125 K≤ ∆𝑇 ≤295 K for a fixed value of Δ𝑇 = +1 5 K on MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively . Fig. 5(c) exhibits the 𝑇-dependence of the background -corrected ANE 21 voltage, 𝑉𝐴𝑁𝐸(𝑇)= [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥, 𝑇)−𝑉𝐴𝑁𝐸( −𝜇0𝐻𝑚𝑎𝑥, 𝑇) 2] and the background -corrected (ANE+ LSSE ) voltage, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥, 𝑇)−𝑉𝐼𝑆𝐻𝐸( −𝜇0𝐻𝑚𝑎𝑥, 𝑇) 2] measured on MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively . It is evident that both |𝑉𝐴𝑁𝐸(𝑇)| and |𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)| increase with decreasing temperature up to T = 200 K below which both of them decrease gradually with further reducing the temperature, resulting in a maximum around 200 K. Furthermore, |𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)|> |𝑉𝐴𝑁𝐸(𝑇)| throughout the measured temperature range, which confirms that both ANE and LSSE contribute towards the voltage measured on the MgO/CoFeCrGa (95nm)/Pt heterostructure. In order to determine the temperature dependence of intrinsic LS SE voltage, we have disentangled the LSSE contribution from the ANE contribution using Eqn. 4 . The right y-scale of Fig. 5(d) shows the temperature dependence of the intrinsic LSSE voltage, 𝑉𝐿𝑆𝑆𝐸(𝑇) obtained by using by Eqn. 4 incorporating the correction factor: (𝐹 1+𝐹), whereas the left y-scale shows the temperature dependence of the voltage difference [𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)−𝑉𝐴𝑁𝐸(𝑇)] without incorporating the aforementioned correction factor, for comparison. A clear distinction can be observed be tween 𝑉𝐿𝑆𝑆𝐸(𝑇) and [𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)−𝑉𝐴𝑁𝐸(𝑇)] in terms of the absolute value as well the nature of the T-dependence, highlighting the importance of the correction factor for accurately determining the intrinsic LSSE contribution. Evidently, |𝑉𝐿𝑆𝑆𝐸(𝑇)| increases with decreasing temperature and shows a broad maximum around 200 K below which it decreases gradually with further lowering the temperature, as shown in Fig. 5(d). To ensure that the observed behavior of 𝑉𝐴𝑁𝐸(𝑇) and 𝑉𝐿𝑆𝑆𝐸(𝑇) are intrinsic to the MgO/ CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt films, respectively, we repeated the same experiments for two more CoFeCrGa films with different thicknesses , namely, 𝑡CoFeCrGa= 50 and 200 nm . 22 Figure 6. (a) and (b) 𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured on the MgO/ CoFeCrGa (𝑡CoFeCrGa) and MgO/CoFeCrGa (𝑡CoFeCrGa)/Pt films for 𝑡CoFeCrGa (CoFeCrGa film thickness )=50,95 and 200 nm at 295 K for Δ𝑇 = +1 5 K. (c) 𝑉𝐴𝑁𝐸(𝜇0𝐻=1 T), 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝜇0𝐻=1 T) and 𝑉𝐿𝑆𝑆𝐸(𝜇0𝐻=1 T) at 295 K plotted as a function of 𝑡CoFeCrGa . (d), (e) and (f) show the comparison of 𝑉𝐴𝑁𝐸(𝑇), 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇) and 𝑉𝐿𝑆𝑆𝐸(𝑇), respectively for different 𝑡CoFeCrGa. The temperature dependent magnetometry and DC electrical transport pro perties of the 𝑡CoFeCrGa= 50 and 200 nm films are displayed in the supplementary information ( Figures S1 and S3). Furthermore, similar to the 𝑡CoFeCrGa= 95 nm film, both 𝑉𝐴𝑁𝐸 and 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸 for 𝑡CoFeCrGa= 50 and 200 nm films scale linearly with Δ𝑇 and |𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸|>|𝑉𝐴𝑁𝐸|, as shown in Figure S5. In Figure S6, we demonstrate 𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops at selected average sample temperatures in the range: 125 K≤ ∆𝑇 ≤295 K for a fixed value of 23 Δ𝑇 = +15 K for 𝑡CoFeCrGa= 50 and 200 nm films . In Figs. 6(a) and (b), we compare 𝑉𝐴𝑁𝐸(𝐻) and 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured on the MgO/CoFeCrGa (𝑡CoFeCrGa) and MgO/ CoFeCrGa (𝑡CoFeCrGa)/Pt films for 𝑡CoFeCrGa (CoFeCrGa film thickness )= 50,95 and 200 nm at 𝑇= 295 K for Δ𝑇 = +15 K. As shown in Figs. 6(c), 𝑉𝐴𝑁𝐸(𝜇0𝐻=1 T), 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝜇0𝐻=1 T) and 𝑉𝐿𝑆𝑆𝐸(𝜇0𝐻=1 T) at 295 K increase with increasing 𝑡CoFeCrGa . In Figs. 6(d), (e) and (f), we compare 𝑉𝐴𝑁𝐸(𝑇), 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇) and 𝑉𝐿𝑆𝑆𝐸(𝑇), respectively for different 𝑡CoFeCrGa. Clearly, the values of all the three quantities: 𝑉𝐴𝑁𝐸, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸 and 𝑉𝐿𝑆𝑆𝐸 are higher for thicker CoFeCrGa films at all temperatures. Furthermore, |𝑉𝐴𝑁𝐸(𝑇)|, |𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(𝑇)| and |𝑉𝐿𝑆𝑆𝐸(𝑇)| exhibit the same behavior for all the three different CoFeCrGa film thicknesses , i.e., all these quantities increase with decreasing temperature from 295 K and show a broad maximum around 200 K, which is followed by a gradual decrease with further lowering the temperature. These observations confirm that the observed behavior of 𝑉𝐴𝑁𝐸(𝑇) and 𝑉𝐿𝑆𝑆𝐸(𝑇) are intrinsic to the MgO/CoFeCrGa and MgO/CoFeCrGa /Pt films, respectively. 3.4. Mechanism of LSSE and ANE at low temperatures Since the density of the thermally generated magnons -driven spin current is proportional to the effective temperature gradient across the CoFeCrGa film through the expression, |𝑱𝑺⃗⃗⃗ |=𝐺↑↓ 2𝜋𝛾ℏ 𝑀𝑆𝑉𝑎𝑘𝐵|𝛁𝑻⃗⃗⃗⃗⃗ |, it is imperative to accurately determine the ef fective temperature difference s between the top and bottom surfaces of the CoFeCrGa film(∆𝑇𝑒𝑓𝑓). The total temperature difference (Δ𝑇) across the MgO/CoFeCrGa /Pt heterostructure can be expressed as a linear combination of temperature drops in the Pt layer , at the Pt/ CoFeCrGa interface, in the CoFeCrGa layer, at the CoFeCrGa /MgO interface, across the GSGG substrate as well as in the N-grease layers (thickness ≈ 1 m) on both sides of the MgO/CoFeCrGa /Pt 24 heterostructure, and can be written as,72 ∆𝑇= ∆𝑇𝑃𝑡+∆𝑇𝑃𝑡 CoFeCrGa+∆𝑇CoFeCrGa+ ∆𝑇CoFeCrGa MgO+∆𝑇MgO+2.∆𝑇N−Grease . Since the thermal resistance of Pt is very small compared to the other contributions and the bulk contributions towards the measured ISHE voltage dominate o ver the interfacial contributions when the thickness of the magnetic film (CoFeCrGa ) is high enough,72 the total temperature difference can be approximately written as, ∆𝑇= ∆𝑇CoFeCrGa+∆𝑇MgO+2.∆𝑇𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 . Figure 7. (a)-(c) Right y-scale: temperature dependence of ∆𝑇𝑒𝑓𝑓 for different 𝑡CoFeCrGa , left y-scale: temperature dependence of the modified LSSE coefficient, 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇) for MgO/ CoFeCrGa (𝑡CoFeCrGa)/Pt(5nm) films for 𝑡CoFeCrGa=50,95 and 200 nm, respectively, fitted with the expression 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓∝ (𝜃𝑆𝐻𝑃𝑡𝐺↑↓ 2𝜋𝑘𝐵 𝐷3/2)𝑇𝑛. (d) -(f) Temperature dependence of the 25 ANE coefficient, 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) for the MgO/CoFeCrGa (𝑡CoFeCrGa) films for 𝑡CoFeCrGa= 50,95 and 200 nm, respectively, fitted with Eqn. 6 . Considering the 4 -slab model, the total thermal resistance between hot and cold plates can be written as, 𝑅𝑇ℎ= 1 𝐴(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒+ 𝑡CoFeCrGa 𝜅CoFeCrGa+𝑡MgO 𝜅MgO), where, 𝑡N−Grease, 𝑡MgO and 𝑡CoFeCrGa are the thicknesses of the grease layers, MgO substrate and the CoFeCrGa layer , respectively; 𝜅N−Grease,𝜅MgO and 𝜅CoFeCrGa are the thermal conductivities of the grease layer s, MgO substrate and the CoFeCrGa layer , respectively, and 𝐴 is the c ross sectional area. Since the rate of heat flow across the entire heterostructure reaches a constant value in the steady state, the effective temperature difference across the CoFeCrGa film can be written as,42 ∆𝑇𝑒𝑓𝑓= ∆𝑇CoFeCrGa = 𝛥𝑇 [1+𝜅𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎 𝑡𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝑀𝑔𝑂 𝜅𝑀𝑔𝑂)] (5) We have measured the temperature dependence of thermal conductivity of bulk CoFeCrGa using the thermal transport option of the PPMS, as shown in the supplement ary information (Figure S4 ). Using the reported values of the thermal conductivities of the Apiezon N -grease ,73 and the MgO crystal74, we have determined the temperature dependence of ∆𝑇𝑒𝑓𝑓 for different 𝑡CoFeCrGa using Eqn. 5, as shown in Figs. 7(a)-(c). Here, we have ignored the interfacial thermal resistances between the N -grease and the hot/cold plates as well as between the sample and N -grease layers .14 Using the T-dependence of ∆𝑇𝑒𝑓𝑓, we have estimated the T-dependence of the modified LSSE coefficient , 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇)= 𝑉𝐿𝑆𝑆𝐸(𝑇) ∆𝑇𝑒𝑓𝑓(𝑇)𝑅𝑦(𝑇)×(𝐿𝑧 𝐿𝑦) for MgO/CoFeCrGa (𝑡CoFeCrGa)/Pt(5nm) films for 𝑡CoFeCrGa=50,95 and 200 nm; where, 𝐿𝑦 (= 3 mm) is the distance between the voltage leads and 𝐿𝑧 = 𝑡CoFeCrGa (see Figs. 7(a)-(c)). Note that we have measured the T- dependence of resistance (𝑅𝑦(𝑇)) between the voltage -leads placed on the Pt layer of the 26 MgO/CoFeCrGa (𝑡CoFeCrGa)/Pt heterostructure s using 4 -point probe configuration. Note that the value of 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇)for our MgO/CoFeCrGa (𝑡CoFeCrGa)/Pt heterostructures are ≈ 12.8,20.5 and 29.8 nV.K−1.Ω−1 at T = 295 K for 𝑡CoFeCrGa=50,95 and 200 nm, respectively, which are higher than that of the half-metallic FM thin films of La 0.7Sr0.3MnO 3 (≈9 nV.K−1.Ω−1 at room temperature )43. As shown in Figs. 7(a)-(c), 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇) for the MgO/CoFeCrGa (𝑡CoFeCrGa)/Pt(5nm) heterostructures for all the three CoFeCrGa film thicknesses increases as T decreases from room temperature and shows a peak around 225 K below which it decreases rapidly with further decrease in temperature. Since the saturation magnetization, 𝑀𝑆≈ 𝑇−1 in the measured temperature range (as shown in Fig. 3(a)) and, 𝑉𝑎∝ 𝑇−3/2, according to the theory of magnon -driven LSSE, |𝑱𝑺⃗⃗⃗ |∝ 𝐺↑↓ 2𝜋𝑘𝐵 𝐷3/2𝑇5/2|𝛁𝑻⃗⃗⃗⃗⃗ |.42 Considering tanh(𝑡𝑃𝑡 2𝜆𝑃𝑡)≈1 for our case and, 𝜆𝑃𝑡 ∝ 𝑇−1,75 according to the Eqn. 1 , the modified LSSE coefficient becomes 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓=𝑉𝐿𝑆𝑆𝐸 ∆𝑇𝑒𝑓𝑓𝑅𝑦𝐿𝑦∝ (𝜃𝑆𝐻𝑃𝑡𝐺↑↓ 2𝜋𝑘𝐵 𝐷3/2)𝑇3/2.42 As shown in Figs. 7(a)-(c), 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇) varies as 𝑇1.41±0.12, 𝑇1.48±0.08 and 𝑇1.49±0.1 for 𝑡CoFeCrGa= 50,95 and 200 nm, respectively in the measured temperature range , which are in good agreement with the theory of thermally generated magnon -driven interfacial spin pumping mechanism42,65,66. Now, let us understand the origin of ANE in our MgO/CoFeCrGa (𝑡CoFeCrGa) films. The transverse thermoelectric coefficient (𝑆𝑥𝑦) is expressed as, 𝑆𝑥𝑦= [𝛼𝑥𝑦− 𝑆𝑥𝑥𝜎𝑥𝑦 𝜎𝑥𝑥], where 𝜎𝑥𝑥 and 𝜎𝑥𝑦 are the longitudinal and transverse electrical conductivit ies which are defined as,9,18,22 𝜎𝑥𝑥= [ 𝜌𝑥𝑥 (𝜌𝑥𝑥)2 + (𝜌𝑥𝑦)2 ] and 𝜎𝑥𝑦= [ − 𝜌𝑥𝑦 (𝜌𝑥𝑥)2 + (𝜌𝑥𝑦)2 ], respectively . Also, 𝛼𝑥𝑦 and 𝑆𝑥𝑥 are the transverse thermoelectric conductivity and longitudinal Seebeck coefficient, which a ccording 27 to the Mott’s relations can be expressed as , 9,15,76 𝛼𝑥𝑦= 𝜋2𝑘𝐵2𝑇 3𝑒(𝜕𝜎𝑥𝑦 𝜕𝐸) 𝐸=𝐸𝐹and 𝑆𝑥𝑥= 𝜋2𝑘𝐵2𝑇 3𝑒𝜎𝑥𝑥(𝜕𝜎𝑥𝑥 𝜕𝐸) 𝐸=𝐸𝐹, respectively, where 𝐸𝐹 is the Fermi energy . Since ANE and anomalous Hall Effect (AHE) share the common physical origin and the AHE follows the power law connecting the anomalous Hall resistivity , 𝜌𝑥𝑦𝐴𝐻𝐸 with the longitudinal electrical resistivity, 𝜌𝑥𝑥 through the expression, 𝜌𝑥𝑦𝐴𝐻𝐸= 𝜆𝑀𝜌𝑥𝑥𝑛,9 where 𝜆 is the spin -orbit coupling constant and 𝑛 is a constant exponent, the anomalous Nernst coefficient can be expressed as,9,15 𝑆𝑥𝑦𝐴𝑁𝐸= 𝜌𝑥𝑥𝑛−1[𝜋2𝑘𝐵2𝑇 3𝑒(𝜕𝜆 𝜕𝐸) 𝐸=𝐸𝐹−(𝑛−1)𝜆𝑆𝑥𝑥]. (6) When n = 1, the extrinsic skew scattering is the predominant mechanism for the anomalous Nernst /Hall transport, whereas n = 2 indicates the intrinsic Berry curvature or, the extrinsic side jump dominated anomalous Nernst/Hall transport25. Using the T-dependences of ANE voltage, 𝑉𝐴𝑁𝐸(𝑇) and ∆𝑇𝑒𝑓𝑓, we have estimated the T-dependence of the ANE coefficient, 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇)= 𝑉𝐴𝑁𝐸(𝑇) ∆𝑇𝑒𝑓𝑓(𝑇)×(𝐿𝑧 𝐿𝑦) for the MgO/CoFeCrGa (𝑡CoFeCrGa) films , as shown in Figs. 7(d)- (f). Similar to the modified LSSE voltage, 𝑉𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇), 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) for the MgO/ CoFeCrGa (𝑡CoFeCrGa) films for all the three CoFeCrGa film thicknesses also increases as T decreases from room temperature and shows a maximum around 225 K below which it decreases rapidly with further decrease in temperature. Interestingly, 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) for the MgO/CoFeCrGa (200 nm) film increases slowly with decreasing temperature from the room temperature and the maximum around 225 K is much broader in contrast to the films with lower thicknesses. Note that the value s of 𝑆𝑥𝑦𝐴𝑁𝐸 for our MgO/CoFeCrGa (𝑡CoFeCrGa) films are ≈ 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 𝑡CoFeCrGa=50,95 and 200 nm, respectively which are nearly two orders of magnitude higher 28 than that of the bulk polycrystalline sample of CoFeCrGa (≈0.018 μV.K−1 at 300 K)14 but, comparable to that of the magnetic Weyl semimetal Co 2MnGa thin films (≈2−3 μV.K−1 at 300 K )77,78. We fitted the 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) data in the temperature range 125 K≤ 𝑇 ≤200 K for our MgO/CoFeCrGa (𝑡CoFeCrGa) films using Eqn. 6 considering 𝜆, (𝜕𝜆 𝜕𝐸) 𝐸=𝐸𝐹, and n as the fitting parameter s. The best fit was obtained for 𝑛 =0.61 ± 0.02,0.68 ± 0.06 and 0.87 ± 0.05, for 𝑡CoFeCrGa=50,95 and 200 nm, respectively which implies that the origin of ANE in our MgO/CoFeCrGa (𝑡CoFeCrGa) films is dominated by the asymmetric skew scattering of charge carriers below 200 K25. Note that, we have also observed skew -scattering dominated ANE in bulk polycrystalline sample of CoFeCrGa ,14 for which 𝑛≈0.78. Next, let us examine the temperature evolution of the anomalous off -diagonal thermoelectric conductivity, 𝛼𝑥𝑦𝐴𝑁𝐸(𝑇). To determine 𝛼𝑥𝑦𝐴𝑁𝐸(𝑇), we have performed the Hall measurements on the MgO/CoFeCrGa (𝑡CoFeCrGa) films . Figs. 8(a)-(c) present the magnetic field dependence of Hall resistivity 𝜌𝑥𝑦(𝐻) of our MgO/CoFeCrGa (𝑡CoFeCrGa) films for 𝑡CoFeCrGa=50,95 and 200 nm, respectively recorded at few selected temperatures in the range: 125 K ≤𝑇 ≤295 K. By subtra cting the ordinary Hall effect (OHE) contribution from 𝜌𝑥𝑦(𝐻), we determined the T-dependence of the anomalous Hall resistivity 𝜌𝑥𝑦𝐴𝐻𝐸(𝑇). The left - y scale s of Figs. 8(d)-(f) exhibit the T -dependence of the anomalous Hall conductivity, |𝜎𝑥𝑦𝐴𝐻𝐸|= [ 𝜌𝑥𝑦𝐴𝐻𝐸 (𝜌𝑥𝑥)2 + (𝜌𝑥𝑦𝐴𝐻𝐸)2 ] of our MgO/CoFeCrGa (𝑡CoFeCrGa) films for 𝑡CoFeCrGa= 50,95 and 200 nm, respectively . Note that |𝜎𝑥𝑦𝐴𝐻𝐸(𝑇)| for our MgO/CoFeCrGa (𝑡CoFeCrGa) films increases almost linearly with decreasing temperature, unlike UCo 0.8Ru0.2Al for which |𝜎𝑥𝑦𝐴𝐻𝐸| is nearly temperature independent at low temperatures79. This implies that 𝜎𝑥𝑦𝐴𝐻𝐸 for our MgO/CoFeCrGa (95nm) film is strongly dependent on the scattering rate, which further 29 supports that the extrinsic mechanisms ( e.g., asymmetric skew scattering) dominate the transverse thermoelectric response of our sample at low temperatures79. Figu re 8. (a)-(c) Magnetic field dependence of Hall resistivity 𝜌𝑥𝑦(𝐻) of our MgO/ CoFeCrGa (𝑡CoFeCrGa) films for 𝑡CoFeCrGa=50,95 and 200 nm, respectively recorded at few selected temperatures in the range: 125 K ≤𝑇 ≤295 K. (d)-(f) Left y -scale: the temperature dependence of the anomalous Hall conductivity, |𝜎𝑥𝑦𝐴𝐻𝐸| of our MgO/CoFeCrGa (𝑡CoFeCrGa) films for 𝑡CoFeCrGa=50,95 and 200 nm, respectively , right y -scale: corresponding temperature variation s of transverse thermoelectric conductivity 𝛼𝑥𝑦𝐴𝑁𝐸. 30 The right y-scale of Figs. 8(d)-(f) illustrates the temperature variation of 𝛼𝑥𝑦𝐴𝑁𝐸 of our MgO/CoFeCrGa (𝑡CoFeCrGa) films for 𝑡CoFeCrGa=50,95 and 200 nm, respectively , which was obtained by i ncorporating the T-dependences of 𝑆𝑥𝑥, 𝑆𝐴𝑁𝐸, 𝜌𝑥𝑥 and 𝜌𝑥𝑦𝐴𝐻𝐸 in the expression ,20,21,80 𝛼𝑥𝑦𝐴𝑁𝐸= 𝑆𝑥𝑦𝐴𝑁𝐸𝜎𝑥𝑥+𝑆𝑥𝑥𝜎𝑥𝑦𝐴𝐻𝐸= [𝑆𝑥𝑦𝐴𝑁𝐸𝜌𝑥𝑥 − 𝑆𝑥𝑥𝜌𝑥𝑦𝐴𝐻𝐸 (𝜌𝑥𝑥)2 + (𝜌𝑥𝑦𝐴𝐻𝐸)2 ]. It is evident that 𝛼𝑥𝑦𝐴𝑁𝐸(𝑇) for all the films shows a maximum around 225 K, similar to 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇). Note that similar to 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇), 𝛼𝑥𝑦𝐴𝑁𝐸(𝑇) for the MgO/CoFeCrGa (200 nm) film increases slowly with decreasing temperature from the room temperature and the maximum around 225 K is much broader in contrast to the films with lower thicknesses. The value s of 𝛼𝑥𝑦𝐴𝑁𝐸 at room temperature (295 K) for our MgO/CoFeCrGa (𝑡CoFeCrGa) films are 0.55,0.77 and 1.4 A.m−1.K−1 for𝑡CoFeCrGa=50,95 and 200 nm, respectively , which are much smaller than that of non- centrosymmetric Kagome ferromagnet UCo 0.8Ru0.2Al79 (≈15 A.m−1.K−1 at 40 K ), Co2MnGa single crystal18 (≈7 A.m−1.K−1 at 300 K ) but closer to that of Co2MnGa thin films78 (≈ 2 A.m−1.K−1 at 300 K) . Next, we focus on the origin of the maximum in both 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓(𝑇) and 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) centered around 225 K. Note that the occurrence of maximum in both LSSE and ANE signals at the same temperature has been observed in other ferromagnetic metallic films, e.g., mixed valent manganites42, iron oxides71. The maximum in the temperature dependent LSSE signal in the magnetically ordered state is commonly observed in different ferro - and ferrimagnets for example , YIG, La0.7Ca0.3MnO 3 etc., which originates as a consequence of the combined effects of boundary scattering and diffusive inelastic magnon -phonon or magnon -magnon scattering processes together with the reduction of magnon population at low temperatures33,42,81. In YIG, the maximum in the LSSE signal is thickness dependent ; it shifts from ≈ 70 K for bulk YIG slab to ≈ 2 00 K for 1 m YIG film.33 In ferromagnetic metals, extrinsic contributions arising 31 from electron -magnon scattering contributes significantly to the anomalous Nernst thermopower.45 In presence of a temperature gradient and external magnetic field, magnons are excited in the bulk of a ferromagnetic material and these thermally generated magnons transfer spin-angular mo menta to the itinerant electrons via electron -magnon scattering as a result of which the itinerant electrons of the ferromagnetic layer get spin polarized and contribute to the ANE.45 Since the observed ANE in our MgO/CoFeCrGa (𝑡CoFeCrGa) films has dominating contributi on from the extrinsic mechanism , the occurrence of maxima in 𝑆𝑥𝑦𝐴𝑁𝐸(𝑇) around 225 K and the subsequent decrease in 𝑆𝑥𝑦𝐴𝑁𝐸 in our MgO/CoFeCrGa (𝑡CoFeCrGa) films can also be attributed to the diffusive inelastic magnon scatterings and reduced magnon population at low temperatures.45 A decrease in the magnon population at low temperatures also reduces electron -magnon scattering which eventually diminishes the population of the spin-polarized itinerant electrons participating in the skew -scattering process. In case of LSSE, the magnon propagation length (〈𝜉〉) of the ferr omagnetic material also plays vital role in addition to the magnon population. 〈𝜉〉 signifies the critical length scale for thermally -generated magnons to develop a spatial gradient of magnon accumulation inside a ferro magnetic film which is one of the crucial factors that governs spin angular momentum transfer to the adjacent HM layer31,33,82. A decrease in 〈𝜉〉 also suppresses the LSSE signal. It was theoretically shown that 〈𝜉〉 of a magnetic material with lattice constant 𝑎0 is related to the effective anisotropy constant ( 𝐾𝑒𝑓𝑓) and the Gilbert damping parameter ( 𝛼), through the relation34,82 〈𝜉〉= 𝑎0 2𝛼.√𝐽𝑒𝑥 2𝐾𝑒𝑓𝑓, , where 𝐽𝑒𝑥 is the strength of the Heisenberg exchange interaction between nearest neighbors . Since 𝐾𝑒𝑓𝑓= 1 2𝑀𝑆𝐻𝐾𝑒𝑓𝑓, the aforementioned expression can be written as, 〈𝜉〉=𝑎0 2𝛼.√𝐽𝑒𝑥 𝜇0𝑀𝑆𝐻𝐾𝑒𝑓𝑓. Thus, 〈𝜉〉 is inversely proportional to 𝛼 as 32 well as the square -root of (𝑀𝑆𝐻𝐾𝑒𝑓𝑓). This implies that the T-evolution of 〈𝜉〉 is related to that of 𝛼, 𝐻𝐾𝑒𝑓𝑓and 𝑀𝑆. As shown in Fig. 3(a), 𝑀𝑆 for our MgO/CoFeCrGa (95nm) film increases with decreasing temperature. Furthermore, 𝐻𝐾𝑒𝑓𝑓 of our MgO/CoFeCrGa (95nm) film for both IP and OOP configurations ( both 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃) increases with decreasing temperature and the increase is more rapid below ≈ 200 K compare d to the temperature range of 200 K ≤𝑇 ≤ 300 K, as indicated in Fig. 3(f). Notably, similar behavior of 𝐻𝐾𝑒𝑓𝑓 has been observed for the MgO/CoFeCrGa (200nm) film (see Figure S2 ). Therefore, both 𝐻𝐾𝑒𝑓𝑓and 𝑀𝑆 tends to suppress 〈𝜉〉 (and hence , 𝑆𝐿𝑆𝑆𝐸𝑒𝑓𝑓) at low temperatures, especially below ≈ 200 K. To comprehend the role of 𝛼 in 〈𝜉〉 and hence, the LSSE signal at low temperatures, we have investigated the spin - dynamic properties of our MgO/CoFeCrGa (95nm)and MgO/CoFeCrGa (95nm)/Pt(5nm) films by employing the broadband ferromagnetic resonance (FMR) measurements . 3.5. Magnetization dynamics and Gilbert damping Figs. 9(a) and (b) display the field -derivative of microwave (MW) power absorption spectra (𝑑𝑃 𝑑𝐻) as a function of the IP DC magnetic field for various frequencies in the range: 4 GHz ≤ 𝑓 ≤18 GHz recorded at T = 250 K for the MgO/CoFeCrGa (95nm) and MgO/ CoFeCrGa (95nm)/Pt(5nm) films, respectively . To extract the resonance field (𝐻𝑟𝑒𝑠) and linewidth (∆𝐻), we fitted the 𝑑𝑃 𝑑𝐻 lineshapes with a linear combination of symmetric and antisymmetric Lorentzian function derivatives as,83 𝑑𝑃 𝑑𝐻= 𝑃𝑆𝑦𝑚∆𝐻 2(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠) [(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻 2)2 ]2+𝑃𝐴𝑠𝑦𝑚(∆𝐻 2)2 −(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2 [(𝐻𝑑𝑐−𝐻𝑟𝑒𝑠)2+(∆𝐻 2)2 ]2+𝑃0 (7) wher e, 𝑃𝑆𝑦𝑚 and 𝑃𝐴𝑠𝑦𝑚 are the coefficients of the symmetric and antisymmetric Lorentzian derivatives, and 𝑃0 is a constant offset parameter. The fitted curves are represented by solid lines in Figs. 9(a) and (b) . To obtain the temperature evolution of the damping parameter, 𝛼(𝑇) 33 for the MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt(5nm) films , we have fitted the ∆𝐻-f curves with the expression,84 ∆𝐻= ∆𝐻0+4𝜋𝛼 𝛾𝜇0𝑓, where, ∆𝐻0 represents the inhomogeneous broadening , 𝛾 2𝜋= 𝑔𝑒𝑓𝑓 𝜇𝐵 ℏ is the gyromagnetic ratio, 𝜇𝐵 is the Bohr magneton, 𝑔𝑒𝑓𝑓 is the Landé g-factor . Figs. 9(c) shows the ∆𝐻-f curves for the MgO/CoFeCrGa (95nm) film at different temperatures fitted with the aforementioned expression. Clearly, the slope of the ∆𝐻-f curves increases with decreasing temperature which implies increase 𝛼 at low temperatures. In Fig. 9(d), we compare the ∆𝐻-f curves for the MgO/CoFeCrGa(95nm) and MgO/CoFeCrGa (95nm)/Pt(5nm) films recorded at T = 250 K. It is evident that ∆𝐻 for MgO/CoFeCrGa (95nm)/Pt(5nm) is higher than that of MgO/CoFeCrGa (95nm)for all the frequencies, which is because of the loss of spin angular momentum in the CoFeCrGa film as a result of spin pumping and can be expressed as,85 [ ∆𝐻CoFeCrGa /𝑃𝑡− ∆𝐻CoFeCrGa]= 𝐺𝑅↑↓(𝑔𝑒𝑓𝑓𝜇𝐵 2𝛾𝑀𝑆𝑡CoFeCrGa)𝑓, where 𝐺𝑅↑↓ is the real component of the interfacial spin mixing conductance (𝐺↑↓). From the fits, we obtained 𝛼CoFeCrGa = (3.6±0.2)×10−2 and 𝛼CoFeCrGa /𝑃𝑡=(4.12±0.1)×10−2 at 250 K for theMgO/CoFeCrGa (95nm), and MgO/ CoFeCrGa (95nm)/Pt(5nm) films , respectively . Clearly, 𝛼CoFeCrGa /𝑃𝑡> 𝛼CoFeCrGa which is caused by additional damping due to the spin pumping effect85. In Fig. 9(e), we compare 𝛼(𝑇) for theMgO/CoFeCrGa (95nm), and MgO/CoFeCrGa (95nm)/Pt(5nm) films . It is evident that 𝛼CoFeCrGa /𝑃𝑡> 𝛼CoFeCrGa at all the temperatures and both 𝛼CoFeCrGa /𝑃𝑡 and 𝛼CoFeCrGa increase with decrease temperature, especially below 225 K. Such increase in 𝛼 and ∆𝐻 at low temperatures can be primarily attributed to the impurity relaxation mechanisms86–88. Since 〈𝜉〉∝ 1 𝛼, an increase in 𝛼 at low temperatures gives rise to decrease in 〈𝜉〉, and hence, the LSSE signal. The increase s in ∆𝐻0 at low temperatures for both MgO/CoFeCrGa (95nm) MgO/ 34 CoFeCrGa (95nm)/Pt(5nm) films (see right y-scale of Fig. 9(f)) also support the occurrence of impurity relaxation at low temperatures89. Figure 9. (a) and (b) Field-derivative of microwave (MW) power absorption spectra (𝑑𝑃 𝑑𝐻) as a function of the IP DC magnetic field for various frequencies in the range: 4 GHz ≤𝑓 ≤ 18 GHz recorded at 250 K for the MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/ Pt(5nm) films, respectively fitted with Eqn. 7 . (c) The ∆𝐻-f curves for the MgO/ CoFeCrGa (95nm) film at different temperatures fitted with ∆𝐻= ∆𝐻0+4𝜋𝛼 𝛾𝜇0𝑓. (d) The comparison of the ∆𝐻-f curves for the MgO/CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/ Pt(5nm) films recorded at 250 K. (e) Comparison of the temperature dependence of damping parameter 𝛼(𝑇) for theMgO/CoFeCrGa (95nm), and MgO/CoFeCrGa (95nm)/Pt(5nm) films . (f) Right y -scale: temperature dependence of ∆𝐻0 for the MgO/CoFeCrGa (95nm) and 35 MgO/CoFeCrGa (95nm)/Pt films, left y-scale : temperature dependence of the real component of the spin mixing conductance 𝐺𝑅↑↓ for the MgO/CoFeCrGa (95nm)/Pt(5nm) film. To have a quantitative understanding of the T-evolution of spin pumping efficiency in the MgO/CoFeCrGa (95nm)/Pt(5nm) film, we estimated 𝐺𝑅↑↓ using the expression,90 𝐺𝑅↑↓= (2𝑒2 ℎ)(2𝜋𝑀𝑆𝑡CoFeCrGa 𝑔𝑒𝑓𝑓𝜇𝐵)[ 𝛼CoFeCrGa /𝑃𝑡− 𝛼CoFeCrGa] where, 𝐺0=(2𝑒2 ℎ) is the conductance quantum, and found that 𝐺𝑅↑↓≈3.25 × 1014 Ω−1m−2 at 300 K which is close to 𝐺𝑅↑↓ =7.5 × 1014 Ω−1m−2 in YIG/Pt91 and 𝐺𝑅↑↓ =5.7 × 1014 Ω−1m−2 in TmIG/Pt bilayers90. As shown in Fig. 9(d), 𝐺𝑅↑↓ for the MgO/CoFeCrGa (95nm)/Pt(5nm) film increases with decreasing temperature, which is consistent with the phenomenological expression,92 𝐺𝑅↑↓∝(𝑇𝐶−𝑇), where 𝑇𝐶= Curie temperature. Furthermore, to confirm the aforementioned behavior of the temperature evolution of 𝛼, we have repeated the broadband FMR measurements on the MgO/CoFeCrGa (200nm)/Pt(5nm) film. Figs. 10(a) display the magnetic fiel d dependence of the (𝑑𝑃 𝑑𝐻) lineshapes in the range: 6 GHz ≤𝑓 ≤24 GHz recorded at T = 250 K for the MgO/CoFeCrGa (200nm)/Pt film, fitted with Eqn. 7 . To obtain the temperature evolution of the damping parameter, 𝛼(𝑇) we have fitted the ∆𝐻-f curves at different temperatures in the range of 140 K ≤𝑇 ≤300 K with the expression,84 ∆𝐻= ∆𝐻0+4𝜋𝛼 𝛾𝜇0𝑓, as shown in Fig. 10(b). Evidently, the slope of the ∆𝐻-f curves increases with decreasing temperature which implies increase 𝛼 at low temperatures. Moreover, in Fig. 10(c), we show the fitting of the f- 𝐻𝑟𝑒𝑠 curves at T = 250 K using Kittel’s equation for magnetic thin films with IP magnetic field,86 which is expressed as, 𝑓= 𝛾𝜇0 2𝜋√𝐻𝑟𝑒𝑠(𝐻𝑟𝑒𝑠+𝑀𝑒𝑓𝑓), where 𝑀𝑒𝑓𝑓 is the effective magnetization. 36 Figure 10. (a) Field-derivative of (𝑑𝑃 𝑑𝐻) as a function of the IP DC magnetic field for various frequencies in the range: 6 GHz ≤𝑓 ≤24 GHz recorded at T = 250 K for the MgO/ CoFeCrGa (200nm)/Pt fitted with Eqn. 7 . (b) The ∆𝐻-f curves for the MgO/ CoFeCrGa (200nm)/Pt film at different temperatures fitted with ∆𝐻= ∆𝐻0+4𝜋𝛼 𝛾𝜇0𝑓. (c) Fitting of the f vs. the resonance field, 𝐻𝑟𝑒𝑠 using the Kittel’s equation at T = 250 K for the MgO/CoFeCrGa (200nm)/Pt film. (d) Left y scale: temperature dependence of damping parameter 𝛼(𝑇) for theMgO/CoFeCrGa (200nm)/Pt, and r ight y-scale: temperature dependence of ∆𝐻0 for the same . The estimated value of 𝑔𝑒𝑓𝑓=(2.09 ±0.01) at 250 K for the MgO/ CoFeCrGa (200nm)/Pt(5nm) film, which is slightly higher than the free electron value ( 𝑔𝑒𝑓𝑓 = 2.002). Note that 𝑔𝑒𝑓𝑓=(2.046 ±0.01) and (2.048 ±0.02) for the MgO/ CoFeCrGa (95nm) and MgO/CoFeCrGa (95nm)/Pt(5nm) films, respectively at 250 K . Finally, 𝛼(𝑇) for the MgO/CoFeCrGa (200nm)/Pt film is shown on the left y-axis of Fig. 10(d). It is evident that 𝛼(𝑇) increases with decreasing temperature, especially below 225 K similar to what we have observed for the MgO/CoFeCrGa (95nm) and MgO/ 37 CoFeCrGa (95nm)/Pt(5nm) films. This observation furthe r confirms the contribution of 𝛼 towards the observed decrease in the LSSE signal in the CoFeCrGa films below the temperature range of 200 -225 K. 4. CONCLUSIONS In summary, we present a comprehensive investigation of the temperature ANE and intrinsic longitudinal spin Seebeck effect (LSSE) in the quaternar y Heusler alloy based SGS thin films of CoFeCrGa grown on MgO substrates. We found that the anomalous Nernst coefficient for the MgO/CoFeCrGa (95 nm) film is ≈ 1.86 μV.K−1 at room temperature which is much higher than the bulk polycrystalline sample of CoFeCrGa (≈0.018 μV.K−1 at 300 K) but comparable to that of the magnetic Weyl semimetal Co 2MnGa thin films ( ≈2−3 μV.K−1 at 300 K ). Furthermore, the LSSE coefficient for our MgO/CoFeCrGa (95nm)/Pt(5nm) heterostructure is ≈20.5 nV.K−1.Ω−1 at 295 K which is twice larger than that of the half- metallic ferromagnet ic La0.7Sr0.3MnO 3 thin films (≈9 nV.K−1.Ω−1 at room temperature ). We have show n that both ANE and LSSE coefficients follow identical temperatu re dependences and exhibit a maximum ≈225 K which is understood as the combined effects of inelastic magnon scatterings and reduced magnon population at low temperatures . Our analys es not only indicate d that the extrinsic skew scattering is the dominating mechanism for ANE in these films but also, provide d critical insight s into the functional form of the observed temperature dependent LSSE at low temperatures . Furthermore, by employing radio frequency transverse susceptibility and broadband ferromagnetic r esonance in combination with the LSSE measurements, we have establish ed a correlation among the observed LSSE signal, magnetic anisotropy and Gilbert damping of the CoFeCrGa thin films which will be beneficial for fabricating tunable and highly efficient s pincaloritronic nanodevices. We believe that our findings will also attract the attention of materials science and spintronics community for 38 further exploration of different Heusler alloys based magnetic thin films and heterostructures co-exhibiting multip le thermo -spin effects with promising efficiencies. ACKNOWLEDGEMENTS HS and MHP acknowledge support from the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award No. DE -FG02 - 07ER46438 . HS thanks the Alexander von Humboldt foundation for a research award and also acknowledges a visiting professorship at IIT Bombay. D.A.A. acknowledges the support of the National Science Foundation under Grant No. ECCS -1952957. DD and RC acknowledge the financial assistance received from DST Nanomission project (DST/NM/TUE/QM -11/2019). SUPPORTING INFORMATION Magnetometry, temperature dependence of electrical resistivity, magnetic field and temperature dependences of transverse susceptibility, magnetic fiel d dependence of ANE and LSSE voltages for the MgO/CoFeCrGa (200 nm) and MgO/CoFeCrGa (50 nm) films. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. 39 REFERENCES (1) Bauer, G. E. 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Lett. 2015 , 106 (5), 52405. 50 Supplementary Information Large thermo -spin effects in Heusler alloy based spin -gapless semiconductor thin film s Amit Chanda1*, Deepika Rani2, Derick DeT ellem1, Noha Alzahrani1, Dario A. Arena1, Sarath Witanachchi1, Ratnamala Chatterjee2, Manh -Huong Phan1 and Hari haran Srikanth1* 1 Department of Physics, University of South Florida, Tampa FL 33620 2 Physics Department, Indian Institute of Technology Delhi, New Delhi - 110016 *Corresponding authors: achanda@usf.edu ; sharihar@usf.edu 51 Figure S 1. (a) and (b) Magnetic field dependence of magnetization, 𝑀(𝐻) of our MgO/CoFeCrGa (200nm) and MgO/CoFeCrGa (50nm)/Pt films. respectively measured at selected temperatures in the range: 125 K ≤𝑇 ≤300 K in presence of an in -plane sweeping magnetic field , (c) and (d) temperature dependence of the saturation magnetization, MS for the same films, respectively. 52 Figure S2. (a) Schematic illustration of the transverse susceptibilbity (TS) measurements. T he bipolar field scan s (+𝐻𝐷𝐶𝑚𝑎𝑥−𝐻𝐷𝐶𝑚𝑎𝑥 +𝐻𝐷𝐶𝑚𝑎𝑥) of the field dependence of TS, ∆𝜒𝑇 𝜒𝑇(𝐻𝐷𝐶) for the MgO/CoFeCrGa (200nm)/Pt(5nm) film measured at T = 300 K and 20 K for the (b) IP (HDC is parallel to the film surface) and (c) OOP ( HDC is perpendicular to the film surface) configurations . (d) Temperature variations of the effective anisotropy fields: 𝐻𝐾𝐼𝑃 and 𝐻𝐾𝑂𝑂𝑃for the MgO/CoFeCrGa (200nm)/Pt film. 53 Figure S3. Temperature dependence of longitudinal resistivity, 𝜌𝑥𝑥(𝑇) for the (a) MgO/ CoFeCrGa (200nm), (b) MgO/CoFeCrGa (95nm) and (c) MgO/CoFeCrGa (200nm) films, respectively in the temperature range: 10 K ≤𝑇 ≤300 K. 54 Figure S4. Temperature variations of thermal conductivity of (a) Apiezon N -grease,1 (b) MgO crystal2 and (c) bulk CoFeCrGa (measured) using the thermal transport option (TTO) of the PPMS. (d) Schematic illustration of the heat flow through N -grease/MgO substrate/CoFeCrGa film/N -grease considering the 4 -slab model. (e) The temperture variation of the effective temperature difference across the MgO/CoFeCrGa(95nm) film estimated from the expression,3 ∆𝑇𝑒𝑓𝑓= ∆𝑇CoFeCrGa = 𝛥𝑇 [1+𝜅𝐶𝑜𝐹𝑒𝐶𝑟𝐺 𝑎 𝑡𝐶𝑜𝐹𝑒𝐶𝑟𝐺𝑎(2𝑡𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 𝜅𝑁−𝐺𝑟𝑒𝑎𝑠𝑒 + 𝑡𝑀𝑔𝑂 𝜅𝑀𝑔𝑂)] 55 Figure S 5. (a) and ( b) The magnetic field dependence of the ANE voltage, 𝑉𝐴𝑁𝐸(𝐻) and ISHE - induced in-plane voltage, 𝑉𝐼𝑆𝐻𝐸(𝐻) measured on the MgO/CoFeCrGa (200nm) and MgO/ CoFeCrGa (200nm)/Pt films, respectively for different values of the temperature difference between the hot ( 𝑇ℎ𝑜𝑡) and cold ( 𝑇𝑐𝑜𝑙𝑑) copper blocks, ∆𝑇= (𝑇ℎ𝑜𝑡−𝑇𝑐𝑜𝑙𝑑) in the range: +5 K≤ ∆𝑇 ≤+18 K at a fixed average sample temperature 𝑇= 𝑇ℎ𝑜𝑡+𝑇𝑐𝑜𝑙𝑑 2 = 295 K. (c) and (d) The ∆𝑇dependence of the background -corrected ANE voltage, 𝑉𝐴𝑁𝐸(Δ𝑇)= [𝑉𝐴𝑁𝐸(+𝜇0𝐻𝑚𝑎𝑥, Δ𝑇)−𝑉𝐴𝑁𝐸( −𝜇0𝐻𝑚𝑎𝑥, Δ𝑇) 2] and the background -corrected (ANE+ LSSE ) volta ge, 𝑉𝐴𝑁𝐸+𝐿𝑆𝑆𝐸(Δ𝑇)= [𝑉𝐼𝑆𝐻𝐸(+𝜇0𝐻𝑚𝑎𝑥, Δ𝑇)−𝑉𝐼𝑆𝐻𝐸( −𝜇0𝐻𝑚𝑎𝑥, Δ𝑇) 2], respectively. 56 Figure S6. (a) and (b) 𝑉𝐴𝑁𝐸(𝐻) hysteresis loops measured at selected average sample temperatures in the temperature range: 125 K≤ ∆𝑇 ≤295 K for a fixed value of Δ𝑇 = +1 5 K on the MgO/CoFeCrGa (200nm) and MgO/CoFeCrGa (500nm) films, respectively . (c) and (d) 𝑉𝐼𝑆𝐻𝐸(𝐻) hysteresis loops measured at selected average sample temperatures in the temperature range: 125 K≤ ∆𝑇 ≤295 K for a fixed value of Δ𝑇 = +1 5 K on the MgO/CoFeCrGa (200nm)/Pt and MgO/CoFeCrGa (50nm)/Pt films, respectively . 57 Reference s (1) Ashworth, T.; Loomer, J. E.; Kreitman, M. M. Thermal Conductivity of Nylons and Apiezon Greases. In Advances in Cryogenic Engineering ; Springer, 1973; pp 271 –279. (2) Grimvall, G. Thermophysical Properties of Materials ; Elsevier, 1999. (3) De, A.; Ghosh, A .; Mandal, R.; Ogale, S.; Nair, S. Temperature Dependence of the Spin Seebeck Effect in a Mixed Valent Manganite. Phys. Rev. Lett. 2020 , 124 (1), 17203.

2023-08-18

Recently, Heusler alloys-based spin gapless semiconductors (SGSs) with high Curie temperature (TC) and sizeable spin polarization have emerged as potential candidates for tunable spintronic applications. We report comprehensive investigation of the temperature dependent ANE and intrinsic longitudinal spin Seebeck effect (LSSE) in CoFeCrGa thin films grown on MgO substrates. Our findings show the anomalous Nernst coefficient for the MgO/CoFeCrGa (95 nm) film is $\cong 1.86$ micro V/K at room temperature which is nearly two orders of magnitude higher than that of the bulk polycrystalline sample of CoFeCrGa (= 0.018 micro V/K) but comparable to that of the magnetic Weyl semimetal Co2MnGa thin film (2-3 micro V/K). Furthermore, the LSSE coefficient for our MgO/CoFeCrGa(95nm)/Pt(5nm) heterostructure is $\cong 20.5$ $\mu$V/K/$\Omega$ at room temperature which is twice larger than that of the half-metallic ferromagnetic La$_{0.7}$Sr$_{0.3}$MnO$_3$ thin films ($\cong$ 20.5 $\mu$V/K/$\Omega$). We show that both ANE and LSSE coefficients follow identical temperature dependences and exhibit a maximum at $\cong$ 225 K which is understood as the combined effects of inelastic magnon scatterings and reduced magnon population at low temperatures. Our analyses not only indicate that the extrinsic skew scattering is the dominating mechanism for ANE in these films but also provide critical insights into the functional form of the observed temperature dependent LSSE at low temperatures. Furthermore, by employing radio frequency transverse susceptibility and broadband ferromagnetic resonance in combination with the LSSE measurements, we establish a correlation among the observed LSSE signal, magnetic anisotropy and Gilbert damping of the CoFeCrGa thin films, which will be beneficial for fabricating tunable and highly efficient Heusler alloys based spincaloritronic nanodevices.

Large thermo-spin effects in Heusler alloy based spin-gapless semiconductor thin films

2308.09843v1

Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation Panchi Lia, Changjian Xiea, Rui Dua,b,, Jingrun Chena,b,, Xiao-Ping Wangc, aSchool of Mathematical Sciences, Soochow University, Suzhou, 215006, China. bMathematical Center for Interdisciplinary Research, Soochow University, Suzhou, 215006, China. cDepartment of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China A B S T R A C T Micromagnetic simulation is an important tool to study various dynamic behaviors of magnetic order in ferromagnetic materials. The underlying model is the Landau-Lifshitz- Gilbert equation, where the magnetization dynamics is driven by the gyromagnetic torque term and the Gilbert damping term. Numerically, considerable progress has been made in the past decades. One of the most popular methods is the Gauss-Seidel projection method developed by Xiao-Ping Wang, Carlos Garc a-Cervera, and Weinan E in 2001. It rst solves a set of heat equations with constant coecients and updates the gyromagnetic term in the Gauss-Seidel manner, and then solves another set of heat equations with constant coecients for the damping term. Afterwards, a projection step is applied to preserve the length con- straint in the pointwise sense. This method has been veried to be unconditionally stable numerically and successfully applied to study magnetization dynamics under various controls. In this paper, we present two improved Gauss-Seidel projection methods with uncondi- tional stability. The rst method updates the gyromagnetic term and the damping term simultaneously and follows by a projection step. The second method introduces two sets of approximate solutions, where we update the gyromagnetic term and the damping term simul- taneously for one set of approximate solutions and apply the projection step to the other set of approximate solutions in an alternating manner. Compared to the original Gauss-Seidel projection method which has to solve heat equations 7 times at each time step, the improved methods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time and second-order accuracy in space are veried by examples in both 1D and 3D. In addi- tion, unconditional stability with respect to both the grid size and the damping parameter is conrmed numerically. Application of both methods to a realistic material is also presented with hysteresis loops and magnetization proles. Compared with the original method, the recorded running times suggest that savings of both methods are about 2 =7 and 4=7 for the same accuracy requirement, respectively. Keywords: Landau-Lifshitz-Gilbert equation, Gauss-Seidel projection method, unconditional stability, micromagnetic simulation 2000 MSC: 35Q99, 65Z05, 65M06 1. Introduction In ferromagnetic materials, the intrinsic magnetic order, known as magnetization M= (M1;M2;M3)T, is modeled by the following Landau-Lifshitz-Gilbert (LLG) equation [1, 2, 3] @M @t= MH  MsM(MH) (1) Corresponding authors e-mail: LiPanchi1994@163.com (Panchi Li), 20184007005@stu.suda.edu.cn (Changjian Xie), durui@suda.edu.cn (Rui Du), jingrunchen@suda.edu.cn (Jingrun Chen), mawang@ust.hk (Xiao-Ping Wang) 1arXiv:1907.11853v1 [math.NA] 27 Jul 2019with the gyromagnetic ratio and jMj=Msthe saturation magnetization. On the right- hand side of (1), the rst term is the gyromagnetic term and the second term is the Gilbert damping term with the dimensionless damping coecient [2]. Note that the gyromagnetic term is a conservative term, whereas the damping term is a dissipative term. The local eld H=F Mis computed from the Landau-Lifshitz energy functional F[M] =1 2Z A M2sjrMj2+ M Ms 20He
M dx+0 2Z R3jrUj2dx; (2) whereAis the exchange constant,A M2sjrMjis the exchange interaction energy;  M Ms is the anisotropy energy, and for simplicity the material is assumed to be uniaxial with  M Ms =Ku M2s(M2 2+M2 3) withKuthe anisotropy constant; 20He
Mis the Zeeman energy due to the external eld with 0the permeability of vacuum. is the volume occupied by the material. The last term in (2) is the energy resulting from the eld induced by the magnetization distribution inside the material. This stray eld Hs=rUwhereU(x) satises U(x) =Z rN(xy)
M(y)dy; (3) whereN(xy) =1 41 jxyjis the Newtonian potential. For convenience, we rescale the original LLG equation (1) by changes of variables t! (0 Ms)1tandx!LxwithLthe diameter of . Dene m=M=Msandh=MsH. The dimensionless LLG equation reads as @m @t=mhm(mh); (4) where h=Q(m2e2+m3e3) +
m+he+hs (5) with dimensionless parameters Q=Ku=(0M2 s) and=A=(0M2 sL2). Here e2= (0;1;0), e3= (0;0;1). Neumann boundary condition is used @m @j@ = 0; (6) whereis the outward unit normal vector on @ . The LLG equation is a weakly nonlinear equation. In the absence of Gilbert damping, = 0, equation (4) is a degenerate equation of parabolic type and is related to the sympletic ow of harmonic maps [4]. In the large damping limit, !1 , equation (4) is related to the heat ow for harmonic maps [5]. It is easy to check that jmj= 1 in the pointwise sense in the evolution. All these properties possesses interesting challenges for designing numerical methods to solve the LLG equation. Meanwhile, micromagnetic simulation is an important tool to study magnetization dynamics of magnetic materials [3, 6]. Over the past decades, there has been increasing progress on numerical methods for the LLG equation; see [7, 8, 9] 2for reviews and references therein. Finite dierence method and nite element method have been used for the spatial discretization. For the temporal discretization, there are explicit schemes such as Runge-Kutta methods [10, 11]. Their stepsizes are subject to strong stability constraint. Another issue is that the length of magnetization cannot be preserved and thus a projection step is needed. Implicit schemes [12, 13, 14] are unconditionally stable and usually can preserve the length of magne- tization automatically. The diculty of implicit schemes is how to solve a nonlinear system of equations at each step. Therefore, semi-implicit methods [15, 16, 17, 18, 19] provide a com- promise between stability and the dicult for solving the equation at each step. A projection step is also needed to preserve the length of magnetization. Among the semi-implicit schemes, the most popular one is the Gauss-Seidel projection method (GSPM) proposed by Wang, Garc a-Cervera, and E [15, 18]. GSPM rst solves a set of heat equations with constant coecients and updates the gyromagnetic term in the Gauss-Seidel manner, and then solves another set of heat equations with constant coecients for the damping term. Afterwards, a projection step is applied to preserve the length of mag- netization. GSPM is rst-order accurate in time and has been veried to be unconditionally stable numerically. In this paper, we present two improved Gauss-Seidel projection methods with uncondi- tional stability. The rst method updates the gyromagnetic term and the damping term simultaneously and follows by a projection step. The second method introduces two sets of approximate solutions, where we update the gyromagnetic term and the damping term simul- taneously for one set of approximate solutions and apply the projection step to the other set of approximate solutions in an alternating manner. Compared to the original Gauss-Seidel projection method, which solves heat equations 7 times at each time step, the improved methods solve heat equations 5 times and 3 times, respectively. First-order accuracy in time and second-order accuracy in space are veried by examples in both 1D and 3D. In addi- tion, unconditional stability with respect to both the grid size and the damping parameter is conrmed numerically. Application of both methods to a realistic material is also presented with hysteresis loops and magnetization proles. Compared with the original method, the recorded running times suggest that savings of both methods are about 2 =7 and 4=7 for the same accuracy requirement, respectively. The rest of the paper is organized as follows. For completeness and comparison, we rst introduce GSPM in Section 2. Two improved GSPMs are presented in Section 3. Detailed numerical tests are given in Section 4, including accuracy check and eciency check in both 1D and 3D, unconditional stability with respect to both the grid size and the damping parameter, hysteresis loops, and magnetization proles. Conclusions are drawn in Section 5. 32. Gauss-Seidel projection method for Landau-Lifshitz-Gilbert equation Before the introduction of the GSPM [15, 18], we rst use the nite dierence method for spatial discretization. Figure 1 shows a schematic picture of spatial grids in 1D. Let i= 0;1;


;M;M + 1,j= 0;1;


;N;N + 1, andk= 0;1;


;K;K + 1 be the indices of grid points in 3D. 0 1𝑥−1 2𝑥1 2𝑥𝑁−1 2𝑥𝑁+1 2𝑥3 2𝑥𝑁−3 2 Fig. 1. Spatial grids in 1D. Nodes x1 2andxN+1 2are ghost points. Second-order centered dierence for
mreads as
hmi;j;k=mi+1;j;k2mi;j;k+mi1;j;k
x2 +mi;j+1;k2mi;j;k+mi;j1;k
y2 +mi;j;k+12mi;j;k+mi;j;k1
z2; (7) where mi;j;k=m((i1 2)
x;(j1 2)
y;(k1 2)
z). For the Neumann boundary condition, a second-order approximation yields m0;j;k=m1;j;k;mM;j;k =mM+1;j;k; j = 1;


;N;k = 1;


;K; mi;0;k=mi;1;k;mi;N;k =mi;N+1;k; i= 1;


;M;k = 1;


;K; mi;j;0=mi;j;1;mi;j;K =mi;j;K +1; i= 1;


;M;j = 1;


;N: To illustrate the main ideas, we rst consider the following simplied equation mt=m
mm(m
m); which can be rewritten as mt=m
mm(m

m) +
m: (8) We split (8) into two equations mt=m
m; (9) mt=
m: (10) However, (9) is still nonlinear. Therefore, we consider a fractional step scheme to solve (9) mmn
t=
hm mn+1=mnmnm 4or mn+1=mnmn(I
t
h)1mn; whereIis the identity matrix. This scheme is subject to strong stability constraint, and thus the implicit Gauss-Seidel scheme is introduced to overcome this issue. Let gn i= (I
t
h)1mn i; i= 1;2;3: (11) We then have 0 @mn+1 1 mn+1 2 mn+1 31 A=0 @mn 1+ (gn 2mn 3gn 3mn 2) mn 2+ (gn 3mn+1 1gn+1 1mn 3) mn 3+ (gn+1 1mn+1 2gn+1 2mn+1 1)1 A: (12) This scheme solve (9) with unconditional stability. (10) is linear heat equation which can be solved easily. However, the splitting scheme (9) - (10) cannot preserve jmj= 1, and thus a projection step needs to be added. For the full LLG equation (4), the GSPM works as follows. Dene h=
m+^f; (13) where ^f=Q(m2e2+m3e3) +he+hs. The original GSPM [15] solves the equation (4) in three steps: Implicit Gauss-Seidel gn i= (I
t
h)1(mn i+
t^fn i); i= 2;3; g i= (I
t
h)1(m i+
t^fn i); i= 1;2; (14) 0 @m 1 m 2 m 31 A=0 @mn 1+ (gn 2mn 3gn 3mn 2) mn 2+ (gn 3m 1g 1mn 3) mn 3+ (g 1m 2g 2m 1)1 A: (15) Heat ow without constraints ^f=Q(m 2e2+m 3e3) +he+hn s; (16) 0 @m 1 m 2 m 31 A=0 @m 1+
t(
hm 1+^f 1) m 2+
t(
hm 2+^f 2) m 3+
t(
hm 3+^f 3)1 A: (17) Projection onto S2 0 @mn+1 1 mn+1 2 mn+1 31 A=1 jmj0 @m 1 m 2 m 31 A: (18) Here the numerical stability of the original GSPM [15] was founded to be independent of gridsizes but depend on the damping parameter . This issue was solved in [18] by replacing (14) and (16) with g i= (I
t
h)1(m i+
t^f i); i= 1;2; 5and ^f=Q(m 2e2+m 3e3) +he+h s; respectively. Update of the stray eld is done using fast Fourier transform [15]. It is easy to see that the GSPM solves 7 linear systems of equations with constant coecients and updates the stray eld using FFT 6 times at each step. 3. Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation Based on the description of the original GSPM in Section 2, we introduce two improved GSPMs for LLG equation. The rst improvement updates both the gyromagnetic term and the damping term simultaneously, termed as Scheme A. The second improvement introduces two sets of approximate solution with one set for implicit Gauss-Seidel step and the other set for projection in an alternating manner, termed as Scheme B. Details are given in below. 3.1. Scheme A The main improvement of Scheme A over the original GSPM is the combination of (13) - (17), or (9) - (10). Implicit-Gauss-Seidel gn i= (I
t
h)1(mn i+
t^fn i); i= 1;2;3; g i= (I
t
h)1(m i+
t^f i); i= 1;2; (19) 0 @m 1 m 2 m 31 A=0 @mn 1(mn 2gn 3mn 3gn 2)(mn 1gn 1+mn 2gn 2+mn 3gn 3)mn 1+gn 1 mn 2(mn 3g 1m 1gn 3)(m 1g 1+mn 2gn 2+mn 3gn 3)mn 2+gn 2 mn 3(m 1g 2m 2g 1)(m 1g 1+m 2g 2+mn 3gn 3)mn 3+gn 31 A:(20) Projection onto S2 0 @mn+1 1 mn+1 2 mn+1 31 A=1 jmj0 @m 1 m 2 m 31 A: (21) It is easy to see that Scheme A solves 5 linear systems of equations with constant coecients and uses FFT 5 times at each step. 3.2. Scheme B The main improvement of Scheme B over Scheme A is the introduction of two sets of approximate solutions, one for (19) - (20) and the other for (21) and the update of these two sets of solutions in an alternating manner. Given the initialized g0 g0 i= (I
t
h)1(m0 i+
t^f0 i); i= 1;2;3; (22) Scheme B works as follows 6Implicit Gauss-Seidel gn+1 i= (I
t
h)1(m i+
t^f i); i= 1;2;3 (23) m 1=mn 1(mn 2gn 3mn 3gn 2)(mn 1gn 1+mn 2gn 2+mn 3gn 3)mn 1+ ((mn 1)2+ (mn 2)2+ (mn 3)2)gn 1 m 2=mn 2(mn 3gn+1 1m 1gn 3)(m 1gn+1 1+mn 2gn 2+mn 3gn 3)mn 2+ ((m 1)2+ (mn 2)2+ (mn 3)2)gn 2 m 3=mn 3(m 1gn+1 2m 2gn+1 1)(m 1gn+1 1+m 2gn+1 2+mn 3gn 3)mn 3+ ((m 1)2+ (m 2)2+ (mn 3)2)gn 3 (24) Projection onto S2 0 @mn+1 1 mn+1 2 mn+1 31 A=1 jmj0 @m 1 m 2 m 31 A: (25) Here one set of approximate solution fmgis updated in the implicit Gauss-Seidel step and the other set of approximate solution fmn+1gis updated in the projection step. Note that (23) is dened only for fmgwhich can be used in two successive temporal steps, and thus only 3 linear systems of equations with constant coecients are solved at each step and 3 FFT executions are used for the stray eld. The length of magnetization can be preserved in the time evolution. The computational cost of GSPM and its improvements comes from solving the linear systems of equations with constant coecients. To summarize, we list the number of linear systems of equations to be solved and the number of FFT executions to be used at each step for the original GSPM [18], Scheme A, and Scheme B in Table 1. The savings represent the ratio between costs of two improved schemes over that of the original GSPM. GSPM Scheme Number of linear systems Saving Execution of FFT Saving Original 7 0 4 0 Scheme A 5 2=7 3 1=4 Scheme B 3 4=7 3 1=4 Table 1. The number of linear systems of equations to be solved and the number of FFT executions to be used at each step for the original GSPM [18], Scheme A, and Scheme B. The savings represent the ratio between costs of two improved schemes over that of the original GSPM. 4. Numerical Experiments In this section, we compare the original GSPM [15, 18], Scheme A, and Scheme B via a series of examples in both 1D and 3D, including accuracy check and eciency check, uncon- ditional stability with respect to both the grid size and the damping parameter, hysteresis 7loops, and magnetization proles. For convenience, we dene ratioi=Time(GSPM)Time(Scheme i) Time(GSPM); fori= A and B, which quanties the improved eciency of Scheme A and Scheme B over the original GSPM [15, 18]. 4.1. Accuracy Test Example 4.1 (1D case). In 1D, we choose the exact solution over the unit interval = [0;1] me= (cos(x) sin(t);sin(x) sin(t);cos(t)); which satises mt=mmxxm(mmxx) +f with x=x2(1x)2, and f=met+memexx+me(memexx). Parameters are = 0:00001 andT= 5:0e2. We rst show the error kmemhk1withmhbeing the numerical solution with respect to the temporal stepsize
tand the spatial stepsize
x. As shown in Figure 2(a) and Fig- ure 2(c), suggested by the least squares tting, both rst-order accuracy in time and second- order accuracy in space are observed. Meanwhile, we record the CPU time as a function of accuracy (error) by varying the temporal stepsize and the spatial stepsize in Figure 2(b) and Figure 2(d), Table 2 and Table 3, respectively. In addition, from Table 2 and Table 3, the saving of Scheme A over GSPM is about 2=7, which equals 15=7, and the saving of Scheme B over GSPM is about 4=7, respectively. This observation is in good agreement with the number of linear systems being solved at each step for these three methods, as shown in Table 1. XXXXXXXXXXCPU time
tT/1250 T/2500 T/5000 T/10000 Reference GSPM 7.7882e-01 1.5445e+00 3.1041e+00 6.2196e+00 - Scheme A 4.8340e-01 9.9000e-01 2.0527e+00 4.4917e+00 - Scheme B 3.3010e-01 6.3969e-01 1.2281e+00 2.5510e+00 - ratio-A 0.38 0.36 0.34 0.28 0.29(2/7) ratio-B 0.58 0.59 0.60 0.59 0.57(4/7) Table 2. Recorded CPU time in 1D with respect to the approximation error when only
tis varied and
x= 1=100. Example 4.2 (3D case). In 3D, we choose the exact solution over = [0;2][0;1][0;0:2] me= (cos(xyz) sin(t);sin(xyz) sin(t);cos(t)); which satises mt=m
mm(m
m) +f 8log(∆t)-12.5 -12 -11.5 -11 -10.5 -10log(error) -12.5-12-11.5-11-10.5-10 GSPM Scheme A Scheme B(a) Temporal accuracy log(error)-12.5 -12 -11.5 -11 -10.5 -10log(time) -1.5-1-0.500.511.52 GSPM Scheme A Scheme B (b) CPU time versus approximation error (
t) log(∆x)-5.1 -5 -4.9 -4.8 -4.7 -4.6log(error) -16-15.9-15.8-15.7-15.6-15.5-15.4-15.3-15.2-15.1-15 GSPM Scheme A Scheme B (c) Spatial accuracy log(error)-16 -15.8 -15.6 -15.4 -15.2 -15log(time) 77.588.599.5 GSPM Scheme A Scheme B (d) CPU time versus approximation error (
x) Fig. 2. Approximation error and CPU time in 1D. (a) Approximation error as a function of the temporal step size; (b) CPU time as a function of the approximation error when
tis varied and
xis xed; (c) Approximation error as a function of the spatial step size; (d) CPU time as a function of the approximation error when
xis varied and
tis xed. with x=x2(1x)2,y=y2(1y)2,z=z2(1z)2andf=met+me
me+me(me
me). Parameters are T= 1:0e05and= 0:01. Like in the 1D case, we rst show the error kmemhk1withmhbeing the numerical solution with respect to the temporal stepsize
tand the spatial stepsize
x. As shown in Figure 3(a) and Figure 3(c), suggested by the least squares tting, both rst-order accuracy in time and second-order accuracy in space are observed. Meanwhile, we record the CPU time as a function of accuracy (error) by varying the temporal stepsize and the spatial stepsize in Figure 3(b) and Figure 3(d), Table 4 and Table 5, respectively. In addition, from Table 4 and Table 5, the saving of Scheme A over GSPM is about 2=7, and the saving of Scheme B over GSPM is about 4=7, respectively. This observation is in good agreement with the number of linear systems being solved at each step for these three methods, as shown in Table 1. It worths mentioning that all these three methods are tested to be unconditionally stable with respect to the spatial gridsize and the temporal stepsize. 9XXXXXXXXXXCPU time
x1/100 1/120 1/140 1/160 Reference GSPM 3.3752e+03 5.2340e+03 9.0334e+03 1.0495e+04 - Scheme A 2.4391e+03 3.7175e+03 6.5149e+03 8.0429e+03 - Scheme B 1.4740e+03 2.2448e+03 3.9152e+03 4.8873e+03 - ratio-A 0.28 0.29 0.28 0.23 0.29(2/7) ratio-B 0.56 0.57 0.57 0.53 0.57(4/7) Table 3. Recorded CPU time in 1D with respect to the approximation error when only
xis varied and
t= 1:0e8. XXXXXXXXXXCPU time
tT/10 T/20 T/40 T/80 Reference GSPM 3.5188e+01 6.8711e+01 1.4146e+02 2.9769e+02 - Scheme A 2.3015e+01 4.3920e+01 8.6831e+01 1.7359e+02 - Scheme B 1.3984e+01 2.6313e+01 5.1928e+01 1.0415e+02 - ratio-A 0.35 0.36 0.39 0.42 0.29(2/7) ratio-B 0.60 0.62 0.63 0.65 0.57(4/7) Table 4. Recorded CPU time in 3D with respect to the approximation error when only
tis varied and the spatial mesh is 1286410. 4.2. Micromagnetic Simulations To compare the performance of Scheme A and Scheme B with GSPM, we have carried out micromagnetic simulations of the full LLG equation with realistic material parameters. In all our following simulations, we consider a thin lm ferromagnet of size = 1 m1m 0:02m with the spatial gridsize 4 nm 4 nm4 nm and the temporal stepsize
t= 1 picosecond. The demagnetization eld (stray eld) is calculated via FFT [15, 18]. 4.2.1. Comparison of hysteresis loops The hysteresis loop is calculated in the following way. First, a positive external eld H0=0His applied and the system is allowed to reach a stable state. Afterwards, the external eld is reduced by a certain amount and the system is relaxed to a stable state again. The process continues until the external eld attains a negative eld of strength H0. Then the external eld starts to increase and the system relaxes until the initial applied external eld H0is approached. In the hysteresis loop, we can monitor the magnetization dynamics and plot the average magnetization at the stable state as a function of the strength of the external eld. The stopping criterion for a steady state is that the relative change of the total energy is less than 107. The applied eld is parallel to the xaxis. The initial state we take is the uniform state and the damping parameter = 0:1. In Figure 4, we compare the average magnetization in the hysteresis loop simulated by GSPM, Scheme A and Scheme B. Proles of the average magnetization of these three methods are in quantitative agreements with approximately the same switch eld 9 ( 0:4) mT. 4.2.2. Comparison of magnetization proles It is tested that GSPM in [15] was unstable with a very small damping parameter and was resolved in [18]. This section is devoted to the unconditional stability of Scheme A and 10log(∆t)-14 -13.5 -13 -12.5 -12 -11.5log(error) -14-13.5-13-12.5-12-11.5 GSPM Scheme A Scheme B(a) Temporal accuracy log(error)-14 -13.5 -13 -12.5 -12 -11.5log(time) 2.533.544.555.56 GSPM Scheme A Scheme B (b) CPU time versus approximation error (
t) The spatial step size log( ∆x)-2.5 -2.4 -2.3 -2.2 -2.1 -2 -1.9 -1.8 -1.7log(error) -32.5-32-31.5-31 GSPM Scheme A Scheme B (c) Spatial accuracy log(error)-32.5 -32 -31.5 -31log(time) 22.533.544.555.56 GSPM Scheme A Scheme B (d) CPU time versus approximation error (
x) Fig. 3. Approximation error and CPU time in 3D. (a) Approximation error as a function of the temporal step size; (b) CPU time as a function of the approximation error when
tis varied and
x=
y=
zis xed; (c) Approximation error as a function of the spatial step size; (d) CPU time as a function of the approximation error when space is varied uniformly and
tis xed. Scheme B with respect to . We consider a thin lm ferromagnet of size 1 m1m0:02m with the spatial gridsize 4 nm 4 nm4 nm and the temporal stepsize is 1 picosecond. Following [18], we consider the full LLG equation with = 0:1 and= 0:01 and without the external eld. The initial state is m0= (0;1;0) ifx2[0;Lx=5][[4Lx=5;Lx] and m0= (1;0;0) otherwise. The nal time is 10 ns. In Figures 5 to 7, we present a color plot of the angle between the in-plane magnetization and the xaxis, and an arrow plot of the in-plane magnetization for the original GSPM [15], Scheme A, and Scheme B, respectively. In these gures, = 0:1 is presented in the top row and = 0:01 is presented in the bottom row; a color plot of the angle between the in-palne magnetization and the xaxis is presented in the left column and an arrow plot of the in-plane magnetization is presented in the right column. 5. Conclusion In this paper, based on the original Gauss-Seidel projection methods, we present two improved Gauss-Seidel projection methods with the rst-order accuracy in time and the second-order accuracy in space. The rst method updates the gyromagnetic term and the 11XXXXXXXXXXCPU time
x1/6 1/8 1/10 1/12 Reference GSPM 2.1066e+01 9.2615e+01 1.9879e+02 3.7820e+02 - Scheme A 1.5278e+01 6.5953e+01 1.4215e+02 2.6725e+02 - Scheme B 8.9698e+00 3.8684e+01 8.4291e+01 1.5977e+02 - ratio-A 0.27 0.29 0.28 0.29 0.29(2/7) ratio-B 0.57 0.58 0.58 0.58 0.57(4/7) Table 5. Recorded CPU time in 3D with respect to the approximation error when only the spatial gridsize is varied with
x=
y=
zand
t= 1:0e09. -50 -40 -30 -20 -10 0 10 20 30 40 50 0 H (mT)-1-0.8-0.6-0.4-0.200.20.40.60.81M/Ms GSPM Scheme A Scheme B Fig. 4. Comparison of hysteresis loops for GSPM, Scheme A and Scheme B. Proles of the av- erage magnetization of these three methods are in quantitative agreements with approximately the same switch eld 9 (0:4) mT . The applied eld is parallel to the xaxis and the initial state is the uniform state. damping term simultaneously and follows by a projection step, which requires to solve heat equations 5 times at each time step. The second method introduces two sets of approximate solutions, where we update the gyromagnetic term and the damping term simultaneously for one set of approximate solutions and apply the projection step to the other set of approximate solutions in an alternating manner. Therefore, only 3 heat equations are needed to be solved at each step. Compared to the original Gauss-Seidel projection method, which solves heat equations 7 times at each step, savings of these two improved methods are about 2 =7 and 4=7, which is veried by both 1D and 3D examples for the same accuracy requirement. In addition, unconditional stability with respect to both the grid size and the damping parameter is conrmed numerically. Application of both methods to a realistic material is also presented with hysteresis loops and magnetization proles. Acknowledgments This work is supported in part by the grants NSFC 21602149 (J. Chen), NSFC 11501399 (R. Du), the Hong Kong Research Grants Council (GRF grants 16302715, 16324416, 16303318 12(a) Angle prole ( = 0:1) 0 0.2 0.4 0.6 0.8 1 x (m)00.20.40.60.81y (m) (b) Magnetization prole ( = 0:1) (c) Angle prole ( = 0:01) 0 0.2 0.4 0.6 0.8 1 x (m)00.20.40.60.81y (m) (d) Magnetization prole ( = 0:01) Fig. 5. Simulation of the full Landau-Lifshitz-Gilbert equation using GSPM without any exter- nal eld. The magnetization on the centered slice of the material in the xyplane is used. Top row:= 0:1; Bottom row: = 0:01. Left column: a color plot of the angle between the in-plane magnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization. 13(a) Angle prole ( = 0:1) 0 0.2 0.4 0.6 0.8 1 x (m)00.20.40.60.81y (m) (b) Magnetization prole ( = 0:1) (c) Angle prole ( = 0:01) 0 0.2 0.4 0.6 0.8 1 x (m)00.20.40.60.81y (m) (d) Magnetization prole ( = 0:01) Fig. 6. Simulation of the full Landau-Lifshitz-Gilbert equation using Scheme A without any external eld. The magnetization on the centered slice of the material in the xyplane is used. Top row: = 0:1; Bottom row: = 0:01. Left column: a color plot of the angle between the in- plane magnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization. 14(a) Angle prole ( = 0:1) 0 0.2 0.4 0.6 0.8 1 x (m)00.20.40.60.81y (m) (b) Magnetization prole ( = 0:1) (c) Angle prole ( = 0:01) 0 0.2 0.4 0.6 0.8 1 x (m)00.20.40.60.81y (m) (d) Magnetization prole ( = 0:01) Fig. 7. Simulation of the full Landau-Lifshitz-Gilbert equation using Scheme B without any external eld. The magnetization on the centered slice of the material in the xyplane is used. Top row: = 0:1; Bottom row: = 0:01. Left column: a color plot of the angle between the in- plane magnetization and the xaxis; Right column: an arrow plot of the in-plane magnetization. 15and NSFC-RGC joint research grant N-HKUST620/15) (X.-P. Wang), and the Innovation Program for postgraduates in Jiangsu province via grant KYCX19 1947 (C. Xie). References [1] L. Landau, E. Lifshitz, On the theory of the dispersion of magetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion 8 (1935) 153{169. [2] T. Gilbert, A lagrangian formulation of gyromagnetic equation of the magnetization eld, Phys. Rev. 100 (1955) 1243{1255. [3] W. F. B. Jr., Micromagnetics, Interscience Tracts on Physics and Astronomy, 1963. [4] P. Sulem, C. Sulem, C. Bardos, On the continuous limit limit for a system of classical spins, Comm. Math. Phys. 107 (1986) 431{454. [5] M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Dierential Geom. 28 (1988) 485{502. [6] I. Zuti c, J. Fabian, S. Das Sarma, Spintronics: Fundamentals and applications, Rev. Mod. Phys. 76 (2004) 323{410. [7] M. Kruzik, A. Prohl, Recent developments in the modeling, analysis, and numerics of ferromagnetism, SIAM Rev. 48 (2006) 439{483. [8] I. Cimr ak, A survey on the numerics and computations for the Landau-Lifshitz equation of micromag- netism, Arch. Comput. Methods Eng. 15 (2008) 277{309. [9] C. J. Garc a-Cervera, Numerical micromagnetics: a review, Bol. Soc. Esp. Mat. Apl. 39 (2007) 103{135. [10] A. Fran cois, J. Pascal, Convergence of a nite element discretization for the Landau-Lifshitz equations in micromagnetism, Math. Models Methods Appl. Sci. 16 (2006) 299{316. [11] A. Romeo, G. Finocchio, M. Carpentieri, L. Torres, G. Consolo, B. Azzerboni, A numerical solution of the magnetization reversal modeling in a permalloy thin lm using fth order runge-kutta method with adaptive step size control, Physica B. 403 (2008) 1163{1194. [12] Y. H, H. N, Implicit solution of the Landau-Lifshitz-Gilbert equation by the Crank-Nicolson method, J. Magn. Soc. Japan 28 (2004) 924{931. [13] S. Bartels, P. Andreas, Convergence of an implicit nite element method for the Landau-Lifshitz-Gilbert equation, SIAM J. Numer. Anal. 44 (2006) 1405{1419. [14] A. Fuwa, T. Ishiwata, M. Tsutsumi, Finite dierence scheme for the Landau-Lifshitz equation, Japan J. Indust. Appl. Math. 29 (2012) 83{110. [15] X. Wang, C. J. Garc a-Cervera, W. E, A gauss-seidel projection method for micromagnetics simulations, J. Comput. Phys. 171 (2001) 357{372. [16] W. E, X. Wang, Numerical methods for the Landau-Lisfshitz equation, SIAM J. Numer. Anal. 38 (2000) 1647{1665. [17] J. Chen, C. Wang, C. Xie, Convergence analysis of a second-order semi-implicit projection method for Landau-Lifshitz equation, arXiv 1902.09740 (2019). [18] C. J. Garc a-Cervera, W. E, Improved gauss-seidel projection method for micromagnetics simulations, IEEE Trans. Magn. 39 (2003) 1766{1770. [19] I. Cimr ak, Error estimates for a semi-implicit numerical scheme solving the Landau-Lifshitz equation with an exchange eld, IMA J. Numer. Anal. (2005) 611{634. 16

2019-07-27

In this paper, we present two improved Gauss-Seidel projection methods with unconditional stability. The first method updates the gyromagnetic term and the damping term simultaneously and follows by a projection step. The second method introduces two sets of approximate solutions, where we update the gyromagnetic term and the damping term simultaneously for one set of approximate solutions and apply the projection step to the other set of approximate solutions in an alternating manner. Compared to the original Gauss-Seidel projection method which has to solve heat equations $7$ times at each time step, the improved methods solve heat equations $5$ times and $3$ times, respectively. First-order accuracy in time and second-order accuracy in space are verified by examples in both 1D and 3D. In addition, unconditional stability with respect to both the grid size and the damping parameter is confirmed numerically. Application of both methods to a realistic material is also presented with hysteresis loops and magnetization profiles. Compared with the original method, the recorded running times suggest that savings of both methods are about $2/7$ and $4/7$ for the same accuracy requirement, respectively.

Two improved Gauss-Seidel projection methods for Landau-Lifshitz-Gilbert equation

1907.11853v1

Noname manuscript No. (will be inserted by the editor) Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling Hedy ATTOUCH
A cha BALHAG
Zaki CHBANI
Hassan RIAHI the date of receipt and acceptance should be inserted later Abstract In a Hilbert setting, we develop fast methods for convex unconstrained optimization. We rely on the asymptotic behavior of an inertial system combining geometric damping with temporal scaling. The convex function to minimize enters the dynamic via its gradient. The dynamic includes three coecients varying with time, one is a viscous damping coecient, the second is attached to the Hessian- driven damping, the third is a time scaling coecient. We study the convergence rate of the values under general conditions involving the damping and the time scale coecients. The obtained results are based on a new Lyapunov analysis and they encompass known results on the subject. We pay particular attention to the case of an asymptotically vanishing viscous damping, which is directly related to the accelerated gradient method of Nesterov. The Hessian-driven damping signi- cantly reduces the oscillatory aspects. As a main result, we obtain an exponential rate of convergence of values without assuming the strong convexity of the objec- tive function. The temporal discretization of these dynamics opens the gate to a large class of inertial optimization algorithms. Keywords damped inertial gradient dynamics; fast convex optimization; Hessian-driven damping; Nesterov accelerated gradient method; time rescaling Mathematics Subject Classication (2010) 37N40, 46N10, 49M30, 65K05, 65K10, 90B50, 90C25. Hedy ATTOUCH IMAG, Univ. Montpellier, CNRS, Montpellier, France hedy.attouch@umontpellier.fr, Supported by COST Action: CA16228 A cha BALHAG
Zaki CHBANI
Hassan RIAHI Cadi Ayyad University S emlalia Faculty of Sciences 40000 Marrakech, Morroco aichabalhag@gmail.com
chbaniz@uca.ac.ma
h-riahi@uca.ac.maarXiv:2009.07620v1 [math.OC] 16 Sep 20202 Hedy ATTOUCH et al. 1 Introduction Throughout the paper, His a real Hilbert space with inner product h
;
iand induced normk
k;andf:H!Ris a convex and dierentiable function. We aim at developping fast numerical methods for solving the optimization problem (P) min x2Hf(x): We denote by argminHfthe set of minimizers of the optimization problem ( P), which is assumed to be non-empty. Our work is part of the active research stream that studies the close link between continuous dissipative dynamical systems and optimization algorithms. In general, the implicit temporal discretization of con- tinuous gradient-based dynamics provides proximal algorithms that benet from similar asymptotic convergence properties, see [28] for a systematic study in the case of rst-order evolution systems, and [5,6,8,11,12,19,20,21] for some recent results concerning second-order evolution equations. The main object of our study is the second-order in time dierential equation (IGS) ;;bx(t) + (t) _x(t) +(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0; where the coecients ;: [t0;+1[!R+take account of the viscous and Hessian- driven damping, respectively, and b:R+!R+is a time scale parameter. We take for granted the existence and uniqueness of the solution of the corresponding Cauchy problem with initial conditions x(t0) =x02H, _x(t0) =v02H. Assuming thatrfis Lipschitz continuous on the bounded sets, and that the coecients are continuously dierentiable, the local existence follows from the nonautonomous version of the Cauchy-Lipschitz theorem, see [24, Prop. 6.2.1]. The global existence then follows from the energy estimates that will be established in the next section. Each of these damping and rescaling terms properly tuned, improves the rate of convergence of the associated dynamics and algorithms. An original aspect of our work is to combine them in the same dynamic. Let us recall some classical facts. 1.1 Damped inertial dynamics and optimization The continuous-time perspective gives a mechanical intuition of the behavior of the trajectories, and a valuable tool to develop a Lyapunov analysis. A rst important work in this perspective is the heavy ball with friction method of B. Polyak [29] (HBF) x(t) + _x(t) +rf(x(t)) = 0: It is a simplied model for a heavy ball (whose mass has been normalized to one) sliding on the graph of the function fto be minimized, and which asymptoti- cally stops under the action of viscous friction, see [14] for further details. In this model, the viscous friction parameter is a xed positive parameter. Due to too much friction (at least asymptotically) involved in this process, replacing the xed viscous coecient with a vanishing viscous coecient ( i.e.which tends to zero as t!+1) gives Nesterov's famous accelerated gradient method [26] [27]. The other two basic ingredients that we will use, namely time rescaling, and Hessian-driven damping have a natural interpretation (cinematic and geometric, respectively) inInertial dynamics with Hessian damping and time rescaling 3 this context. We will come back to these points later. Precisely, we seek to develop fast rst-order methods based on the temporal discretization of damped inertial dynamics. By fast we mean that, for a general convex function f, and for each trajectory of the system, the convergence rate of the values f(x(t))infHfwhich is obtained is optimal ( i.e.is achieved of nearly achieved in the worst case). The importance of simple rst-order methods, and in particular gradient-based and proximal algorithms, comes from the applicability of these algorithms to a wide range of large-scale problems arising from machine learning and/or engineering. 1.1.1 The viscous damping parameter (t). A signicant number of recent studies have focused on the case (t) = t,= 0 (without Hessian-driven damping), and b= 1 (without time rescaling), that is (AVD)x(t) + t_x(t) +rf(x(t)) = 0: This dynamic involves an Asymptotically Vanishing Damping coecient (hence the terminology), a key property to obtain fast convergence for a general convex functionf. In [32], Su, Boyd and Cand es showed that for = 3 the above system can be seen as a continuous version of the accelerated gradient method of Nesterov [26,27] with f(x(t))minHf=O(1 t2) ast!+1. The importance of the parame- terwas put to the fore by Attouch, Chbani, Peypouquet and Redont [9] and May [25]. They showed that, for >3, one can pass from capital Oestimates to small o. Moreover, when >3, each trajectory converges weakly, and its limit belongs to argminf1. Recent research considered the case of a general damping coecient (
) (see [4,7]), thus providing a complete picture of the convergence rates for (AVD): f(x(t))minHf=O(1=t2) when3, andf(x(t))minHf=O 1=t2 3 when 3, see [7,10] and Apidopoulos, Aujol and Dossal [3]. 1.1.2 The Hessian-driven damping parameter (t). The inertial system (DIN) ;x(t) + _x(t) +r2f(x(t)) _x(t) +rf(x(t)) = 0; was introduced by Alvarez, Attouch, Bolte, and Redont in [2]. In line with (HBF), it contains a xed positive friction coecient . As a main property, the introduc- tion of the Hessian-driven damping makes it possible to neutralize the transversal oscillations likely to occur with (HBF), as observed in [2]. The need to take a geometric damping adapted to fhad already been observed by Alvarez [1] who considered the inertial system x(t) +D_x(t) +rf(x(t)) = 0; whereD:H!H is a linear positive denite anisotropic operator. But still this damping operator is xed. For a general convex function, the Hessian-driven damp- ing in (DIN) ;performs a similar operation in a closed-loop adaptive way. (DIN) stands shortly for Dynamical Inertial Newton, and refers to the link with the 1Recall that for = 3 the convergence of the trajectories is an open question4 Hedy ATTOUCH et al. Levenberg-Marquardt regularization of the continuous Newton method. Recent studies have been devoted to the study of the inertial dynamic x(t) + t_x(t) +r2f(x(t)) _x(t) +rf(x(t)) = 0; which combines asymptotic vanishing damping with Hessian-driven damping [17]. 1.1.3 The time rescaling parameter b(t). In the context of non-autonomous dissipative dynamic systems, reparameteriza- tion in time is a simple and universal means to accelerate the convergence of trajectories. This is where the coecient b(t) comes in as a factor of rf(x(t)). In [11] [12], in the case of general coecients (
) andb(
) without the Hessian damping, the authors made in-depth study. In the case (t) = t, they proved that under appropriate conditions on andb(
),f(x(t))minHf=O(1 t2b(t)). Hence a clear improvement of the convergence rate by taking b(t)!+1ast!+1. 1.2 From damped inertial dynamics to proximal-gradient inertial algorithms Let's review some classical facts concerning the close link between continuous dissipative inertial dynamic systems and the corresponding algorithms obtained by temporal discretization. Let us insist on the fact that, when the temporal scalingb(t)!+1ast!+1, the transposition of the results to the discrete case naturally leads to consider an implicit temporal discretization, i.e.inertial proximal algorithms. The reason is that, since b(t) is in front of the gradient, the application of the gradient descent lemma would require taking a step size that tends to zero. On the other hand, the corresponding proximal algorithms involve a proximal coecient which tends to innity (large step proximal algorithms). 1.2.1 The case without the Hessian-driven damping The implicit discretization of (IGS) ;0;bgives the Inertial Proximal algorithm (IP)k;k( yk=xk+k(xkxk1) xk+1= proxkf(yk) wherekis non-negative and kis positive. Recall that for any >0, the proximity operator proxf:H!H is dened by the following formula: for every x2H proxf(x) := argmin2H f() +1 2kxk2 : Equivalently, proxfis the resolvent of index of the maximally monotone opera- tor@f. When passing to the implicit discrete case, we can take f:H!R[f+1g a convex lower semicontinuous and proper function. Let us list some of the main results concerning the convergence properties of the algorithm (IP)k;k: 1. Casek>0 andk= 1 k. When= 3, the (IP)13=k;algorithm has a similar structure to the original Nesterov accelerated gradient algorithm[26],Inertial dynamics with Hessian damping and time rescaling 5 just replace the gradient step with a proximal step. Passing from the gradient to the proximal step was carried out by G uler [22,23], then by Beck and Teboulle [18] for structured optimization. A decisive step was taken by Attouch and Peypouquet in [16] proving that, when >3,f(xk)minHf=o1 k2
. The subcritical case <3 was examined by Apidopoulos, Aujol, and Dossal [3] and Attouch, Chbani, and Riahi [10] with the rate of convergence rate of values f(xk)minHf=O 1 k2 3 . 2. For a general k, the convergence properties of (IP)k;were analyzed by Attouch and Cabot [5], then by Attouch, Cabot, Chbani, and Riahi [6], in the presence of perturbations. The convergence rates are then expressed using the sequence ( tk) which is linked to ( k) by the formula tk:= 1 +P+1 i=kQi j=kj. Under growth conditions on tk, it is proved that f(xk)minHf=O(1 t2 k). This last results covers the special case k= 1 kwhen3. 3. For a general k, Attouch, Chbani, and Riahi rst considered in [11] the casek= 1 k. They proved that under a growth condition on k, we have the estimate f(xk)minHf=O(1 k2k). This result is an improvement of the one discussed previously in [16], because when k=kwith 0<<3, we pass from O(1 k2) toO(1 k2+). Recently, in [13] the authors analyzed the algorithm (IP)k;kfor general kandk. By including the expression of tkpreviously used in [5,6], they proved that f(xk)minHf=O1=t2 kk1
under certain conditions on k andk. They obtained f(xk)minHf=o1=t2 kk
, which gives a global view of of the convergence rate with small o, encompassing [5,13]. 1.2.2 The case with the Hessian-driven damping Recent studies have been devoted to the inertial dynamic x(t) + t_x(t) +r2f(x(t)) _x(t) +rf(x(t)) = 0; which combines asymptotic vanishing viscous damping with Hessian-driven damp- ing. The corresponding algorithms involve a correcting term in the Nesterov ac- celerated gradient method which reduces the oscillatory aspects, see Attouch- Peypouquet-Redont [17], Attouch-Chbani-Fadili-Riahi [8], Shi-Du-Jordan-Su [30]. The case of monotone inclusions has been considered by Attouch and L aszl o [15]. 1.3 Contents The paper is organized as follows. In section 2, we develop a new Lyapunov anal- ysis for the continuous dynamic (IGS) ;;b. In Theorem 1, we provide a system of conditions on the damping parameters (
) and(
), and on the temporal scaling parameter b(
) giving fast convergence of the values. Then, in sections 3 and 4, we present two dierent types of growth conditions for the damping and tempo- ral scaling parameters, respectively based on the functions andp , and which satisfy the conditions of Theorem 1. In doing so, we encompass most existing re- sults and provide new results, including linear convergence rates without assuming strong convexity. This will also allow us to explain the choice of certain coecients in the associated algorithms, questions which have remained mysterious and only6 Hedy ATTOUCH et al. justied by the simplication of often complicated calculations. In section 5, we specialize our results to certain model situations and give numerical illustrations. Finally, we conclude the paper by highlighting its original aspects. 2 Convergence rate of the values. General abstract result We will establish a general result concerning the convergence rate of the values veried by the solution trajectories x(
) of the second-order evolution equation (IGS) ;;b x(t) + (t) _x(t) +(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0: The variable parameters (
),(
) andb(
) take into account the damping, and temporal rescaling eects. They are assumed to be continuously dierentiable. To analyze the asymptotic behavior of the solutions trajectories of the evolution system (IGS) ;;b, we will use Lyapunov's analysis. It is a classic and powerful tool which consists in building an associated energy-like function which decreases along the trajectories. The determination of such a Lyapunov function is in general a delicate problem. Based on previous works, we know the global structure of such a Lyapunov function. It is a weighted sum of the potential, kinetic and anchor functions. We will introduce coecients in this function that are a priori unknown, and which will be identied during the calculation to verify the property of decay. Our approach takes advantage of the technics recently developed in [4], [17], [12]. 2.1 The general case Letx(
) be a solution trajectory of (IGS) ;;b. Givenz2argminHf, we introduce the Lyapunov function t7!E(t) dened by E(t) :=c(t)2b(t)(f(x(t))f(z))+(t)(t)2 2kv(t)k2+(t) 2kx(t)zk2;(1) wherev(t) :=x(t)z+1 (t)(_x(t) +(t)rf(x(t))): The four variable coecients c(t);(t);(t);(t) will be adjusted during the calcu- lation. According to the classical derivation chain rule, we obtain d dtE(t) =d dt c2(t)b(t) (f(x(t))f(z))+c(t)2b(t)hrf(x(t));_x(t)i +1 2d dt((t)2(t))kv(t)k2+(t)2(t)h_v(t);v(t)i +1 2_(t)kx(t)zk2+(t)h_x(t);x(t)zi:Inertial dynamics with Hessian damping and time rescaling 7 From now, without ambiguity, to shorten formulas, we omit the variable t. According to the denition of v, and the equation (IGS) ;;b, we have _v= _x_ 2(_x+rf(x))+1 d dt(_x+rf(x)) = _x_ 2(_x+rf(x))+1  x+r2f(x) _x+_rf(x) = _x_ 2(_x+rf(x))+1  _xbrf(x) +_rf(x)
= 1_ 2  _x+_ _ 2b  rf(x): Therefore, h_v;vi= 1_ 2  _x+_ _ 2b  rf(x); xz+1 (_x+rf(x)) = 1_ 2  h_x; xzi+ 1_ 2 1 k_xk2 + 1_ 2   +_ _ 2b 1  hrf(x);_xi +_ _ 2b  hrf(x); xzi+_ _ 2b   krf(x)k2: According to the denition of v(t), after developing kv(t)k2, we get kvk2=kxzk2+1 2 k_xk2+2krf(x)k2 +2 h_x; xzi +2 hrf(x); xzi+2 2hrf(x);_xi: Collecting the above results, we obtain d dtE(t) =d dt c2b (f(x)f(z))+c2bhrf(x);_xi+1 2_kxzk2+h_x;xzi +1 2d dt(2) kxzk2+1 2 k_xk2+2krf(x)k2 +2 h_x; xzi +1 2d dt(2)2 hrf(x); xzi+2 2hrf(x);_xi +2 1_ 2  h_x; xzi+ 1_ 2 1 k_xk2 +2 1_ 2   +_ _ 2b 1  hrf(x);_xi +2_ _ 2b  hrf(x); xzi+_ _ 2b   krf(x)k2 : In the second member of the above formula, let us examine the terms that contain hrf(x); xzi. By grouping these terms, we obtain the following expression  d dt(2) +2_ _ 2b  hrf(x); xzi:8 Hedy ATTOUCH et al. To majorize it, we use the convex subgradient inequality hrf(x);xzif(x) f(z);and we make a rst hypothesis d dt(2)+2_ _ 2b  0:Therefore, d dtE(t) d dt c2b + d dt(2) +2_ _ 2b  (f(x)f(z)) + c2b+ 2d dt(2) + 1_ 2  +_ _ 2b  hrf(x);_xi +1 d dt(2) +2 1_ 2  + h_x; xzi +1 2 d dt(2) +_ kxzk2+1 22d dt(2) + 1_ 2  k_xk2 + 2 22d dt(2) +_ _ 2b  krf(x)k2: (2) To getd dtE(t)0, we are led to make the following assumptions: (i) d dt(2) +2_ _ 2b  0 (ii)d dt c2b + d dt(2) +2_ _ 2b  0; (iii)c2b+ 2d dt(2) + 1_ 2  +_ _ 2b  = 0; (iv)1 d dt(2) +2 1_ 2  += 0; (v)d dt(2) +_0; (vi)1 22d dt(2) + 1_ 2  0; (vii)2 22d dt(2) +_ _ 2b  0: After simplication, we get the following equivalent system of conditions: A: Lyapunov system of inequalities involving c(t);(t);(t);(t): (i)d dt()b0 (ii)d dt c2b+
b0; (iii)b(c2) +( ) +d dt() = 0; (iv)d dt() +( )+= 0; (v)d dt(2+)0; (vi)_+ 2( )0; (vii) _+ 2_b
 0:Inertial dynamics with Hessian damping and time rescaling 9 Let's simplify this system by eliminating the variable . From (iv) we get= d dt()( ), that we replace in ( v), and recall that is prescribed to be nonnegative. Now observe that the unkown function ccan also be eliminated. Indeed, it enters the above system via the variable bc2, which according to ( iii) is equal to bc2=b( )d dt():Replacing in ( ii), which is the only other equation involving bc2, we obtain the equivalent system involving only the variables(t);(t). B: Lyapunov system of inequalities involving the variables: (t);(t) (i)d dt()b0; (ii)d dt(b+ )d2 dt2()b0; (iii)b( )d dt()0; (iv)d dt() +( )0; (v)d dt d dt() +
0; (vi)_+ 2( )0; (vii) _+ 2_b
 0: Then, the variables andcare obtained by using the formulas =d dt()( ) bc2=b( )d dt(): Thus, under the above conditions, the function E(
) is nonnegative and nonincreas- ing. Therefore, for every tt0,E(t)E(t0);which implies that c2(t)b(t)f(x(t))min Hf)E(t0): Therefore, as t!+1 f(x(t))min Hf=O1 c2(t)b(t) : Moreover, by integrating (2) we obtain the following integral estimates: a) On the values: Z+1 t0 (t)b(t)(t)d dt c2(t)b(t) +(t)(t)(t) f(x(t))inf Hf dt< +1; where we use the equality:  d dt c2b
+ d dt(2) +2_ _ 2b  =bd dt c2b+
and the fact that, according to ( ii), this quantity is nonnegative.10 Hedy ATTOUCH et al. b) On the norm of the gradients: Z+1 t0q(t)krf(x(t))k2dt< +1: whereqis the nonnegative weight function dened by q(t) :=(t)(t)_(t)(t) (t)+b(t)_(t) 2(t) 22(t)d dt(2)(t) =b(t)(t)(t)1 2d dt(2)(t): (3) We can now state the following Theorem, which summarizes the above results. Theorem 1 Letf:H!Rbe a convex dierentiable function with argminHf6=;. Letx(
)be a solution trajectory of (IGS) ;;b x(t) + (t) _x(t) +(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0: Suppose that (
),(
), andb(
), areC1functions on [t0;+1[such that there exists auxiliary functions c(t);(t);(t);(t)that satisfy the conditions (i)(vii)above. Set E(t) :=c(t)2b(t)(f(x(t))f(z))+(t)(t)2 2kv(t)k2+(t) 2kx(t)zk2;(4) withz2argminHfandv(t) =x(t)z+1 (t)(_x(t) +(t)rf(x(t))). Then,t7!E(t)is a nonincreasing function. As a consequence, for all tt0, (i)f(x(t))min HfE(t0) c2(t)b(t); (5) (ii)Z+1 t0 (t)b(t)(t)d dt c2b+
(t) f(x(t))inf Hf dt< +1; (6) (iii)Z+1 t0 b(t)(t)(t)1 2d dt 2
(t) krf(x(t))k2dt< +1: (7) 2.2 Solving system ( i)(vii) The system of inequalities ( i)(vii) of Theorem 1 may seem complicated at rst glance. Indeed, we will see that it simplies notably in the classical situations. Moreover, it makes it possible to unify the existing results, and discover new interesting cases. We will present two dierent types of solutions to this system, respectively based on the following functions: p (t) = expZt t0 (u)du ; (8) and (t) =p (t)Z+1 tdu p (u): (9) The use of has been considered in a series of articles that we will retrieve as a special case of our approach, see [4], [5], [7], [12]. Using p will lead to new results, see section 4.Inertial dynamics with Hessian damping and time rescaling 11 3 Results based on the function In this section, we will systematically assume that condition ( H0) is satised. (H0)Z+1 t0ds p(s)<+1: Under (H0), the function (
) is well dened. It can be equally dened as the solution of the linear non autonomous dierential equation _ (t) (t) (t) + 1 = 0; (10) which satises the limit condition lim t!+1 (t) p (t)= 0. 3.1 The case without the Hessian, i.e.0 The dynamic writes (IGS) ;0;b x(t) + (t) _x(t) +b(t)rf(x(t)) = 0: To solve the system ( i)(vii) of Theorem 1, we choose 0; c(t) = (t); (t) =1 (t); (t) = (t)2: According to (10), we can easily verify that conditions ( i);(iii)(vii) are satised, and (ii) becomes d dt (t)2b(t) (t)b(t)0: After dividing by (t), and using (10), we obtain the condition 0 (t)_b(t)(32 (t) (t))b(t): This leads to the following result obtained by Attouch, Chbani and Riahi in [12]. Theorem 2 [12, Theorem 2.1] Suppose that for all tt0 (t)_b(t)b(t)(32 (t) (t)); (11) where is dened from by(9). Letx: [t0;+1[!H be a solution trajectory of (IGS) ;0;b. Givenz2argminHf, set E(t) := 2 (t)b(t)(f(x(t))f(z))+1 2kx(t)z+ (t) _x(t)k2: (12) Then,t7!E(t)is a nonincreasing function. As a consequence, as t!+1 f(x(t))min Hf=O1 (t)2b(t) : (13) Precisely, for all tt0 f(x(t))min HfC (t)2b(t); (14) withC= (t0)2b(t0)(f(x(t0))minHf)+d(x(t0);argminf)2+ (t0)2k_x(t0)k2: Moreover, Z+1 t0 (t) b(t)(32 (t) (t)) (t)_b(t) (f(x(t))min Hf)dt< +1: Remark 1 Whenb1, condition (11) reduces to (t) (t)3 2, introduced in [4].12 Hedy ATTOUCH et al. 3.2 Combining Nesterov acceleration with Hessian damping Let us specialize our results in the case (t)>0, and (t) = t. We are in the case of a vanishing damping coecient ( i.e. (t)!0 ast!+1). According to Su, Boyd and Cand es [32], the case = 3 corresponds to a continuous version of the accelerated gradient method of Nesterov. Taking  > 3 improves in many ways the convergence properties of this dynamic, see section 1.1.1. Here, it is combined with the Hessian-driven damping and temporal rescaling. This situation was rst considered by Attouch, Chbani, Fadili and Riahi in [8]. Then the dynamic writes (IGS)=t;;b x(t) + t_x(t) +(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0: Elementary calculus gives that ( H0) is satised as soon as >1. In this case, (t) =t 1: After [8], let us introduce the following quantity which will simplify the formulas: w(t) :=b(t)_(t)(t) t: (15) The following result will be obtained as a consequence of our general abstract The- orem 1. Precisely, we will show that under an appropriate choice of the functions c(t);(t);(t);(t), the conditions ( i)(vii) of Theorem 1 are satised. Theorem 3 [8, Theorem 1] Letx: [t0;+1[!H be a solution trajectory of (IGS)=t;;b x(t) + t_x(t) +(t)r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0: Suppose that >1, and that the following growth conditions are satised: for tt0 (G2)b(t)>_(t) +(t) t; (G3)t_w(t)(3)w(t): Then,w(t) :=b(t)_(t)(t) tis positive and (i)f(x(t))min Hf=O1 t2w(t) ast!+1; (ii)Z+1 t0t (3)w(t)t_w(t) (f(x(t))min Hf)dt< +1; (iii)Z+1 t0t2(t)w(t)krf(x(t))k2dt< +1: Proof Take(t) = (t)2; (t) =1 (t); (t)0;and c(t)2=1 (1)2t b(t) tb(t)(t)t_(t)
: (16) This formula for c(t) will appear naturally during the calculation. Note that the condition (G2) ensures that the second member of the above expression is positive, which makes sense to think of it as a square. Let us verify that the conditions ( i) and (iv);(v);(vi);(vii) are satised. This is a direct consequence of the formula (10) and the condition ( G2):Inertial dynamics with Hessian damping and time rescaling 13 (i)d dt()b=d dtb=1 1d dt(t)tb
=t 1 _+ tb 0. (iv)d dt() +( )+=_ + 1
=_ + 1 = 0. (v) Since21 and1, we haved dt(2+) = 0. (vi)_+ 2( )= 2 _ + 2( 2) = 2( _ + 1 ) = 0. (vii)_+ 2_b
= 2(_ + ( _b)) = 22(_b+ t)0: Let's go to the conditions ( ii) and (iii). The condition ( iii) gives the formula (16) forc(t). Then replacing c(t)2by this value in ( ii) gives the condition ( G3). Note then thatb(t)c(t)2=1 (1)2t2!(t), which gives the convergence rate of the values f(x(t))min Hf=O1 t2w(t) : Let us consider the integral estimate for the values. According to the denition (16) forc2band the denition of w, we have bd dt c2b+ =1 1tbd dt1 1t2w(t) +1 1t =t (1)2 (1)b2wt_w(1)(_+ t) =t (1)2 (3)wt_w : According to Theorem 1 ( ii) Z+1 t0t (3)w(t)t_w(t) (f(x(t))min Hf)dt< +1: Moreover, since 2= 1, the formula giving the weighting coecient q(t) in the integral formula simplies, and we get q(t) =(t)(t)(t) _(t)(t) 2(t)+b(t) (t)_(t) (t)! =(t) (t) (t)_ (t) +b(t) (t)_(t) (t) =(t) (t)2!(t): According to Theorem 1 ( iii) Z+1 t0t2(t)w(t)krf(x(t))k2dt< +1 which gives the announced convergence rates. u t Remark 2 Take= 0. Then, according to the denition (15) of w, we havew=b, and the conditions of Theorem 3 reduce to t_b(t)(3)b(t)0 fort2[t0;+1[:14 Hedy ATTOUCH et al. We recover the condition introduced in [12, Corollary 3.4]. Under this condition, each solution trajectory xof (IGS)=t;0;b x(t) + t_x(t) +b(t)rf(x(t)) = 0; satises f(x(t))min Hf=O1 t2b(t) ast!+1: 3.3 The case (t) = t,constant Due to its practical importance, consider the case (t) = t,(t)whereis a xed positive constant. In this case, the dynamic (IGS) ;;bis written as follows x(t) + t_x(t) +r2f(x(t)) _x(t) +b(t)rf(x(t)) = 0: (17) The set of conditions ( G2), (G3) boils down to: for tt0 (G2)b(t)> t; (G3)t_w(t)(3)w(t); wherew(t) =b(t) t. Therefore, b(
) must satisfy the dierential inequality td dt b(t) t (3) b(t) t : Equivalently td dtb(t)(3)b(t) +(2)1 t0: Let us integrate this linear dierential equation. Set b(t) =k(t)t3wherek(
) is an auxiliary function to determine. We obtain d dt k(t) t2 0; which gives k(t) = t2+d(t) withd(
) nonincreasing. Finally, b(t) = t+d(t)t3; withd(
) a nonincreasing function to be chosen arbitrarily. In summary, we get the following result: Proposition 1 Letx: [t0;+1[!H be a solution trajectory of x(t) + t_x(t) +r2f(x(t)) _x(t) + t+d(t)t3 rf(x(t)) = 0 (18) whered(
)is a nonincreasing positive function. Then, the following properties are sat- ised: (i)f(x(t))min Hf=O1 t1d(t) ast!+1; (ii)Z+1 t0_d(t)t1(f(x(t))inf Hf)dt< +1: (iii)Z+1 t0t1d(t)krf(x(t))k2dt< +1: Proof According to the denition of w(t) andb(t), we have the equalities t2w(t) =t2 b(t) t =t2d(t)t3=t1d(t). Then apply Theorem 3. u tInertial dynamics with Hessian damping and time rescaling 15 3.4 Particular cases According to Theorem 3 and Proposition 1, let us discuss the role and the impor- tance of the scaling coecient b(t) in front of the gradient term. a)The rst inertial dynamic system based on the Nesterov method, and which includes a damping term driven by the Hessian, was considered by Attouch, Pey- pouquet, and Redont in [17]. This corresponds to b(t)1, which gives: x(t) + t_x(t) +r2f(x(t)) _x(t) +rf(x(t)) = 0: In this case, we have w(t) = 1 t, and we immediately get that ( G2), (G3) are satised by taking >3 andt> . This corresponds to take d(t) =1 t3 t2, which is nonincreasing when t2 3. Corollary 1 [17, Theorem 1.10, Proposition 1.11] Suppose that  >3and >0. Letx: [t0;+1[!H be a solution trajectory of x(t) + t_x(t) +r2f(x(t)) _x(t) +rf(x(t)) = 0: (19) Then, (i)f(x(t))min Hf=O1 t2 ast!+1; (ii)Z+1 t0t(f(x(t))inf Hf)dt< +1; (iii)Z+1 t0t2krf(x(t))k2dt< +1: b)Another important situation is obtained by taking d(t) =1 t3. This is the limiting case where the following two properties are satised: d(
) is nonincreasing, and the coecient of rf(x(t)) is bounded. This oers the possibility of obtaining similar results for the explicit temporal discretized dynamics, that is to say the gradient algorithms. Precisely, we obtain the dynamic system considered by Shi, Du, Jordan, and Su in [30], and Attouch, Chbani, Fadili, and Riahi in [8]. Corollary 2 [8, Theorem 3], [30, Theorem 5] Suppose that 3. Letx: [t0;+1[!H be a solution trajectory of x(t) + t_x(t) +r2f(x(t)) _x(t) + 1 + t rf(x(t)) = 0 (20) Then, the conclusions of Theorem 3are satised: (i)f(x(t))min Hf=O1 t2 ast!+1; (ii)When>3;Z+1 t0t(f(x(t))inf Hf)dt< +1: (iii)Z+1 t0t2krf(x(t))k2dt< +1:16 Hedy ATTOUCH et al. Note that (20) has a slight advantage over (19): the growth conditions are valid fort>0, while for (19) one has to take t> . Accordingly, the estimates involve the quantity1 t2instead of1 t2(1 t). c)Taked(t) =1 tswiths >0. According to Proposition 1, for any solution trajectory x: [t0;+1[!H of x(t) + t_x(t) +r2f(x(t)) _x(t) + t+t3s rf(x(t)) = 0 (21) we have: (i)f(x(t))min Hf=O1 t1s ast!+1; (ii)Z+1 t0ts2(f(x(t))inf Hf)dt< +1;Z+1 t0ts1krf(x(t))k2dt< +1: 4 Results based on the function p In this section, we examine another set of growth conditions for the damping and rescaling parameters that guarantee the existence of solutions to the system (i)(vii) of Theorem 1. In the following theorems, the Lyapunov analysis and the convergence rates are formulated using the function p : [t0;+1[!R+dened by p (t) := expZt t0 (s)ds : In Theorems 2 and 3, in line with the previous articles devoted to these questions (see [4], [7], [12]), the convergence rate of the values was formulated using the function (t) =p (t)R+1 t1 p (s)ds. In fact, each of the two functions p and captures the properties of the viscous damping coecient (
), but their growths are signicantly dierent. To illustrate this, in the model case (t) = t, > 1, we havep (t) =t t0
, while (t) =t 1. Therefore, p grows faster than ast!+1, and we can expect to get better convergence rates when formulating them using p . Moreover, p makes sense and allows to analyze the case 1, while does not. Thus, we will see that the approach based on p provides results that cannot be captured by the approach based on . To illustrate this, we start with a simple situation, then we consider the general case. 4.1 A model situation Consider the system (IGS) ;0;b x(t) + (t) _x(t) +b(t)rf(x(t)) = 0 with (t) = 0(t) +1 p0(t)and p 0(t) = expZt t0 0(s)ds : Choose 0; c(t) = p 0(t); (t) =1 p0(t); (t) = p 0(t)2:Inertial dynamics with Hessian damping and time rescaling 17 According to _ p0(t) = 0(t)p0(t), we can easily verify that the conditions ( i);(iii) (vii) of Theorem 1 are satised, and ( ii) becomesd dtp0(t)2b(t)
p0(t)b(t)0: Then, a direct application of Theorem 1 gives the following result. Theorem 4 Suppose that for all tt0 p0(t)_b(t) + 2 0(t)p0(t)1 b(t)0: (22) Letx: [t0;+1[!H be a solution trajectory of (IGS) ;0;b. Then, ast!+1 f(x(t))min Hf=O1 p0(t)2b(t) : (23) Moreover,R+1 t0p0(t) 1(2 0(t)p0(t))p0(t)_b(t) (f(x(t))minHf)dt< +1: Remark 3 Let us rewrite the linear dierential inequality (22) as follows: _b(t) b(t)1 p0(t)2_ p0(t) p0(t): A solution corresponding to equality is b(t) = p 0(t)2exphZt t01 p0(s) dsi . In the case 0(t)(t) = t, 0<< 1,t0= 1, we have p 0(t) =t;which gives b(t) =t2expht11 1i : Therefore, for 0 << 1;and for this choice of b, (23) gives f(x(t))min Hf=O1 exph t1 1i : (24) Thus, we obtain an exponential convergence rate in a situation that cannot be covered by the approach. 4.2 The general case, with the Hessian-driven damping Theorem 5 Letf:H!Rbe a convex function of class C1such that argminHf6=;. Suppose that (
);(
)areC1functions and b(
)is aC2function which is nondecreasing. Suppose that randmare positive parameters which satisfy 0<r1 3and2rm 1r. Suppose that the following growth conditions are satised: for tt0 (H1)0(t)0; (H2)0(t)2(t)(m+r) (t)
1 2_0(t) + m(1r)
(t)2(t)0; (H3)b(t)_(t) +(t) (t) (t)
0; (H4)d dt (w+) (t)(t)b(t)(t)0:18 Hedy ATTOUCH et al. where 0(t) :=(12(r+m)) (t) +(t)
(t)_(t); (25) (t) :=m (t) +1 3_b(t) b(t): (26) w(t) =b(t)_(t) +(t)(t) + (12r2m) (t)(t): (27) Then, for each solution trajectory of x: [t0;+1[!H of(IGS) ;;b, we have, (i)f(x(t))min Hf=O 1 p (t)2rw(t)b(t)2 3! ast!+1 (28) (ii)Z+1 t0p2r (t)(t) f(x(t))inf Hf dt< +1; (29) (iii)Z+1 t0 p2r (t)b1 3(t)(t)d dt p2r b2 32
(t) krf(x(t))k2dt< +1:(30) Here(t) := 3(t)2(r+m) (t) w(t)_w(t)2(1rm) (t). Proof According to Theorem 1, it suces to show that, under the hypothesis (H1)(H4);there exists c;;; which satisfy the conditions ( i)(vii) of Theorem 1. To perform the corresponding derivative calculation, let's start by establishing some preliminary results. lnp (t) =Rt t0 (s)ds, which by derivation gives_p p = , that is to say _ p = p : According to the denition of , d dt p2r b2 3 = 2p2r b2 3 r 1 3_b b (31) = 2((r+m) ): (32) Let us show that the following choice of the unknown parameters c;;; satises the conditions ( i)(vii) of Theorem 1: :=p2r b2 3;  :=m +1 3_b b;  :=0; and c2b:=w:= b_++ (12r2m)  ; (33) where0has been dened in (25). We underline that under condition ( H3); c2b= b_+ |{z} 0+2 (1rm)|{z} 0  0: Also, according to (32), we have _= 2(r+m) 
:Inertial dynamics with Hessian damping and time rescaling 19 (i)d dt()b=_+(_+_)b =h _+ 2 (r+m)  +_bi because _= 2((r+m) ) = 0(b_+ ) : (34) Sincebis nondecreasing, then 0;so by (H1) and (H3);we get d dt()b= 0(b_+ )|{z} 0 0 (ii) According to the derivation chain rule and ( H4), we conclude that d dt c2b +d dt()b=d dt(w)+d dt()b =d dt(w+)b0: (iii)b(c2) +( ) +d dt() = 0 results from (33). (iv) According to the derivation chain rule, (31), and the denition of  d dt() +( )+=_+_+( )+ = 2(r+m) 
+_+( )+ = _(12r2m) +
 +: For this quantity to be equal to zero, we therefore take =0, where0is dened in (26). (v) According to our choice =0, we have (v)()d dt (2+0) 0: Let's compute this quantity. According to the derivation chain rule and (32) d dt (2+0) = _0+_(2+0) + 2_ = 21 2_0+ (2+0)((r+m) )+ _ = 21 2_0+0(r+m) 2
+ 0+(r+m) 
+ _
 : By denition of 0, we have0+ _= (12(r+m)) + . Therefore d dt (2+0) = 21 2_0+0((r+m) 2)+ 2(1(r+m)) : So, (v) is satised under the condition 1 2_0+0(r+m) 2
+ 21(r+m)
0; which is precisely ( H2).20 Hedy ATTOUCH et al. (vi) Let's compute _+ 2( )= 2 r 1 3_b b+ (m1) +1 3_b b = 2(r+m1) : According to the assumption m1r, this quantity is less or equal than zero. We have (vii)()(_+ 2( _b))0: According to condition H3and the assumption m1r;we conclude _+ 2( _b)= 2 (r+m) b
= 2h (b_+ )|{z} 0 (1rm)|{z} 0i 0: So, (vii) is satised According to Theorem 1, we obtain (28)-(29)-(30) which completes the proof. u t 4.3 The case without the Hessian Let us specialize the previous results in the case = 0, i.e.without the Hessian: (IGS) ;0;b x(t) + (t) _x(t) +b(t)rf(x(t)) = 0: Theorem 6 Suppose that the conditions (H1)and(H2)of Theorem 5are satised. Then, for each solution trajectory x: [t0;+1[!H of(IGS) ;0;b, we have, as t!+1 f(x(t))min Hf=O 1 p (t)2rb(t)1 3! : (35) Moreover, when m> 2r Z+1 t0p (t)2rb(t)1 3 (t) f(x(t))inf Hf dt< +1: (36) Proof Conditions (H1) and (H2) in Theorem 5 remain unchanged since they are independent of . We just need to verify ( H4), because (H3) is written b(t)0 and becomes obvious. Since = 0, we have (H4)()d dt b (t)(t)b(t)(t)0: According to d dt b (t)(t)b(t)(t) =_(t)b(t) +(t)_b(t)(t)b(t) m +_b(t) 3b(t) =b(t)1=3 d dt (t)b(t)2=3 m (t) (t)b(t)2=3 =b(t)1=3 d dt p (t)2r m (t) p (t)2r = (2rm) (t)b(t)1=3p (t)2r0 since 2rm; we conclude that ( H4) holds, which completes the proof. u t Next, we show that the condition ( H2) on the coecients (
) andb(
) can be formulated in simpler form which is useful in practice.Inertial dynamics with Hessian damping and time rescaling 21 Theorem 7 The conclusions of Theorem 6remain true when we replace (H2)by (H+ 2)(t) (t)(r+m) (t)
2(t) +12(r+m)
(t)
+1 2(t)0, and assume moreover that b(
)is log-concave, i.e.,d2 dt2(ln(b(t)))0. Proof According to Theorem 6, it suces to show that ( H2) is satised under the hypothesis (H+ 2). By denition of , we have 2(t)(m+r) (t)
= (mr) (t) +2 3_b(t) b(t) : So (H2) can be written equivalently as A0, where A=:0(t) (mr) (t) +2 3_b(t) b(t) 1 2_0(t) + m+r1 (t)2(t): (37) A calculation similar to the one above gives 0(t) = (12rm) (t)m (t) +(t) (t)_(t); = (12rm) (t) +1 3_b(t) b(t) (t)_(t): (38) In (37), let's replace 0(
) by its formulation (38), we obtain A=1 2d2 dt2(t)1 2d dt (t) (12rm) (t) +1 3_b(t) b(t) _(t) (mr) (t) +2 3_b(t) b(t) + m+r1 (t)2(t) + (mr) (t) +2 3_b(t) b(t) (12rm) (t) +1 3_b(t) b(t) (t): Set B:= m+r1 (t)2(t) + (mr) (t) +2 3_b(t) b(t) (12rm) (t) +1 3_b(t) b(t) (t); then we have (by omitting the variable tto shorten the formulas) B=h (m+r1) + (mr) (t) +2 3_b b (12rm) +1 3_b bi =h (m+r1) + r +1 3_b b+ (12r) +2 3_b bi =h (m+r1) 2+ (m+ 1r)+2+ r +1 3_b b (12r) +2 3_b bi = r +1 3_b b (12r) +2 3_b b : ReplacingBinA, we obtain A=(t) (t)(m+r) (t) 2(t) + (12(m+r)) (t) +1 2d2 dt2(t) +C(t) (39)22 Hedy ATTOUCH et al. where C(t) :=_(t) (mr) (t) +2 3_b(t) b(t) 1 2d dt (t) (12rm) (t) +1 3_b(t) b(t) : Let us show that C(t) is nonnegative. After replacing (t) by its value m (t)+1 3_b(t) b(t), and developing, we get C(t) =m_ (t) (t)(13r)1 6(4m2r+ 1)_ (t)_b(t) b(t) 1 6d2 dt2(ln(b(t))) (1 + 2(m2r)) (t) + 2_b(t) b(t) : By assumption, m2r0, 13r0, (
) is nonincreasing, b(
) is nondecreasing, andd2 dt2(ln(b(t)))0. We conclude that C(t)0. According to (39), we obtain A(t) (t)(m+r) (t) 2(t) + (12(m+r)) (t) +1 2d2 dt2(t): The condition (H+ 2) expresses that the second member of the above inequality is nonnegative. Therefore ( H+ 2) implies (H2), which gives the claim. u t 4.4 Comparing the two approaches As we have already underlined, Theorems 2 and 7 are based on the Lyapunov analysis of the dynamic (IGS) ;0;busing the functions andp , respectively. As such, they lead to signicantly dierent growth conditions on the coecients of the dynamic. Precisely, using the following example, we will show that Theorem 7 better captures the case where bhas an exponential growth. Take b(t) =etqand (t) = t1qwith=q> 0; q2(0;1): a)First, let us show that the condition ( H+ 2) of Theorem 7 is satised. We have 1 2(t) +(t) (t)(m+r) (t) 2(t) + (12(m+r)) (t) = (q)3 m+1 31 3r5 32r1 t33q+1 2q m+1 3 (1q)(2q)1 t3q which is nonnegative because of the hypothesis r1 3andq<1. b)Let us now examine the growth condition used in Theorem 2: (t)_b(t)b(t) 32 (t)(t) where(t) :=p(t)Z+1 tds p(s): (40) Herept) =e(tqtq 0). Therefore (t) =etqZ+1 tesq ds, which gives (t)_b(t)b(t) 32 (t)(t) = 3etq qtq1etqZ+1 tesq ds1 :Inertial dynamics with Hessian damping and time rescaling 23 Let us analyze the sign of the above quantity, which is the same as D(t) :=qtq1etqZ+1 tesq ds1 =qtq1etqZ+1 td ds esq1 qs1qds1 After integration by parts, we get D(t) :=1 q1 +1q qtq1etqZ+1 tesq1 sqds>1 q1 >0: Therefore, the condition (40) is not satised. 5 Illustration of the results Let us particularize our results in some important special cases, and compare them with the existing litterature. We do not detail the proofs which result from the direct applications of the previous theorems and the classical dierential calculus. 5.1 The case b(t) =p(t)3p0. Recall that p(t) = expZt t0 (s)ds . We start with results in [7] concerning the rate of convergence of values in the case b(t) =c0p(t)3p0withp00 andc00. In this case, the system (IGS) ;0;bbecomes: x(t) + (t) _x(t) +c0exp 3p0Zt t0 (s)ds rf(x(t) = 0: (41) Observe that_b(t) 3b(t)=p0 (t) and0(t) = (m+p0)(12rm+p0) 2(t)_ (t)
. Therefore, conditions ( H1) and (H2) of Theorem 6 become after simplication: (H1) [(p0r) + (1rm)] 2(t)_ (t)0; (H2) 2(p0r)(1 + 2(p0r)) 3(t)2(1 + 3(p0r)) (t)_ (t) + (t)0. Sincem1r, instead of (H1), it suces to verify (H+ 1) p0r
2(t)_ (t)0. Theorem 8 Let : [t0;+1)!R+be a nonincreasing and twice continuously dier- entiable function. Suppose that there exists r2(0;1 3 such that (t)2min(0;p0r)2 3(t)on[t0;+1): (42) Then, for each solution trajectory x(
)of(41), we have as t!+1 f(x(t))min Hf=O1 p(t)2r+p0 : (43)24 Hedy ATTOUCH et al. Proof To prove the claim, we use Theorem 5 and distinguish two cases: ?Supposepr0, then (42) implies (t)0, and since is a nonincreasing, we also have _ (t)0; thus both conditions ( H+ 1) and (H2) are satised. ?Supposepr<0, then (42) becomes (t)(2pr)2 3(t) on [t0;+1): (44) Since (
) is a positive and nonincreasing, lim t!+1 (t) =`exists and is equal to zero. Otherwise, by integrating (44) on [ t0;t] fort>t 0, we would have _ (t)_ (t0)2(pr)2Zt t0 (s)3ds2(pr)2`3(tt0): This in turn gives lim t!+1_ (t) = +1, which implies lim t!+1 (t) = +1, that is a contradiction. Then, multiply (44) by _ (t). Since (
) is nonincreasing, we obtain (t)_ (t)2(pr)2 3(t)_ (t)()1 2d dt(_ (t)2)(pr)2 2d dt( 4(t)): By integrating this inequality from ttoT >t , we get _ (T)2_ (t)2(pr)2( 4(T) 4(t)); LettingT!+1;and using lim T!+1 (T) = 0, we obtain _ 2(t)(pr)2 4(t); which is equivalent to j_ (t)jjplj 2(t):Since _ (t)0 andp < r , this gives _ (t)(rp) 2(t);8t>t 0, that is (H+ 1):We have [(pr) + (1rm)] 2(t)_ (t) =2(pr)2 3(t) + (t)|{z} 0 by (44)+2 (13r+ 3p)|{z} 0 sincep<r (t)(pr) 2(t)_ (t)|{z} 0 by (H+ 1)
0: Therefore, (H+ 1) and (H2) are satised. Applying Theorem 5, we conclude. u t As a particular case of Theorem 8, with p0= 0, we obtain the following result. Theorem 9 [7, Theorem 2.1] Let (
)be a nonncreasing function of class C2, and x(
)a solution trajectory of x(t) + (t) _x(t) +c0rf(x(t) = 0: (45) Suppose that (Hr; )9r>0such that2r2 3(t) + (t)0fortlarge enough. Then,f(x(t))minHf=O e2 min(r;1 3)Rt t0 (s)ds ast!+1. Remark 4 The case (t) =1 t(lnt), for 01, was developed in [7]. In that case condition (H3; ) writes as 2(lnt)2+ 3lnt+(+ 1)2r2(lnt)2(1); which is satised for any r1 and any te.Inertial dynamics with Hessian damping and time rescaling 25 {If= 1, thenp(t) = expZt t01 s(lns)ds = exp Zlnt lnt0du x! =lnt lnt0; and forr=1 3, we getf(x(t))minHf=O 1 (lnt)2 3 : {If 0<1, thenp(t) = exp Zlnt lnt01 udu! = exp 1 1(lnt)1(lnt0)1
 ; and, forr=1 3, we also get f(x(t))minHf=O 1 exp 2 3(1)(lnt)1 : 5.2 The case b(t) =c0tqand (t) = t. Whenb(t) =c0tqand (t) = twhere > 0 andq0, we rst observe that p(t) = expRt t0 (s)ds =t t0
:The second-order continuous system becomes: x(t) + t_x(t) +c0tqrf(x(t)) = 0: (46) Applying Theorem 8, we obtain the following new result. Theorem 10 Letx(
)be a solution trajectory of (46) with>1andq0. Suppose that1<3 +q. Then, f(x(t))min Hf=O1 t2+q 3 ;ast!+1: (47) Remark 5 Takingq= 0, a direct application of the above result covers the results obtained in [9,32] (case 3), and in [3,10], (case 3). It suces to take (t) = tandr=1 . More precisely, we get : {if 0<3 thenf(x(t))minHf=O(t2 3), {if>3 thenf(x(t))minHf=O(1 t2). 5.3 The case b(t) =etqand (t) = t1q. Suppose that 0;0q1 and>0. This will allow us to obtain the following exponential convergence rate of the values. Theorem 11 Letx: [t0;+1[!H be a solution trajectory of x(t) + t1q_x(t) +etq rf(x(t)) = 0: (48) Suppose that q, then, ast!+1 f(x(t))min Hf=O e2+q 3tq :26 Hedy ATTOUCH et al. Remark 6 a) Forq== 0, (48) reduces to the system initiated in [32], i.e. x(t) + t_x(t) +rf(x(t)) = 0: Just assuming >0, we obtain lim t!+1 f(x(t))min Hf = 0. b) Forq=1 2we get {If, thenf(x(t))minHf=O e2(2+) 3)p t : {If, thenf(x(t))minHf=O e2p t : c) Forq= 1, direct application of Theorem 11 gives: Corollary 3 (Linear convergence) Letx: [t0;+1[!H be a solution trajectory of x(t) +_x(t) +etrf(x(t)) = 0: (49) If;thenf(x(t))minHf=O e2+ 3t : Let us illustrate these results. Take f(x1;x2) :=1 2 x2 1+x2 2
ln(x1x2), which is a strongly convex function. Trajectories of x(t) +_x(t) +etrf(x(t)) +cetr2f(x(t)) _x(t) = 0; corresponding to dierent values of the parameters ,,, andc, are plotted in Figure 12. The parameter cshows the importance of the Hessian-damping. Fig. 1 Evolution of f(x(t))minffor solutions of (49), (50), and f(x1;x2) =1 2 x2 1+x2 2
ln(x1x2). 2From Scilab version 6.1.0 http://www.scilab.org as an open source softwareInertial dynamics with Hessian damping and time rescaling 27 (t)(t)b(t) f(x(t))minf Reference Cte 0 1 O t1
(1964) [29] Cte Cte 1 O t1
(2002) [2] =t 0 1O t2 3 if 0<3 O t2
if3(2019) [10] (2014) [32] =t Cte 1O t2
if3;> 0 (2016) [17] (t) 0b(t)O p(t)R+1 t(p(s))1ds2 (b(t))1 wherep(t) := expRt t0 (s)ds (2019) [10] =t(t)b(t)O  t2b(t)_(t)(t) t1! (2020) [8] Fig. 2 Convergence rate of f(x(t))minffor instances of Theorem 1 and general f. 5.4 Numerical comparison Figure 2 summarizes our convergence results, according to the behavior of the parameters (t),(t),b(t). Let's comment on them and compare them, separately considering fto be strongly convex or not. 5.4.1 Strongly convex case Suppose that fiss-strongly convex. Following Polyak's [29], the system x(t) + 2ps_x(t) +rf(x(t)) = 0 (50) provides the linear convergence rate f(x(t))infHfCepst, see also [31, The- orem 2.2]. In the presence of an additional Hessian-driven damping term x(t) + 2ps_x(t) +r2f(x(t)) _x(t) +rf(x(t)) = 0 (0) (51) a related linear rate of convergence can be found in [8, Theorem 7]. Let us insist on the fact that, in Corollary 3, we obtain a linear convergence rate for a general convex dierentiable function f. In Figure 1, for the strongly convex function f(x1;x2) =1 2 x2 1+x2 2
ln(x1x2);we can observe that some values of give a better speed of convergence of f(x(t))minf. We can also note that for correctly set, the system (49) provides a better linear convergence rate than the system (50). 5.4.2 Non-strongly convex case We illustrate our results on the following simple example of a non strongly convex minimization problem, with non unique solutions. min R2f(x1;x2) =1 2(x1+ 103x2)2: (52) From Figure 3 we get the following properties: a) The convergence rate of the values is in accordance with Figure 2. b) The system (49) is best for its linear convergence of values. c) The Hessian-driven damping reduces the oscillations of the trajectories.28 Hedy ATTOUCH et al. Fig. 3 Evolution of f(x(t))minffor systems in Figure 2, and f(x1;x2) =1 2 x2 1+ 103x2 2
. 6 Conclusion, perspectives Our study is one of the rst works to simultaneously consider the combination of three basic techniques for the design of fast converging inertial dynamics in con- vex optimization: general viscous damping (and especially asymptotic vanishing damping in relation to the Nesterov accelerated gradient method), Hessian-driven damping which has a spectacular eect on the reduction of the oscillatory aspects (especially for ill-conditionned minimization problems), and temporal rescaling. We have introduced a system of equations-inequations whose solutions provide the coecients of a general Lyapunov functions for these dynamics. We have been able to encompass most of the existing results and nd new solutions for this sys- tem, thus providing new Lyapunov functions. Also, we have been able to explain the mysterious coecients which have been used in recent algorithmic develope- ments, and which were just justied until now by the simplication of complicated calculations. Finally, by playing on fast rescaling methods, we have obtained linear convergence results for general convex functions. This work provides a basis for the development of corresponding algorithmic results. References 1. F. Alvarez, On the minimizing property of a second-order dissipative system in Hilbert spaces, SIAM J. Control Optim., 38 (4) (2000), 1102-1119. 2. F. Alvarez, H. Attouch, J. Bolte, P. Redont, A second-order gradient-like dissipative dy- namical system with Hessian-driven damping, J. Math. Pures Appl., 81 (2002), 747{779. 3. V. Apidopoulos, J.-F. Aujol, Ch. Dossal, Convergence rate of inertial Forward-Backward algorithm beyond Nesterov's rule, Math. Program., 180 (2020) , 137{156. 4. H. Attouch, A. Cabot, Asymptotic stabilization of inertial gradient dynamics with time- dependent viscosity, J. Dierential Equations, 263 (2017), 5412{5458. 5. H. Attouch, A., Cabot, Convergence rates of inertial forward-backward algorithms, SIAM J. Optim., 28 (2018), 849{874. 6. H. Attouch, A. Cabot, Z. Chbani, H. Riahi, Accelerated forward-backward algorithms with perturbations, J. Optim. Theory Appl., 179 (2018), 1{36.Inertial dynamics with Hessian damping and time rescaling 29 7. H. Attouch, A. Cabot, Z. Chbani, H. Riahi, Rate of convergence of inertial gradient dy- namics with time-dependent viscous damping coecient, Evol. Equ. Control Theory, 7 (2018), 353{371. 8. H. Attouch, Z. Chbani, J. Fadili, H. Riahi, First-order optimization algorithms via inertial systems with Hessian driven damping, (2019) HAL-02193846. 9. H. Attouch, Z. Chbani, J. Peypouquet, P. Redont, Fast convergence of inertial dynam- ics and algorithms with asymptotic vanishing damping, Math. Program., (2019) DOI: 10.1007/s10107-016-0992-8. 10. H. Attouch, Z. Chbani, H. Riahi, Rate of convergence of the Nesterov accelerated gradient method in the subcritical case 3, ESAIM Control Optim. Calc. Var., 25(2) (2019), https://doi.org/10.1051/cocv/2017083 11. H. Attouch, Z. Chbani, H. Riahi, Fast proximal methods via time scaling of damped inertial dynamics, SIAM J. Optim., 29 (2019), 2227{2256. 12. H. Attouch, Z. Chbani, H. Riahi, Fast convex optimization via time scaling of damped inertial gradient dynamics, Pure and Applied Functional Analysis, (2019), DOI: 10.1080/02331934.2020.1764953. 13. H. Attouch, Z. Chbani, H. Riahi, Convergence rates of inertial proximal algorithms with general extrapolation and proximal coecients, Vietnam J. Math. 48 (2020), 247{276, https://doi.org/10.1007/s10013-020-00399-y 14. H. Attouch, X. Goudou, P. Redont, The heavy ball with friction method. The continuous dynamical system. Commun. Contemp. Math., 2(1) (2000), 1{34. 15. H. Attouch, S.C. L aszl o, Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators, (2020), https://hal.archives-ouvertes.fr/hal-02549730. 16. H. Attouch, J. Peypouquet, The rate of convergence of Nesterov's accelerated forward- backward method is actually faster than 1 =k2, SIAM J. Optim., 26 (2016), 1824{1834. 17. H. Attouch, J. Peypouquet, P. Redont, Fast convex minimization via inertial dynamics with Hessian driven damping, J. Dierential Equations, 261 (2016), 5734{5783. 18. A. Beck, M. Teboulle, A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sciences, 2(1) (2009), 183{202. 19. R. I. Bot , E. R. Csetnek, Second order forward-backward dynamical systems for monotone inclusion problems, SIAM J. Control Optim., 54 (2016), 1423-1443. 20. R. I. Bot , E. R. Csetnek, S.C. L aszl o, Approaching nonsmooth nonconvex minimization through second order proximal-gradient dynamical systems, J. Evol. Equ., 18(3) (2018), 1291{1318. 21. R. I. Bot , E. R. Csetnek, S.C. L aszl o, Tikhonov regularization of a second order dynamical system with Hessian damping, Math. Program., DOI:10.1007/s10107-020-01528-8. 22. O. G uler, On the convergence of the proximal point algorithm for convex optimization, SIAM J. Control Optim., 29 (1991), 403{419. 23. O. G uler, New proximal point algorithms for convex minimization, SIAM Journal on Optimization, 2 (4) (1992), 649{664. 24. A. Haraux, Syst emes Dynamiques Dissipatifs et Applications, Recherches en Math ematiques Appliqu ees 17, Masson, Paris, 1991. 25. R. May, Asymptotic for a second order evolution equation with convex potential and vanishing damping term, Turkish Journal of Mathematics, 41 (3) (2016), 681{685. 26. Y. Nesterov, A method of solving a convex programming problem with convergence rate O(1=k2), Soviet Mathematics Doklady, 27 (1983), 372{376. 27. Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course. Springer Science+Business Media New York (2004). 28. J. Peypouquet, S. Sorin, Evolution equations for maximal monotone operators: asymptotic analysis in continuous and discrete time, J. Convex Anal, 17 (3-4) (2010), 1113{1163. 29. B.T. Polyak, Some methods of speeding up the convergence of iteration methods, U.S.S.R. Comput. Math. Math. Phys., 4 (1964), 1{17. 30. B. Shi, S.S Du, M.I. Jordan, W.J. Su, Understanding the acceleration phenomenon via high-resolution dierential equations, arXiv:submit/2440124[cs.LG] 21 Oct 2018. 31. W. Siegel, Accelerated rst-order methods: Dierential equations and Lyapunov functions. arXiv:1903.05671v1 [math.OC] (2019). 32. W.J. Su, S. Boyd, E.J Cand es, A dierential equation for modeling Nesterov's accelerated gradient method: theory and insights,Neural Information Processing Systems, 27 (2014), 2510{2518.

2020-09-16

In a Hilbert setting, we develop fast methods for convex unconstrained optimization. We rely on the asymptotic behavior of an inertial system combining geometric damping with temporal scaling. The convex function to minimize enters the dynamic via its gradient. The dynamic includes three coefficients varying with time, one is a viscous damping coefficient, the second is attached to the Hessian-driven damping, the third is a time scaling coefficient. We study the convergence rate of the values under general conditions involving the damping and the time scale coefficients. The obtained results are based on a new Lyapunov analysis and they encompass known results on the subject. We pay particular attention to the case of an asymptotically vanishing viscous damping, which is directly related to the accelerated gradient method of Nesterov. The Hessian-driven damping significantly reduces the oscillatory aspects. As a main result, we obtain an exponential rate of convergence of values without assuming the strong convexity of the objective function. The temporal discretization of these dynamics opens the gate to a large class of inertial optimization algorithms.

Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling

2009.07620v1

All-Optical Study of Tunable Ultrafast Spin Dynamics in [Co/Pd]-NiFe Systems: The Role of Spin-Twist Structure on Gilbert Damping Chandrima Banerjee,1Semanti Pal,1Martina Ahlberg,2T. N. Anh Nguyen,3, 4Johan Akerman,2, 4and Anjan Barman1, 1Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sec. III, Salt Lake, Kolkata 700 098, India 2Department of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden 3Laboratory of Magnetism and Superconductivity, Institute of Materials Science, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam. 4Department of Materials and Nano Physics, School of Information and Communication Technology, KTH Royal Institute of Technology, Electrum 229, SE-16440 Kista, Sweden (Dated: April 12, 2016) We investigate optically induced ultrafast magnetization dynamics in [Co(0.5 nm)/Pd(1 nm)] 5/NiFe( t) exchange-spring samples with tilted perpendicular magnetic anisotropy using a time- resolved magneto-optical Kerr eect magnetometer. The competition between the out-of-plane anisotropy of the hard layer, the in-plane anisotropy of the soft layer and the applied bias eld reor- ganizes the spins in the soft layer, which are modied further with the variation in t. The spin-wave spectrum, the ultrafast demagnetization time, and the extracted damping coecient all depend on the spin distribution in the soft layer, while the latter two also depend on the spin-orbit coupling between the Co and Pd layers. The spin-wave spectra change from multimode to single-mode as t increases. At the maximum eld reached in this study, H=2.5 kOe, the damping shows a nonmono- tonic dependence on twith a minimum at t= 7.5 nm. For t<7.5 nm, intrinsic eects dominate, whereas for t>7.5 nm, extrinsic eects govern the damping mechanisms. I. INTRODUCTION Nonuniform magnetic structures, including exchange bias (ferromagnet/antiferromagnet)3,24and exchange- spring (ferromagnet/ferromagnet)5{8systems, have recently been explored extensively on account of their intrinsic advantages for applications in both permanent magnets and recording media. Exchange-spring (ES) magnets are systems of exchanged-coupled hard and soft magnetic layers that behave as a single magnet. Here, the high saturation magnetization ( Ms) of the soft phase and the high anisotropy ( Hk) of the hard phase result in a large increase in the maximum energy product. This makes them useful as permanent magnets in energy ap- plications such as engines or generators in miniaturized devices. On the other hand, for spintronic applications, the soft phase is used to improve the writability of the magnetic media, which in turn is stabilized by the magnetic conguration of the hard layer. Consequently, a wealth of research has been devoted to investigating the static and dynamic magnetic properties, including the switching behavior and exchange coupling strength, in ES systems. In case of ES systems with tilted anisotropy, the hard and soft phases consist of materials with out-of-plane (OOP) and in-plane (IP) anisotropies, respectively. This combination results in a canting of the magnetization of the soft layer with a wide and tunable range of tilt angles. The advantage of such a hybrid anisotropy sys- tem is that it is neither plagued by the poor writability and thermal instability of systems with IP anisotropy, nor does it lead to very high switching elds, as in OOPsystems. As a result, these materials provide additional degrees of freedom to control the magnetization dynam- ics in magnetic nanostructures, and hint at potential applications in novel spintronic devices utilizing the spin-transfer torque (STT) eect|such as spin-torque oscillators (STOs)25,26and STT-MRAMs. So far, numerous studies have been performed on such systems where the exchange coupling between the hard and soft layers has been tailored by varying the layer thickness,12,13layer composition,19number of repeats,15and interfacial anisotropy.13The litera- ture describes investigations of domain structure and other static magnetic properties for [Co/Pd]/Co,14 [Co/Pd]/NiFe,12,14,19,21[Co/Pd]/CoFeB,14,15,20 [Co/Pd]-Co-Pd-NiFe,13[Co/Ni]/NiFe,4and CoCrPt- Ni11|these systems being studied with static mag- netometry, magnetic force microscopy (MFM), and micromagnetic simulations. The magnetization dy- namics in such systems have also been measured using Brillouin light scattering (BLS)19,20and ferromagnetic resonance (FMR)21experiments, where the spin-wave (SW) modes have been investigated by varying the thick- ness of the soft layer and changing the conguration of the hard layer. In any process involving magnetization dynamics, the Gilbert damping constant ( ) plays a key role in optimizing writing speeds and controlling power consumption. For example, in case of STT-MRAM and magnonic devices, low facilitates a lower writing current and the longer propagation of SWs, whereas a higheris desirable for increasing the reversal rates and the coherent reversal of magnetic elements, which are required for data storage devices.arXiv:1604.02998v1 [cond-mat.mtrl-sci] 11 Apr 20162 46810121416350400450500 )sf( emit noitazitengameDt (nm)(d)(a) -202 600 1200 1800-2-10 Kerr rotation (a rb. unit) Time (ps)(b)0 10 20 30 Power (arb. unit) Frequency (GHz)(c) (b) -202 60012001800-2-10Kerrrotation(arb.unit) Time(ps) Figure 1. (color online) (a) Schematic of the two-color pump- probe measurement of the time-resolved magnetization dy- namics of exchange-spring systems. The bias eld is applied with a small angle to the normal of the sample plane. (b) Typical time-resolved Kerr rotation data revealing ultrafast demagnetization, fast and slow relaxations, and precession of magnetization for the exchange-spring system with t= 7.5 nm at H= 2.5 kOe. (c) FFT spectrum of the background- subtracted time-resolved Kerr rotation. (d) Variation of de- magnetization time with t. In this paper, we present all-optical excitation and de- tection of magnetization dynamics in [Co(0.5 nm)/Pd(1 nm)] 5/NiFe( t) tilted anisotropy ES systems, with varying soft layer thickness ( t), using a time-resolved magneto- optical Kerr eect (TR-MOKE) magnetometer. The dy- namical magnetic behavior of similar systems has previ- ously been studied using BLS19and FMR21measure- ments. However, a detailed study of the precessional magnetization dynamics and relaxation processes in such composite hard/soft systems is yet to be carried out. The advantage of implementing TR-MOKE is that here the magnetization dynamics can be measured on dier- ent time scales and the damping is measured directly in the time domain, and is therefore more reliable. We investigate the ultrafast magnetization dynamics over pi- cosecond and picosecond time scales. The ultrafast de- magnetization is examined and found to change due to the modied spin structure in the soft layer for dierent tvalues. The extracted SW spectra are strongly depen- dent on t. An extensive study of the damping coecient reveals that the extrinsic contribution to the damping is more dominant in the higher thickness regime, while intrinsic mechanisms govern the behavior at lower thick- nesses.II. EXPERIMENTAL DETAILS A. Sample fabrication The samples were fabricated using dc mag- netron sputtering and have the following structure: Ta(5nm)/Pd(3nm)/[Co(0.5nm)/Pd(1nm)] 5=Ni80Fe20(t) /Ta(5nm), where t= 4{20 nm. The chamber base pres- sure was below 3 108Torr, while the Ar work pressure was 2 and 5 mTorr for the Ta, NiFe and Co, Pd layers, respectively. The samples were deposited at room temperature on naturally oxidized Si(100) substrates. The 5 nm Ta seed layer was used to induce fcc-(111) orientation in the Pd layer, which improves the perpendicular magnetic anisotropy of the Co/Pd multilayers; a Ta cap layer was used to avoid oxidation, which has been reported in previous studies.12{14The layer thicknesses are determined from the deposition time and calibrated deposition rates. B. Measurement technique To investigate the precessional frequency and damp- ing of these samples, the magnetization dynamics were measured by using an all-optical time-resolved magneto- optical Kerr eect (TR-MOKE) magnetometer2based on a two-color optical pump-probe experiment. The mea- surement geometry is shown in Fig. 1(a). The magne- tization dynamics were excited by laser pulses of wave- length () 400 nm (pulse width = 100 fs, repetition rate = 80 MHz) of about 16 mJ/cm2 uence and probed by laser pulses with = 800 nm (pulse width = 88 fs, rep- etition rate = 80 MHz) of about 2 mJ/cm2 uence. The pump and probe beams are focused using the same micro- scope objective with N.A. of 0.65 in a collinear geometry. The probe beam is tightly focused to a spot of about 800 nm on the sample surface and, as a result, the pump becomes slightly defocused in the same plane to a spot of about 1 m. The probe beam is carefully aligned at the centre of the pump beam with slightly larger spot size. Hence, the dynamic response is probed from a ho- mogeneously excited volume. The bias eld was tilted at around 15to the sample normal (and its projection along the sample normal is referred to as Hin this ar- ticle) in order to have a nite demagnetizing eld along the direction of the pump beam. This eld is eventually modied by the pump pulse which induces precessional magnetization dynamics in the samples. The Kerr rota- tion of the probe beam, back-re ected from the sample surface, is measured by an optical bridge detector us- ing phase sensitive detection techniques, as a function of the time-delay between the pump and probe beams. Fig- ure 1(b) presents typical time-resolved Kerr rotation data from the ES sample with t= 7.5 nm at a bias eld H= 2.5 kOe. The data shows a fast demagnetization within 500 fs and a fast remagnetization within 8 ps, followed by a slow remagnetization within 1800 ps. The precessional3 (b) 010 20 30 0 1 2 Power (arb. unit) Kerr Rotation(arb. unit) Frequency (GHz)4.5 nm 5.5 nm 7.5 nm 8 nm 15 nm Time (ns)20 nm B A NiFe (t = 20 nm) Co/Pd 1 -1 Normalized Mz Co/Pd NiFe (t = 6 nm) Co/Pd NiFe ( t = 10 nm) (a) Figure 2. (color online) (a) Background-subtracted time- resolved Kerr rotation and the corresponding FFT spectra for samples with dierent tvalues at H= 2.5 kOe. The black lines show the t according to Eq. 1. (b) Simulated static magnetic congurations for samples with t= 20, 10, and 6 nm with a bias eld H= 2.5 kOe in the experimental conguration. The simulated samples are not to scale. The color map is shown at the bottom of the gure. dynamics appear as an oscillatory signal above the slowly decaying part of the time-resolved Kerr rotation data. This part was further analyzed and a fast Fourier trans- form (FFT) was performed to extract the corresponding SW modes, as presented in Fig. 1(c).III. RESULTS AND DISCUSSIONS In order to closely observe the ultrafast demagnetiza- tion and fast remagnetization, we recorded the transient MOKE signals for delay times up to 30 ps at a resolution of 50 fs. In Fig. 1(d), the demagnetization times are plot- ted as a function of t. We observe that the demagnetiza- tion is fastest in the thinnest NiFe layer ( t= 4 nm) and increases sharply with the increase in t, becoming con- stant at 500 fs at t= 5 nm. At t= 10 nm, it decreases drastically to 400 fs and remains constant for further in- creases in t. For t<5 nm, the laser beam penetrates to the Co/Pd layer. In this regime, the large spin-orbit coupling of Pd enhances the spin- ip rate, resulting in a faster demagnetization process. As tincreases, the top NiFe layer is primarily probed. Here, the spin congura- tion across the NiFe layer, which is further aected by the competition between the in-plane and the out-of-plane anisotropies of the NiFe and [Co/Pd] layers, governs the demagnetization process. Qualitatively, ultrafast demag- netization can be understood by direct transfer of spin angular momentum between neighboring domains10,23. which may be explained as follows: For t>8 nm, the magnetization orientation in the NiFe layer varies over a wide range of angles across the lm thickness, where the magnetization gradually rotates from nearly perpendicu- lar at the Co/Pd and NiFe interface to nearly parallel to the surface plane in the topmost NiFe layer. Such a spin structure across the NiFe layer thickness can be seen as a network of several magnetic sublayers, where the spin ori- entation in each sublayer deviates from that of the neigh- boring sublayer. This canted spin structure accelerates the spin- ip scattering between the neighboring sublay- ers and thus results in a shorter demagnetization time, similar to the work reported by Vodungbo et al.23On the other hand, for 5 nm <t<8 nm, the strong out-of-plane anisotropy of the Co/Pd layer forces the magnetization in the NiFe lm towards the direction perpendicular to the surface plane, giving rise to a uniform spin structure. The strong coupling reduces the transfer rate of spin an- gular momentum and causes the demagnetization time to increase. To investigate the variation of the precessional dynamics with t, we further recorded the time-resolved data for a maximum duration of 2 ns at a resolution of 10 ps. Fig- ure 2(a) shows the background-subtracted time-resolved Kerr rotation data for dierent values of tatH= 2.5 kOe and the corresponding fast Fourier transform (FFT) power spectra. Four distinct peaks are observed in the power spectrum of t= 20 nm, which reduces to two for t= 15 nm. This is probably due to the rela- tive decrease in the nonuniformity of the magnetization across the NiFe thickness, which agrees with the varia- tion in the demagnetization time, as described earlier. To conrm this, we simulated the static magnetic congura- tions of these samples in a eld of H= 2:5 kOe using the LLG micromagnetic simulator.18Simulations were per- formed by discretizing the samples in arrays of cuboidal4 cells with two-dimensional periodic boundary conditions applied within the sample plane. The simulations assume the Co/Pd multilayer as an eective medium16with sat- uration magnetization Ms= 690 emu/cc, exchange sti- ness constant A= 1.3erg/cm, and anisotropy constant Ku1= 5.8 Merg/cc along the (001) direction, while the material parameters used for the NiFe layer were Ms= 800 emu/cc, A= 1.3erg/cm, and Ku1= 0.17The in- terlayer exchange between Co/Pd and NiFe is set to 1.3 erg/cm and the gyromagnetic ratio = 18.1 MHz Oe1 is used for both layers. The Co/Pd layer was discretized into cells of dimension 5 52 nm3and the NiFe layer was discretized into cells of dimension 5 51 nm3. The results are presented in Fig. 2(b) for t= 20, 10, and 6 nm samples. The nonuniform spin structure is promi- nent in the NiFe layer of the t= 20 nm sample, which modies the SW spectrum of this sample, giving rise to the new modes.17With the reduction of t, the spin struc- ture in the NiFe layer gradually becomes more uniform, while at t= 7.5 nm it is completely uniform over the whole thickness prole. Hence, for low values of t, the power spectra shows a single peak due to the collective precession of the whole stack. The variation in precession frequency with tis plotted in Fig. 3(a). The frequency of the most intense mode shows a slow decrease down to t= 7.5 nm, below which it increases sharply down to the lowest thickness t= 4.5 nm, exhibiting precessional dy- namics. This mode is basically the uniform mode of the system and follows Kittel's equation.9The variation in frequency depicts the evolution of the eective anisotropy from OOP to IP with increasing t, which is in agreement with previously reported results.21For lower t, the sys- tem displays an OOP easy axis, owing to the strong OOP anisotropy of the Co/Pd multilayer. This is manifested as a sharp increase in the frequency with decreasing t below 7.5 nm. For greater thicknesses, the eect of the perpendicular anisotropy of the Co/Pd multilayer gradu- ally decreases and the eect of the in-plane NiFe becomes more prominent. These two anisotropies cancel near t= 7.5 nm, resulting in a minimum frequency, as shown in Fig. 3(a). To extract the damping coecient, the time domain data was tted with an exponentially damped harmonic function given by Eq. 1. M(t) =M(0)et sin(2ft) (1) where the relaxation time is related to the Gilbert damping coecient by the relation = 1/(2f). Here, fis the experimentally obtained precession fre- quency and is the initial phase of the oscillation. The tted data for various values of tis shown by the solid black lines in Fig. 2. We did not extract a value of fort = 15 and 20 nm due to the occurrence of multimode os- cillations, which may lead to an erroneous estimate of the damping. The extracted values are plotted against tin Fig. 3(b) for two dierent eld values of 2.5 and 1.3 kOe. The evolution of as a function of tdepends signicantly 5 6 7 8 9 100.0160.0200.0240.0280.032 5 10 15 2041216 t(nm) t(nm)(a) (b)Frequency(GHz)Figure 3. (color online) Evolution of (a) spin-wave frequency and (b) Gilbert damping constant as a function of tat 1.3 kOe (green circles) and 2.5 kOe (violet circles). onH, as can be seen from the gure. This is because of the dierent mechanisms responsible for determining the damping in dierent samples, as will be discussed later. An interesting trend in the vs.tplot is observed forH= 2.5 kOe. For 10 nm t7.5 nm,decreases with decreasing tand reaches a minimum value of about 0.014 for t= 7.5 nm. Below this thickness, increases monotonically and reaches a value of about 0.022 for the lowest thickness. This variation of is somewhat cor- related with the variation of precession frequency with thickness. In the thinner regime, we probe both the NiFe layer and a fraction of the Co/Pd multilayer and the relative contribution from the latter increases as t decreases. The occurrence of a single mode oscillation points towards a collective precession of the stack, which may be considered a medium with eective magnetic pa- rameters consisting of both NiFe and Co/Pd layers. The variation in damping may be related to the variation in the anisotropy of the material. The competing IP and OOP anisotropies of the NiFe and Co/Pd layers lead to the appearance of a minimum in the damping. The damping in this system may have multiple contributions, namely (a) dephasing of the uniform mode in the spin- twist structure1(b) interfacial d-dhybridization at the Co/Pd interface16, and (c) spin pumping into the Pd layer.22The rst is an extrinsic mechanism and is dom- inant in samples with higher NiFe thicknesses, while the other two mechanisms are intrinsic damping mechanisms. Fort>7.5 nm, due to the nonuniformity of the spin distribution, the dominant mode undergoes dynamic de- phasing and the damping thus increasescompared to the magnetically uniform samples. With the increase in NiFe thickness, the nonuniformity of spin distribution and the consequent mode dephasing across its thickness increases, leading to an increase in the damping value. Hence, in samples with higher tvalues, dephasing is the dominant mechanism, while at lower tvalues|i.e., when the con- tribution from the Co/Pd multilayer is dominant|the spin-orbit coupling and spin pumping eects dominate. At intermediate tvalues, the extrinsic and intrinsic ef- fects compete with each other, leading to a minimum in the damping. However, the damping increases mono- tonically with tin a lower eld of H=1.3 kOe. For a deeper understanding of this eect, we have measured 5 24681012140.0120.0160.0200.0240.0280.0324 56789100.0140.0210.0280.0350.042(b) 5nm 5.5nm 6.5nm 7nm/s61537F requency (GHz)(a) 10nm 8.5nm 8nm 7.5nm 7nm/s61537F requency (GHz) Figure 4. (color online) Dependence of Gilbert damping co- ecient on soft layer thickness ( t) for (a) 7{10 nm and (b) 5{7 nm, respectively. as a function of precession frequency f. Figures 4(a){(b) show the variation of with f. Two dierent regimes in the thickness are presented in (a) and (b) to show the rate of variation more clearly. For 10 nm t7 nm, decreases strongly with the decrease in fand the rate of variation remains nearly constant with t. This is the sig- nature of extrinsic damping generated by the nonuniform spin distribution. However, for t= 6.5 nm, the rate falls drastically and for t5.5 nm,becomes nearly indepen- dent of t, which indicates that purely intrinsic damping is operating in this regime. This conrms the competition between two dierent types of damping mechanisms in these samples. The study demonstrates that various aspects of ul- trafast magnetization dynamics|namely demagnetiza- tion time, precession frequency, number of modes, and damping|are in uenced by the spin distribution in the soft magnetic layer, as well as by the properties of the hard layer. By changing the thickness of the soft layer, the relative contributions of these factors can be tuned eectively. This enables ecient control of the damp- ing and other magnetic properties over a broad range, and will hence be very useful for potential applications in spintronic and magnonic devices.IV. CONCLUSION In summary, we have employed the time-resolved MOKE technique to measure the evolution of ul- trafast magnetization dynamics in exchange-coupled [Co/Pd] 5/NiFe( t) multilayers, with varying NiFe layer thicknesses, by applying an out-of-plane bias magnetic eld. The coupling of a high-anisotropy multilayer with a soft layer allows broad control over the spin struc- ture, and consequently other dynamic magnetic prop- erties which are strongly dependent on t. The ultra- fast demagnetization displayed a strong variation with t. The reason for this was ascribed to the chiral-spin- structure-dependent spin- ip scattering in the top NiFe layer, as well as to interfacial 3 d-4dhybridization of Co/Pd layer. The precessional dynamics showed mul- tiple spin-wave modes for t= 20 nm and 15 nm, whereas a single spin-wave mode is observed for thinner NiFe lay- ers following the change in the magnetization prole with decreasing t. The precession frequency and the damp- ing show strong variation with the thickness of the NiFe layer. The changes in frequency are understood in terms of the modication of the anisotropy of the system, while the variation in damping originates from the competition between intrinsic and extrinsic mechanisms, which are somewhat related to the anisotropy. The observed dy- namics will be important for understanding the utiliza- tion of tilted anisotropy materials in devices such as spin- transfer torque MRAM and spin-torque nano-oscillators. V. ACKNOWLEDGEMENTS We acknowledge nancial support from the G oran Gustafsson Foundation, the Swedish Research Coun- cil (VR), the Knut and Alice Wallenberg Foundation (KAW), and the Swedish Foundation for Strategic Re- search (SSF). This work was also supported by the Euro- pean Research Council (ERC) under the European Com- munity's Seventh Framework Programme (FP/2007{ 2013)/ERC Grant 307144 "MUSTANG". AB acknowl- edges the nancial support from the Department of Sci- ence and Technology, Government of India (Grant no. SR/NM/NS-09/2011(G)) and S. N. Bose National Centre for Basic Sciences, India (Grant no. SNB/AB/12-13/96). C.B. thanks CSIR for the senior research fellowship. abarman@bose.res.in 1A. Barman and S. Barman. Dynamic dephasing of mag- netization precession in arrays of thin magnetic elements. Phys. Rev. B , 79:144415, 2009. 2A. Barman and A. Haldar. Chapter One - Time-Domain Study of Magnetization Dynamics in Magnetic Thin Films and Micro- and Nanostructures. volume 65 of Solid State Phys. , pages 1{108. Academic Press, 2014.3R. E. Camley, B. V. McGrath, R. J. Astalos, R. L. Stamps, J.-V. Kim, and L. Wee. Magnetization dynamics: A study of the ferromagnet/antiferromagnet interface and exchange biasing. J. Vac. Sci. Technol. A , 17:1335, 1999. 4S. Chung, S. M. Mohseni, V. Fallahi, T. N. A. Nguyen, N. Benatmane, R. K. Dumas, and J. Akerman. 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2016-04-11

We investigate optically induced ultrafast magnetization dynamics in [Co(0.5 nm)/Pd(1 nm)]x5/NiFe(t) exchange-spring samples with tilted perpendicular magnetic anisotropy using a time-resolved magneto-optical Kerr effect magnetometer. The competition between the out-of-plane anisotropy of the hard layer, the in-plane anisotropy of the soft layer and the applied bias field reorganizes the spins in the soft layer, which are modified further with the variation in t. The spin-wave spectrum, the ultrafast demagnetization time, and the extracted damping coefficient all depend on the spin distribution in the soft layer, while the latter two also depend on the spin-orbit coupling between the Co and Pd layers. The spin-wave spectra change from multimode to single-mode as t increases. At the maximum field reached in this study, H=2.5 kOe, the damping shows a nonmonotonic dependence on t with a minimum at t=7.5 nm. For t<7.5 nm, intrinsic effects dominate, whereas for t>7.5 nm, extrinsic effects govern the damping mechanisms.

All-Optical Study of Tunable Ultrafast Spin Dynamics in [Co/Pd]-NiFe Systems: The Role of Spin-Twist Structure on Gilbert Damping

1604.02998v1

MNRAS 000, 000{000 (0000) Preprint 10 June 2021 Compiled using MNRAS L ATEX style le v3.0 Grammage of cosmic rays in the proximity of supernova remnants embedded in a partially ionized medium S. Recchia1;2?, D. Galli3, L. Nava4, M. Padovani3, S. Gabici5, A. Marcowith6, V. Ptuskin7, G. Morlino3 1Dipartimento di Fisica, Universit a di Torino, via P. Giuria 1, 10125 Torino, Italy 2Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy 3INAF{Osservatorio Astrosico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy 4INAF-Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807 Merate, Italy 5Universit e de Paris, CNRS, Astroparticule et Cosmologie, F-75006 Paris, France 6Laboratoire Univers et particules de Montpellier, Universit e Montpellier/CNRS, F-34095 Montpellier, France 7Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation, 108840, Troitsk, Moscow, Russia Accepted XXX. Received YYY; in original form ZZZ ABSTRACT We investigate the damping of Alfv en waves generated by the cosmic ray resonant streaming instability in the context of the cosmic ray escape and propagation in the proximity of supernova remnants. We consider ion-neutral damping, turbulent damp- ing and non linear Landau damping in the warm ionized and warm neutral phases of the interstellar medium. For the ion-neutral damping, up-to-date damping coecients are used. We investigate in particular whether the self-connement of cosmic rays nearby sources can appreciably aect the grammage. We show that the ion-neutral damping and the turbulent damping eectively limit the residence time of cosmic rays in the source proximity, so that the grammage accumulated near sources is found to be negligible. Contrary to previous results, this also happens in the most extreme scenario where ion-neutral damping is less eective, namely in a medium with only neutral helium and fully ionized hydrogen. Therefore, the standard picture, in which CR secondaries are produced during the whole time spent by cosmic rays throughout the Galactic disk, need not to be deeply revisited. Key words: 1 INTRODUCTION The most popular hypothesis for the origin of Galactic cos- mic rays (CRs) invokes supernova remnants (SNRs) as the main sources of such particles (see e.g. Blasi 2013; Gabici et al. 2019). In this scenario, which in the last decades had become a paradigm, CR diusion plays a central role. Diu- sion is the key ingredient at the base of the diusive shock acceleration of particles at SNRs (e.g. Drury 1983). Diu- sion also aects the escape of CRs from the acceleration site and the subsequent propagation in the source region, with prominent implications for -ray observations (Aharonian & Atoyan 1996; Gabici et al. 2009; Casanova et al. 2010; Ohira et al. 2011; Nava & Gabici 2013). Finally, diusion determines the connement time of CRs in the Galaxy, thus aecting the observed spectrum and the abundances of sec- ondary spallation nuclei and of unstable isotopes (Ptuskin & Soutoul 1998; Wiedenbeck et al. 2007). ?E-mail: sarah.recchia@unito.itThe diusion of CRs is thought to be mostly due to the resonant scattering o plasma waves whose wavelength is comparable to the particle's Larmor radius rL= mpc2=eB, wherempis the proton mass, Bis the magnetic eld strength and the Lorentz factor (see e.g. Skilling 1975a). The magneto-hydrodynamic (MHD) turbulence relevant for CR propagation is composed of incompressible Alfv enic and compressible (fast and slow) magnetosonic uctuations (Cho & Lazarian 2002; Fornieri et al. 2021). MHD turbulence is ubiquitous in the interstellar space and may be injected by astrophysical sources (see e.g. Mac Low & Klessen (2004)) but also by CRs themselves. The active role of CRs in pro- ducing the waves responsible for their scattering has been widely recognized (see e.g. Wentzel 1974; Skilling 1975b; Ce- sarsky 1980; Amato 2011). In fact, spatial gradients in the CR density, as those found in the source vicinity, lead to the excitation of Alfv en waves at the resonant scale (Ptuskin et al. 2008). This process, called resonant streaming insta- bility , produces waves that propagate along magnetic eld lines in the direction of decreasing CR density. ©0000 The AuthorsarXiv:2106.04948v1 [astro-ph.HE] 9 Jun 20212S. Recchia et al. The density of Alfv en waves that scatter CRs is lim- ited by several damping processes. The most relevant are: (i) ion-neutral damping in a partially ionized medium (Kul- srud & Pearce 1969; Kulsrud & Cesarsky 1971; Zweibel & Shull 1982); ( ii) turbulent damping, due to the interaction of a wave with counter-propagating Alfv en wave packets. Such waves may be the result of a background turbulence injected on large scales and cascading to the small scales (we indicate this damping as FG, after Farmer & Goldreich 2004); ( iii) non-linear Landau (NLL) damping, due to the interaction of background thermal ions with the beat of two interfering Alfv en waves (see e.g. Felice & Kulsrud 2001; Wiener et al. 2013). The relative importance of these eects depends sig- nicantly on the physical conditions and chemical composi- tion of the ambient medium. A few other collisionless and collisional damping processes can impact magnetohydrody- namical wave propagation in a partially ionized gas but they mostly aect high-wavenumber perturbations (Yan & Lazar- ian 2004). Recently, it has been suggested that dust grains may also contribute to the damping of Alfv en waves (Squire et al. 2021). In this paper we investigate the escape of CRs from SNRs, and their subsequent self connement in the source region, as due to the interplay between the generation of Alfv en waves by CR streaming instability, and the damp- ing process mentioned above. Our main goal is to establish whether the self-connement of CRs nearby sources can ap- preciably aect the grammage accumulated by these parti- cles. In fact, if this is the case, a signicant fraction of CR secondaries would be produced in the vicinity of CR sources, and not during the time spent by CRs in the Galactic disk, as commonly assumed. This would constitute a profound modication of the standard view of CR transport in the Galaxy (see, e.g. D'Angelo et al. 2016). In particular, we focus on the CR propagation in partially ionized phases of the interstellar medium (ISM), showing that the ion-neutral and FG damping can signicantly aect the residence time of CRs nearby their sources. We nd that, for typical condi- tions, the grammage accumulated by CRs in the vicinity of sources is negligible compared to that accumulated during the time spent in the Galaxy. Even in the case of a medium made of fully ionized H and neutral He, the combination of ion-neutral and turbulent damping can substantially aect the connement time1. This paper is organized as follows: in Sec. 2 we describe the damping of Alfv en waves by ion-neutral collisions in vari- ous partially ionized phases of the ISM, and by other damp- ing mechanisms; in Sec. 3 we illustrate the equations and the setup of our model of CR escape and propagation in the proximity of SNRs, the time dependent CR spectrum and diusion coecient, the residence time of CRs in the source proximity and the implications on the grammage; in Sec. 4 we describe our results; and nally in Sec. 5 we draw our conclusions. 1The case of a fully neutral (atomic or partially molecular) medium and of a diuse molecular medium (see, e.g. Brahimi et al. 2020) are not treated here, since the lling factor of such phases is small, but we report the ion-neutral damping rate for such media for the sake of completeness. The case of a fully ion- ized medium has been extensively treated by Nava et al. (2019).2 DAMPING OF ALFV EN WAVES 2.1 Ion-neutral damping The Galaxy is composed, for most of its volume, by three ISM phases, namely the warm neutral medium (WNM, ll- ing factor25%), warm ionized medium (WIM, lling fac- tor25%) and hot ionized medium (HIM, lling factor 50%, see e.g. Ferri ere 2001; Ferri ere 2019). The physi- cal characteristics of these phases are summarised in Ta- ble 1 (from Jean et al. 2009, see also Ferri ere 2001; Fer- ri ere 2019). The physical characteristics of the cold neu- tral medium (CNM) and the diuse medium (DiM) are also listed for completeness, while their lling factor is .1% (Fer- ri ere 2001; Ferri ere 2019). In the regions where neutrals are present, like the WNM and the WIM, the rate of ion-neutral damping depends on the amount and chemical species of the colliding particles. In the WNM and WIM the ions are H+, while neutrals are He atoms (with a H/He ratio of 10%) and H atoms with a fraction that varies from phase to phase. The main processes of momentum transfer (mt) be- tween ions and neutrals are elastic scattering by induced dipole, and charge exchange (ce). In the former case, domi- nant at low collision energies, the incoming ion is de ected by the dipole electric eld induced in the neutral species, according to its polarizability (Langevin scattering); in the latter case the incoming ion takes one or more electrons from the neutral species, which becomes an ambient ion. The fric- tion force per unit volume Fiexerted on an ion iis thus the sum of Fi;mt+Fi;ce. With the exception of collisions between an ion and a neutral of the same species, as in the important case of col- lisions of H+ions with H atoms (see Sec. A1), the two pro- cesses are well separated in energy. At low collision energies elastic scattering dominates, and the friction force is Fi;mt=ninninhmtviin(unui); (1) whereniandnnare the ion and neutral densities, uiand unare the ion and neutral velocities, inis the reduced mass of the colliding particles, mtis the momentum trans- fer (hereafter m.t.) cross section, and the brackets denote an average over the relative velocity of the colliding particles. At high collision energies (above 102eV), the dom- inant contribution to the transfer of momentum is charge exchange A++ B!A + B+: (2) If the charge exchange rate coecient is approximately in- dependent of temperature, and there is no net backward- forward asymmetry in the scattering process (two conditions generally well satised), Draine (1986) has shown that the friction force on the ions takes the form Fi;ce=ninnhceviinm2 nunm2 iui mn+mi; (3) whereceis the charge exchange (hereafter c.e.) cross sec- tion, andmn(i)the mass of the neutral (ion). The collisional rate coecients hmtviinandhceviin are often estimated from the values given by Kulsrud & Ce- sarsky (1971) or Zweibel & Shull (1982) for H+{ H collisions (e.g. D'Angelo et al. 2016; Nava et al. 2016; Brahimi et al. 2020). The rate coecients for collisions between various MNRAS 000, 000{000 (0000)3 Table 1. ISM phases and parameters adopted in this work. Tis the gas temperature, Bthe interstellar magnetic eld, nthe total gas density,fthe ionisation fraction, the helium fraction and Linjthe injection scale of the background magnetic turbulence. T(K)B(G)n(cm3) neutral ion f  L inj(pc) WIM 8000 5 0.35 H, He H+0.60.9 00.1 He H+1 0.1 50 WNM 8000 5 0.35 H, He H+7103510200.1 50 CNM 80 5 35 H, He C+41041030.1 1-50 DiM 50 5 300 H 2, He C+1040.1 1-50 HIM 10650:01 - H+1.0 0.0 100 species of ions and neutrals adopted in this study are de- scribed in detail in Sec. A1. For elastic collisions, they have been taken from the compilation by Pinto & Galli (2008); for charge exchange, they have been calculated from the most updated available cross sections. Ion-neutral collisions are one of the dominant damping processes for Alfv en waves propagating in a partially ionized medium (see Piddington 1956; Kulsrud & Pearce 1969). In the case of elastic ion-neutral collisions, (Eq. 1), the disper- sion relation for Alfv en waves in this case is !(!2!2 k) +iin[(1 +)!2!2 k] = 0; (4) where!is the frequency of the wave, !k=kvA;iis the wavevector in units of the Alfv en speed of the ions vA;i=Bp4mini; (5) inis the ion-neutral collision frequency in=mn mi+mnhmtviinnn; (6) andis the ion-to-neutral mass density ratio =mini mnnn: (7) Notice that is a small quantity in the WNM and CNM but not in the WIM2. The dispersion relation Eq. (4) is a cubic equation for the wave frequency !(with real and imaginary parts) as a function of the real wavenumber !k. Writing!=<(!) iin d, where in d>0 is the ion-neutral damping rate, and substituting in Eq. (4), one obtains (Zweibel & Shull 1982) !2 k=2in d in2in d[(1 +)in2in d]2; (8) which implies 0 <in d<in=2. If1, then in d!2 kin 2[!2 k+ (1 +)22 in]: (9) Alfv en waves resonantly excited by CR protons have fre- quency!kvA;i=rLThus, the frequency is related to the 2To be precise, the dispersion relation Eq. (4) is valid only if the friction force is proportional to the ion-neutral relative speed unui, as in the case of momentum transfer by elastic collisions. However, we use the same relation also in the case of charge ex- change, simply replacing hmtviinwithhceviin.kinetic energy of the CR proton E= mpc2as !keBv A;i E: (10) The eective Alfv en velocity, vA=<(!)=k, felt by CRs de- pends on the coupling between ions and neutrals. In general, the following asymptotic behavior can be identied: Low wavenumber, !kin: at large CR energy ions and neutrals are well coupled; the total density is n=nH+nHe+niand the Alfv en speed relevant for CRs resonant with the waves is vA;n=Bp4m pn; (11) where1:4 is the mean molecular weight, and in d/E2; High wavenumber, !kin: at small CR energy ions and neutrals are weakly coupled and ion-neutral damping is most eective. The Alfv en speed is the one in the ions, vA;i, and in dconst. Notice that if <1=8 there is a range of wavenumbers for which the waves do not propagate in a partially ionized medium (Zweibel & Shull 1982). This is marked as a shaded region in Fig. 1-2. On the other hand, such non-propagation band is found in the absence of CRs propagating in the par- tially ionized medium. Recently it has been suggested (Re- ville et al. 2021) that taking into account the presence of CRs may allow for the propagation of waves in that band. Introducing the fraction of ionized gas fand the helium- to-hydrogen ratio , f=ni nH+ni;  =nHe nH+ni; (12) Eq. (6) becomes in=1f 1 + ~mihmtvii;H+4 4 + ~mihmtvii;Hen 1 +:(13) where ~mi=mi=mp. In the following, the standard value = 0:1 is assumed, but the case = 0 is also considered for illustrative purposes and for a comparison with the results of D'Angelo et al. (2016), who neglect the contribution of helium to ion-neutral damping. 2.1.1 WIM and WNM In this case H is partially ionized and the dominant ion is H+ ( ~mi= 1). Therefore =nH+=(nH+4nHe) =f=(1f+4), MNRAS 000, 000{000 (0000)4S. Recchia et al. i.e.= 0:005{0:05 and 0.75{9 for the WNM and the WIM, respectively. The ion-neutral collision frequency is in=1f 2hmtviH+;H+4 5hmtviH+;Hen 1 +:(14) Fig. 1 shows the damping rate for waves resonant with CRs of energyE, as a function of the CR energy E. Notice the non-propagation band found in the WNM ( <1=8). 2.1.2 CNM and DiM In this case H is neutral and the dominant ion is C+( ~mi= 12), with fractional abundance nC+=nH(0:4{1)103. Therefore= 12nC+=(nH+ 4nHe)(3{9)103and in=1 13hmtviC+;H+ 4hmtviC+;Hen 1 +: (15) Fig. 2 shows the damping rate for waves resonant with CRs of energyE, as a function of the CR energy E. Also in this case non-propagation regions are found. 2.2 Wave cascade and turbulent damping The turbulent damping (FG) of self-generated Alfv en waves is due to their interaction with a pre-existing background turbulence. Such turbulence may be injected by astrophys- ical sources (see e.g. Mac Low & Klessen 2004) with a tur- bulent velocity vturband on scales, Linj, much larger than the CR Larmor radius. For waves in resonance with particles with a given energy E, the damping rate, that accounts for the anisotropy of the turbulent cascade, has been derived by Farmer & Goldreich (2004); Yan & Lazarian (2004) and reads FG d=v3 turb=Linj rLvA1=2 ; (16) wherevAis the eective Alfv en speed felt by CRs, as dened in Sec. 2.1. We take the turbulence as trans-Alfv enic at the injection scale, namely vturb=vA;n(at large scales waves are in the low wavenumber regime, where ions and neutrals are well coupled, as illustrated in Sec. 2.1). This is the likely situation if the turbulence is mainly injected by old SNRs, with a forward shock becoming trans-sonic and trans- Alfv enic. The FG damping rate is shown in Fig. 1 for the WIM and WNM. In highly neutral media, such as the the WNM, CNM and DiM, the background turbulence responsible for the FG damping can be damped by ion-neutral friction at a scale, lmin= 1=kmin(Xu et al. 2015, 2016; Lazarian 2016; Brahimi et al. 2020). Correspondingly, there is a minimum particle energy,Emin, such that rL(Emin) =lmin, below which the FG damping cannot aect the self-generated Alfv en waves (Brahimi et al. 2020): 1 lmin=L1=2 inj2in vA;n3=2r 1 +vA;n 2inLinj: (17) In Fig. 1 the FG damping rate for the WNM is truncated atEmin. In the WIM the cascade rate is found to be always larger than the ion-neutral damping rate and there is no Emin.2.3 Non-linear Landau damping The non-linear Landau (NLL) damping is caused by the in- teraction between the beat of two Alfv en waves and the ther- mal (at temperature T) ions in the background medium. The damping rate for resonant waves is given by (Kulsrud 1978; Wiener et al. 2013) NLL d=1 2s  2kBT mpI(kres) rL; (18) wherekBis the Boltzmann constant and I(kres) is the wave energy density (see Sec. 3 below for the denition) at the resonant wavenumber kres= 1=rL. 3 COSMIC RAY PROPAGATION IN THE PROXIMITY OF SNRS We consider the escape of CRs from a SNR and the sub- sequent propagation in the source proximity. The propaga- tion region is assumed to be embedded in a turbulent mag- netic eld, with a large scale ordered component of strength B0. CRs are scattered by Alfv en waves, which constitute a turbulent magnetic eld background of relative amplitude B=B 0, wherekis the wavenumber. We only consider waves that propagate along the uniform background eld B0. In the limit of B=B 01, which is the one relevant for the cases treated in this paper, the CR diusion along eld lines can be treated in the quasi-linear regime, with a diusion coecient given by Berezinskii et al. (1990) and Kulsrud (2005) D(E) =4cr L(E) 3I(kres) kres=1=rL=DB(E) I(kres); (19) wherecis the speed of light, I(kres) =B(kres)2=B2 0is the wave energy density calculated at the resonant wavenumber kres= 1=rL, andDB(E) = (4=3)crL(E) is the Bohm diu- sion coecient. We also assume that the dominant source of Alfv enic turbulence is produced by the CR resonant stream- ing instability. In our model we adopt the ux tube approximation for the CR transport along B0(see, e.g., Ptuskin et al. 2008), and we neglect the diusion across eld lines, which is sup- pressed in the B=B 01 regime (see e.g. Drury 1983; Casse et al. 2002). Thus, we do not address the perpen- dicular evolution of the ux tube (see, e.g., Nava & Gabici 2013) and any possible CR feedback on it and, in general, on the ISM dynamics (see, e.g., Schroer et al. 2020). Such one-dimensional model for the CR propagation is applicable for distances from the source below the coherence length, Lc, of the background magnetic turbulence, i.e. the scale below which the magnetic ux tube is roughly preserved (see, e.g., Casse et al. 2002). When particles diuse away from the source at distances larger than Lc, diusion becomes 3-D and the CR density drops quickly. In the Galactic disk, Lcis estimated observa- tionally and may range from few pc to 100 pc, depending on the ISM phase (see, e.g., Nava & Gabici 2013, and refer- ences therein). We follow the approach proposed by Nava et al. (2016, 2019), and we determine: ( i) the escape time of CRs of a MNRAS 000, 000{000 (0000)5 Figure 1. Damping rates in dand FG d(ion-neutral and turbulent) of Alfv en waves in the WNM ( left-hand panel ) and WIM ( right-hand panel ) vs. CR energy E. Dierent colors are used for dierent values of the hydrogen ionization fraction f. Unless stated otherwise, a standard= 0:1 He abundance is assumed. The considered parameters for the WNM and WIM are given in Table 1. In the left-hand panel the dotted lines refers to ion-neutral damping, while the dashed lines to the FG damping. The last are truncated to the minimum energy,Emin, below which the background turbulence is damped by ion-neutral friction before reaching the scale relevant for damping self-generated waves resonant with particle of energy <E min(see Sec. 2.2). The shaded regions represent the range of non-propagation of Alfv 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 the case where the contribution to damping from He is neglected ( = 0). The dashed lines refer to the FG damping, which in the case of WIM is found to depend little on the ionization fraction f. Figure 2. Ion-neutral damping rate in dof Alfv en waves in the CNM (left-hand panel) and DiM (right-hand panel) as a function of the CR energy E. Dierent colors are used for dierent values of the C+abundance. The parameters adopted for the CNM and DiM are given in Table 1. The shaded regions represent the range of non-propagation of Alfv en waves (see Sec. 2.1). given energy from the remnant; ( ii) the time dependent evo- lution of the CR cloud and of the self-generated diusion coecient after escape; ( iii) the time spent by CRs in the source vicinity and the corresponding grammage. We focus on the warm ionized and warm neutral phases of the ISM. As shown in the following sections, the results depend sig- nicantly on the considered ISM phase, and in particular on the amount and type of neutral and ionized atoms in the background medium.3.1 Transport equations and initial/boundary conditions The coupled CR and wave transport equations read (Nava et al. 2016, 2019) @PCR @t+vA@PCR @z=@ @zDB I@PCR @z (20) and @I @t+vA@I @z= 2( CRd)I+Q: (21) MNRAS 000, 000{000 (0000)6S. Recchia et al. HerevAis an eective Alfv en velocity that takes into ac- count the coupling between ions and neutrals (see Sec. 2.1), whilePCRis the partial pressure of CRs of momentum p normalized to the magnetic eld pressure PCR=4 3cp4f(p) B2 0=8: (22) The term 2 CRIgives the wave growth due to the CR streaming instability and can be expressed as 2CRI=vA@PCR @z: (23) The term dencompasses the relevant wave damping rates, which are described in detail in Sec. 2. The term Qrepre- sents the possible injection of turbulence from an external source, other than the CR streaming. Here it is taken as Q= 2 dI0, whereI0is a parameter such that typical val- ues of the Galactic CR diusion coecient are recovered at large distances from the source (see, e.g., Strong et al. 2007). In this way, when the streaming instability is not relevant, diusion is regulated by the background turbulence. Notice that in these equations we neglected adiabatic losses that arises when Alfv en speed varies in space (see, e.g., Brahimi et al. 2020), since we are considering a homogeneous ISM around the SNR. The coordinate zis taken along B0andz= 0 refers to the centre of the CR source. The equations are solved numerically with a nite dierence explicit method, using the initial conditions PCR=P0 CR ifz<R esc(E); 0 ifz>R esc(E); and I=I0 everywhere: (24) HereResc(E) is the size of the region lled by CRs at the time of escape, and P0 CRis the initial CR pressure inside this region. The method used to determine the escape radius and time for particles of energy Eis described in detail in Nava et al. (2016, 2019). As for the initial condition for the waves, it is also possible to choose II0forz<R esc(E) in order to mimic Bohm diusion inside the source, as proposed by Malkov et al. (2013). However, as discussed by Nava et al. (2019), dierent choices of Iinside the source have little im- pact on the nal solution. As boundary conditions we impose a symmetric CR distribution at z= 0, and PCR= 0; I =I0 atz&Lc: (25) The 1-D model used in this paper is valid only up to a dis- tance from the source given by the coherence length Lcof the magnetic eld. At larger distances the diusion becomes 3-D and the CR density quickly drops, a behavior that can be described in terms of a free escape boundary at Lc. The value ofLcis constrained from observations to be .100 pc but its value is matter of debate and is likely phase- and position-dependent in our galaxy. We take the free escape boundary at z&Lc, and we check that this assumption is not signicantly aecting the results, for instance changing Lcto 10Lc.4 RESULTS In what follows we discuss the release of CRs of energy E from a SNR, giving an estimate of the age and radius of the source at the moment of escape. Moreover, we investigate the propagation of runaway CRs in the source proximity, with particular focus on the CR spectrum and self-generated dif- fusion coecient. Finally, we estimate the residence time of CRs in the source proximity and we infer general implica- tions for the grammage accumulated in that region. We treat these aspects in the cases of a WIM or a WNM surrounding the remnant, with special emphasis on the role played by the ion-neutral damping of Alfv en waves. In our calculations we assume that: ( i) runaway CRs have a total energy ECR= 1050erg, a power-law spectrum in energy between 1 GeV{5 PeV with spectral index = 2:0; and ( ii) the typical ISM diusion coecient is D(E) = D0(E=10 GeV)0:5withD0= 1028cm2s1(see, e.g., Strong et al. 2007). 4.1 Escape of cosmic rays: source age and radius The age and radius of escape of CRs of energy Eis estimated as follows (Nava et al. 2016, 2019): we dene the half-time, t1=2(E;R) of a CR cloud as the time after which half of the CRs of a given energy, initially conned within a region of sizeR, have escaped that region. The evolution of such CR cloud is studied by solving Eq. (20) and Eq. (21) with initial conditions given by Eq. (24) and boundary conditions given by Eq. (25). The radius of a SNR expanding in a homogeneous medium of density ncan be estimated as (Truelove & McKee 1999) RSNR(t) = 0:5E51 n1=5" 10:09M5=6 ej E1=2 51n1=3tkyr#2=5 t2=5 kyrpc; (26) whereE51is the supernova explosion energy in units of 1051erg,nis the total density of the ambient ISM in cm3, Mejis the mass of the supernova ejecta in solar masses, andtkyris the SNR age in kyr. This equation is valid in the adiabatic phase of the expansion, which starts af- ter86M5=6 ejE1=2 51n1=3yr. Here we take E51= 1 and Mej= 1:4, while the gas density depends on the medium and is reported in Table 1. At earlier times an approx- imate expression for the free expansion phase has to be used (Chevalier 1982). The adiabatic phase stops at roughly 1:4104E3=14 51n4=7yr, when the radiative phase starts (Cio et al. 1988). At this epoch we also assume that the acceleration of CRs becomes ineective and that all CRs are instantly released (the validity of the assumption of instant release may depend on the conditions in the shock precursor, as discussed by Brahimi et al. 2020). The escape time, tesc(E), of particles with energy Eis estimated as the time such that t1=2(E;R SNR) equals the age of a SNR of radius RSNR. This is the timescale over which waves can grow. The dependence of t1=2(E;R) on the initial radius Rat xed energy E, and varying the back- ground diusion coecient and the CR spectral index, has been extensively explored by Nava et al. (2016, 2019), and we refer the reader to these works for a detailed discussion. Here we brie y summarize the most relevant points. At small MNRAS 000, 000{000 (0000)7 Rthe CR gradient is large and the amplication of waves is very eective. In this regime, the NLL damping, that scales with the wave energy density I(k), can play an important role and may dominate over ion-neutral and FG damping at small enough radii. At intermediate R, ion-neutral damp- ing dominates in most cases. In fact, as shown in Fig. 1, the ion-neutral damping rate is larger than the FG rate at least up to particle energies of 10 TeV, with the only ex- ception of a WIM with fully ionized hydrogen and 10% of neutral helium. At large R, the CR gradient is reduced and the streaming instability is less eective. In this case the expansion of the CR bubble is determined by the back- ground turbulence and the test particle limit is recovered, witht1=2/R2=D0. Fig. 3 and Fig. 4 show the SNR age and radius at the time of escape of CRs, as a function of the CR energy,in the case of a WNM and WIM respectively. We conrm the nd- ings by Nava et al. (2016, 2019) and the qualitative results that particles with higher energy escape earlier than the low energy particles (see, e.g., Gabici 2011). In the case of the WIM, Rescandtesctend to decrease with a larger neutral density. The eect of ion-neutral damp- ing is especially visible in the energy range 1{10 TeV, where the ion-neutral damping rate starts to decrease from a roughly constant value at dierent particle energies, de- pending on the value of the ionized hydrogen fraction fand on the helium-to-hydrogen ratio , as shown in Fig. 1. The drop of in dcorresponds to an increase of tesc, and therefore to a better CR connement. The situation is more subtle in the case of the WNM. In fact, here the connement appears to be slightly more eective for a smaller value of f, a result opposite to what happens in the WIM. This can be explained taking into ac- count that, while the in dis practically the same at energies below10 TeV for f71035102, the eective Alfv en speed felt by CRs, which at these energies is roughly that of ions, as illustrated in Sec. 2, is a factor of 2{3 larger forf= 7103. This re ects on an increase by the same amount of the FG damping rate (above the cut-o energy, see Sec. 2.2), as shown in Fig. 1, but also of the growth rate term, which is proportional to vA. On the other hand, at energies in the range 10 GeV{1 TeV, the FG damping is always subdominant compared to the ion-neutral damping, and the net eect of this change of vAis a slightly better connement at smaller fas a result of an enhanced wave growth rate. Also in the case of a WIM the eective vAis a bit larger at smallerfbelow1 TeV, but here the increase is only by a factor of1:2 and cannot overcome the eect of ion- neutral damping. 4.2 Cosmic ray spectra and diusion coecient The spectrum of runaway CRs and the corresponding self- generated turbulence depend signicantly on the distance from the source and on the time. In Fig. 5 and Fig. 6 we show the spectrum of CRs that have already escaped to a distance of 50 pc from the centre of the remnant at dierent times, fromt= 6103yr tot= 105yr, and for the WNM ( f= 0:05,= 0:1) and the WIM ( f= 0:9,= 0:1), respectively. To remove the eect of the simple 1-D geometry that we are adopting we multiply PCRbyR2 esc(E)=R2 SNR(t). For eachcase we also show the ratio between the self-generated and background diusion coecients. In the same gures we also show the spectrum and diu- sion coecient at the position of the shock of CRs that have escaped at earlier times (and are considered as decoupled from the accelerator) as well as the spectrum of particle still conned in the accelerator, at dierent ages of the remnant fromt= 6103yr tot= 3104yr. The spectrum of the conned particles is estimated by assuming that the remnant provides1050erg in CRs with a /E2spectrum that ex- tends from1 GeV to5 PeV, with an exponential cuto at the energy of the particles that escape at the considered age. The SNR is assumed to stop accelerating particles at the onset of the radiative phase, which for the typical den- sity of the WIM and WNM takes place at trad3104yr. The corresponding shock radius is Rrad22 pc. In all cases, at each time and location, the spectrum exhibit a sharp rise at low energies and a peak followed by a nearly power-law like behaviour at higher energies. Since high energy particles escape earlier and diuse faster, they reach a given distance at earlier times, as shown for the spectra at 50 pc, where the peak moves at lower energies with time. At energies lower than the peak, particles have not yet reached that position, giving the sharp rise. At larger energies, particles have diused over a bigger distance, giving a spectrum steeper than the spectrum released from the SNR, here assumed to be /E2. A similar behaviour is observed with the spectra at the shock. Here the position at which the spectra are shown varies with time, since it is given by the shock radius at a given age. This radius is always <50 pc for the cases illustrated here. Comparing the spectra at the shock and at 50 pc for a given age, it is evident that the peak is at lower energy in the former case. This again is the result of an earlier escape and faster diusion of high energy particles. As for theD=D 0ratio as a function of energy, it is possi- ble to infer from the plots that, at a given time, the diusion coecient is equal to its background value at low energies, where particles have not yet escaped or not yet reached the position where Dis calculated, and at high energies, where the turbulence produced by such particle is being damped. At intermediate energies, and roughly corresponding to the peak in the spectrum, the diusion coecient is suppressed compared to D0due to CR streaming instability. The ener- gies at which the suppression is evident move to lower values with time. A more complex situation can be observed in some cases, for instance at t= 104yr at the shock and in the TeV range, for the WIM. Here Dstarts to deviate from D0 at50 GeV, then, above 500 GeV,Dbegins to ap- proach again D0, but above3 TeV the level of turbulence increases again, with the result of a suppression of D. This is due to the energy dependence of the escape radius, which becomes steeper above few TeV. This results in the fact that, above that energy, the escape radius becomes small enough as to eectively excite the streaming instability. In fact the density of runaway CRs is /1=R2 esc. The more ecient self- connement at this high energies also re ects in a hardening of the spectrum (see also Nava et al. 2019 for a discussion). MNRAS 000, 000{000 (0000)8S. Recchia et al. 102103104105 101102103104 1 10 100 -- Resc- tesctesc [yr] Resc [pc] E [GeV]WNM f=0.007 f=0.05 104105106 10110210310-210-1tres [yr] X [g/cm2] E [GeV]WNM f=0.007 f=0.05 Figure 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: Residence time of CRs in a region of 100 pc around the source, as a function of the CR energy, and the corresponding grammage. Dierent colors are used for dierent values of the hydrogen ionization fraction f, while the helium-to-hydrogen ratio = 0:1. The parameters of the WNM are listed in Table 1. 103104105 101102103104 1 10 100 -- Resc - tesctesc [yr] Resc [pc] E [GeV]WIM f=0.6 f=0.9 f=0.9, χ =0.0 104105106 10110210310-210-1100tres [yr] X [g/cm2] E [GeV]WIM f=0.6 f=0.9 f=1 f=1, χ=0 f=1, χ=0, 20 pc Figure 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: Residence time of CRs in a region of 100 pc around the source, as a function of the CR energy, and the corresponding grammage. The cyan 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). Dierent colors are used for dierent values of the hydrogen ionization fraction fand helium-to-hydrogen ratio . The parameters of the WIM are listed in Table 1. 4.3 Residence time and grammage The grammage accumulated by CRs close to their sources, and its relevance compared to the grammage accumulated while diusing in the whole Galaxy, is related to the resi- dence time in the proximity of the source. A formal deter- mination of the grammage should be done by solving the CR transport equation for nuclei with the inclusion of spallation contributions (see, e.g., Berezinskii et al. 1990; Ptuskin & Soutoul 1990). Here we adopt the following approach: we assume that N0particles of energy Eare instantly injected by a source located in a region of constant gas density =mpnand are subject to free escape at a boundary located at a distance Lcfrom the source. At a given time t, the total number of particles that are still in the region is Nin(t) while the number of escaped particles is Nesc=N0Nin(t). Particles that cross the boundary at time thave accumulated the grammageX(t) =ctand the average grammage gained byall escaped particles from t= 0 tot=1is given by hXi=1 N0Z1 0ctdNesc dtdt (27) =c N0Z1 0Nin(t)dtlim t!1(tNin(t)) =c res: The residence time resis given by res=1 N0Z1 0Nin(t)dtlim t!1(tNin(t)) : (28) The number of particles with a given energy contained in the region at a given time can be calculated from the CR pressure (see Sec. 3): Nin(t) =N0 RescP0 CRZLc 0PCR(z;t;E )dz; (29) where we took into account that CRs on energy Eare ini- tially released from a region of size Resc(E). MNRAS 000, 000{000 (0000)9 WNM 10-210-1100 10110210310450 pc6e+3 1e+4 3e+4 6e+4 1e+5f(E)E2 x (Resc/Rsh)2 [eV/cm3] E [GeV]10-310-210-1100101102 101102103104shock 6e+3 1e+43e+4f(E)E2 x (Resc/Rsh)2 [eV/cm3] E [GeV] 1 101102103104D/D0 E [GeV] 1 101102103104D/D0 E [GeV] Figure 5. Case of WNM ( f= 0:05,= 0:1): CR spectrum and ratio D=D 0as a function of energy, where D0(E) = 1028(E=10 GeV)0:5cm2s1. The dierent colors refer to dierent ages in yr (as marked). The left panels refer to a location at 50 pc from the center of the SNR, while the right panels refer to the shock position. In this case also the spectrum of CRs still conned in the accelerator is shown (dashed lines). Fig. 3 and 4 show the residence time of CRs in a re- gion of100 pc around the remnant, for the WNM and the WIM, respectively, and for dierent values of the neu- tral fraction, and the corresponding grammage. Even in the (not very plausible) case of a fully ionized WIM ( f= 1:0, = 0:0), the grammage is nearly two orders of magnitudes smaller than that accumulated in the disk (see e.g. Jones et al. 2001; Gabici et al. 2019). This is due to the action of the ion-neutral and FG damping. The presence of a 10% of neutral helium (which is less eective at wave damping compared to hydrogen) noticeably reduces the CR residence time compared to the case of a fully ionized background medium, above few tens GeV. These results are at odds with what previously sug- gested by D'Angelo et al. (2016), namely that ion-neutral damping is totally negligible in the case of only neutral he- lium, since the c.e. cross section is very small. However, as shown in Sec. 2, not only c.e. but also m.t. has to be taken into account in the ion-neutral damping. In fact, D'Angelo et al. (2016) nd a grammage nearly a factor of ten larger than what we nd. This is in part due to their assumption of ion-neutral damping of waves with neutral helium. In ad-dition, they assume a 20% acceleration eciency, while we assume 10%, which enhances the eect of streaming insta- bility, and they do not include the FG damping. Finally, D'Angelo et al. (2016) assume that CRs at all energies are released when the SNR radius is 20 pc, a scenario implying a smaller release radius at low energies compared to our cal- culation, which translates in an enhancement of the stream- ing instability (and the CR connement) at energies below 1 TeV, as shown in Fig. 4. Any addition of neutral hydrogen compared to the case of only neutral helium further reduces the connement time. Correspondingly, the CR grammage accumulated in the source proximity results to be well below that inferred from observations. A similar result was found by Nava et al. (2019) for the HIM, where the residence time tends to be larger than in a partially ionized medium, but the ISM den- sity is lower (0:01 cm3). 5 CONCLUSIONS We followed the method and the setup proposed by Nava et al. (2016, 2019) to investigated the escape of CRs from MNRAS 000, 000{000 (0000)10 S. Recchia et al. WIM 10-310-210-1100 10110210310450 pc 6e+31e+4 3e+4 6e+4 1e+5f(E)E2 x (Resc/Rsh)2 [eV/cm3] E [GeV]10-310-210-1100101102 101102103104shock 6e+31e+43e+4f(E)E2 x (Resc/Rsh)2 [eV/cm3] E [GeV] 1 101102103104D/D0 E [GeV] 1 101102103104D/D0 E [GeV] Figure 6. Case of WIM ( f= 0:9,= 0:1): CR spectrum and ratio D=D 0as a function of energy, where D0(E) = 1028(E=10 GeV)0:5cm2s1. The dierent colors refer to dierent ages in yr (as marked). The left panels refer to a location at 50 pc from the center of the SNR, while the right panels refer to the shock position. In this case also the spectrum of CRs still conned in the accelerator is shown (dashed lines). SNRs embedded in a WNM and WIM, and the CR self con- nement in the source proximity. Our main objective was to determine whether, in realistic situations, the grammage accumulated by CRs in the source region could become com- parable to that inferred from observations. Ifthiswasthe case, In this case, the standard picture in which CR secon- daries are produced during the whole time spent by cosmic rays throughout the disk of the Galaxy, should be profoundly revisited. We conrm the results found by Nava et al. (2016, 2019) that CRs escaping from SNRs drive the excitation of Alfv en waves through the resonant CR streaming instability, which results in a suppression of the diusion coecient and in the CR self-connement in the source region. The SNR radius at which CRs of a given energy leave the source, Resc(E), results to be a decreasing function of the energy, and regu- lates the CR streaming instability through a dilution factor /1=R2 esc. We found that the growth of self-generated Alfv en waves, and consequently the residence time of CRs in the source region, is signicantly limited by several damping processes, especially by the FG and ion-neutral damping.In particular, for the ion-neutral damping of Alfv en waves we have used up-to-date damping coecients, based on ac- curate experimental or theoretical determinations of the mo- mentum transfer and charge exchange cross sections between ions and neutrals of dierent species. In the 1-D geometry adopted in our calculation a sup- pression of the diusion coecient is found within a distance form the source of the order of the magnetic eld coherence lengthLc50100 pc, for a timescale that can be as large as105yr at10 GeV. A smaller value for Lcwould make the CR propagation to become 3-D closer to the remnant, thus reducing the residence time compared to our results. We conclude that ion-neutral damping strongly limits the CR grammage that can be accumulated in the source region. Under the conditions typically met in the WIM and WNM of the Galactic disk, the CR source grammage is found to be negligible compared to that inferred form ob- servation. A similar result was found for the HIM by Nava et al. (2019), where the CR residence time is typically larger than in a partially ionized medium but the ISM density is more than a factor ten smaller. This makes alternative sce- narios for the interpretation of quantities such as the B/C MNRAS 000, 000{000 (0000)11 ratio, in which an important contribution to the production of secondaries comes from the source region, less attractive. Our results on the residence time (and grammage) may change with the inclusion of the contribution of other sources of turbulence to the CR connement. In particular, it has recently been suggested that another another CR-induced instability, the so called non-resonant streaming instability , which is mediated by the CR current and plays a crucial role in the acceleration of CRs (Bell 2004), may signicantly en- hance the CR self-connement in the source region above TeV energies (Schroer et al. 2020). Further investigations are thus needed in order to rmly establish the importance amount of CR grammage accumulated in the vicinity of sources by very-high energy particles. REFERENCES Aharonian F. A., Atoyan A. M., 1996, A&A, 309, 917 Amato E., 2011, Mem. Soc. Astr. It., 82, 806 Bell A. R., 2004, Monthly Notices of the Royal Astronomical So- ciety, 353, 550 Berezinskii V. S., Bulanov S. V., Dogiel V. A., Ptuskin V. S., 1990, Astrophysics of Cosmic Rays Blasi P., 2013, A&ARv, 21, 70 Brahimi L., Marcowith A., Ptuskin V. S., 2020, A&A, 633, A72 Brennan M. H., Morrow R., 1971, J. Phys. 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P., 2013, ApJ, 767, 87 Xu S., Lazarian A., Yan H., 2015, ApJ, 810, 44 Xu S., Yan H., Lazarian A., 2016, ApJ, 826, 166 Yan H., Lazarian A., 2004, ApJ, 614, 757 Zweibel E. G., Shull J. M., 1982, ApJ, 259, 859 APPENDIX A: COLLISIONAL COEFFICIENTS A1 Collisions of H+with H and He atoms In the WNM and the WIM the relevant damping processes for Alfv en waves are collisions of H+with either H or He atoms. The case of H+{ H collisions is special because the proton elastically scattered by the H atom is indistinguish- able from the recoiling proton produced by charge exchange (Krsti c & Schultz 1999; Glassgold et al. 2005; Schultz et al. 2008, 2016). Therefore, the elastic scattering and the charge exchange channels for collisions of H+with H, as for any col- lision between ions and their parent gas, cannot in general be separated. Only at collision energies Ecm&1 eV can the MNRAS 000, 000{000 (0000)12 S. Recchia et al. forward elastic and the backward charge exchange be ap- proximately separated. In this limit, mt2ce(Dalgarno 1958), a frequently used approximation (see e.g. Kulsrud & Cesarsky 1971). Fig. A1 shows the m.t. cross section for H+{ H colli- sions (Krsti c & Schultz 1999; Glassgold et al. 2005), and the corresponding rate coecient. The gure also shows the m.t. cross sections and rate coecients derived from either charge exchange and elastic scattering in the distinguishable particle approach (Schultz et al. 2008, 2016), and the exper- imental determination of the m. t. cross section obtained by Brennan & Morrow (1971) by measuring the attenuation of Alfv en waves propagating in a partially-ionized hydrogen plasma atEcm5 eV. Also shown in Fig. A1 are the rate co- ecients used by Kulsrud & Cesarsky (1971) and Zweibel & Shull (1982), and frequently adopted also for other species by scaling the collision rate with the ratio mn=miof the neutral and ion masses (see e.g. (Zweibel & Shull 1982)). As shown in the following, this approximation is not generally correct, since the rate coecients in the case of other species exhibit a dierent dependence on the temperature compared to the H+{ H collisions case. Fig. A2 shows the m.t. (Krsti c & Schultz 1999, 2006) and c.e. (Loreau et al. 2018) cross sections for H+{ He col- lisions, and the corresponding rate coecients. Charge ex- change contributes to the transfer of momentum above col- lision energies103eV, corresponding to relative velocities of500 km s1, much larger than typical thermal or Alfv en speeds in the WNM and WIM. A2 Collisions of C+with H and He atoms In the CNM and DiM, ion-neutral damping is dominated by the collisions between neutral hydrogen and ionized carbon. Fig. A3 shows the cross sections and reaction rates for m.t. and c.e. in the case of collisions of C+ions with H atoms. For collisions of C+ions with He atoms, no theoretical or experimental are available. The m.t. transfer rate coecient, according to the Langevin theory, is hmtviC+;He= 1:33 109cm3s1(Pinto & Galli 2008). As in the case of H+{ He collisions, the large dierence with the rate coecients used in this work resides in the assumption that the rate coecient is the same for all species. Figure A1. Top panel: cross sections for collisions of H+with H vs. collision energy in the center-of-mass frame Ecm. Momentum transfer cross section, from Krsti c & Schultz (1999), Glassgold et al. (2005), and Schultz et al. (2008); charge exchange cross section, from Hodges & Breig (1991), and Schultz et al. (2008); dashed line: Langevin m.t. cross section. Experimental data for the c.e. cross section: Newman et al. (1982) ( lled triangles ). The empty circle shows the value of the m.t. cross section measured by Brennan & Morrow (1971) Bottom panel: Collisional rate co- ecient for momentum transfer and charge exchange. The lled and empty circles show the rate coecient adopted by Kulsrud & Cesarsky (1971) and Zweibel & Shull (1982), respectively. MNRAS 000, 000{000 (0000)13 Figure A2. Top panel: cross sections for collisions of H+with He vs. collision energy in the center-of-mass frame Ecm. Momen- tum transfer cross section, from Krsti c & Schultz (1999, 2006); Langevin m.t. cross section ( dashed line ); charge exchange cross section, from Loreau et al. (2018); and ionization cross section, from Shah & Gilbody (1985). Experimental data for the c.e. cross section: Martin et al. (1981) ( empty triangles ), Shah & Gilbody (1985) ( lled triangles ), Shah et al. (1989) ( empty squares ), and Kusakabe et al. (2011) ( lled squares ). Experimental data for the ionization cross section: Shah & Gilbody (1985) ( empty cir- cles).Bottom panel: Collisional rate coecient for Langevin m.t. (dashed line ), m.t. and c.e. ( solid lines ). Figure A3. Top panel: cross sections for collisions of C+with H vs. collision energy in the center-of-mass frame Ecm: m.t. cross section computed by Flower & Pineau des Forets (1995) ( lled triangles ); t, from Pinto & Galli (2008) ( solid line ); Langevin m.t. cross section ( dashed line ). Experimental data for the c.e. cross section: Phaneuf et al. (1978) ( empty circles ), Goe et al. (1979) ( empty squares ) and Stancil et al. (1998) ( lled triangles ); c.e. cross section recommended by Stancil et al. (1998) ( solid line ). Bottom panel: Collisional rate coecients: Langevin m.t. ( dashed line), m.t. and c.e.( solid lines ). MNRAS 000, 000{000 (0000)

2021-06-09

We investigate the damping of Alfv\'en waves generated by the cosmic ray resonant streaming instability in the context of the cosmic ray escape and propagation in the proximity of supernova remnants. We consider ion-neutral damping, turbulent damping and non linear Landau damping in the warm ionized and warm neutral phases of the interstellar medium. For the ion-neutral damping, up-to-date damping coefficients are used. We investigate in particular whether the self-confinement of cosmic rays nearby sources can appreciably affect the grammage. We show that the ion-neutral damping and the turbulent damping effectively limit the residence time of cosmic rays in the source proximity, so that the grammage accumulated near sources is found to be negligible. Contrary to previous results, this also happens in the most extreme scenario where ion-neutral damping is less effective, namely in a medium with only neutral helium and fully ionized hydrogen. Therefore, the standard picture, in which CR secondaries are produced during the whole time spent by cosmic rays throughout the Galactic disk, need not to be deeply revisited.

Grammage of cosmic rays in the proximity of supernova remnants embedded in a partially ionized medium

2106.04948v1

arXiv:2209.07908v1 [quant-ph] 16 Sep 2022Pseudo- PTsymmetric Dirac equation : effect of a new mean spin angular momentum operator on Gilbert damping Y. Bouguerra, S. Mehani, K. Bechane and M. Maamache Laboratoire de Physique Quantique et Syst` emes Dynamiques , Facult´ e des Sciences, Universit´ e Ferhat Abbas S´ etif 1, S ´ etif 19000, Algeria P. -A. Hervieux Institut de Physique et Chimie des Mat´ eriaux de Strasbourg , CNRS and Universit´ e de Strasbourg BP 43, F-67034 Strasbour g, France (Dated: September 19, 2022) Abstract The pseudo- PTsymmetric Dirac equation is proposed and analyzed by using a non-unitary Foldy-Wouthuysen transformations. A new spin operator PTsymmetric expectation value (called the mean spin operator) for an electron interacting with a ti me-dependent electromagnetic field is obtained. Weshowthatspinmagnetization -whichisthequan tityusuallymeasuredexperimentally - is not described by the standard spin operator but by this ne w mean spin operator to properly describe magnetization dynamics in ferromagnetic materia ls and the corresponding equation of motion is compatible with the phenomenological model of the Landau-Lifshitz-Gilbert equation (LLG). PACS numbers: Pseudo PT Symmetry; Non-Hermitian Dirac equation ; Foldy-Wouthuysen transformation; Landau-Lifshitz-Gilbert equation 1In the field of micromagnetism ,which provides the physical framework for understanding and simulating ferromagnetic materials ,there is a fundamental unsolved problem which is the microscopic origin of the intrinsic Gilbert damping. However, this d amping mechanism has been introduced phenomenologically by T. L. Gilbert in 1955 for de scribing the spatial and temporal evolution of the magnetization (known as the LLG equ ation), a vector field which determines the properties of ferromagnetic materials on the sub-micron length scale [1]. Let us stress that this equation leads to the conservation of th e magnetization modulus. This phenomenological model has since been validated by numerous e xperimental data and constitutes the foundation of micromagnetism [2]. Moreover, magn etic damping plays a crucial role in the operation of magnetic devices .The scattering theory can be used to compute the Gilbert damping tensor [3]. Spinisaquantumconcept [4]thatarisesnaturallyfromtheDiracth eoryandisassociated with the operator ˆΣD≡1 2iα∧α= σ0 0σ whereα= 0σ σ0 andσare the usual 2×2 Pauli matrices [5]. For a classical version of the spin (see Supplementary Materials). Usually, spin magnetization M(r,t) (the quantity which is experimentally measured) is defined as the expectation value of the spin angular mo mentum given by µB/angbracketleftBig ΨD|ˆΣD|ΨD/angbracketrightBig withµB≡e/planckover2pi1 2mthe Bohr’s magneton ( e <0) and where ΨDisasolution of the Dirac equation. Indeed, in most magnetic materials the orbita l moment is quenched and therefore magnetism is only due to the spins [6]. While for a free electron the spin angular momentum in the Heisenberg picture is not a constant of motion/parenleftBigg dˆΣD dt/ne}ationslash= 0/parenrightBigg [5], there exists another spin operatorˆΣD ,considered to be a constant of motion ,(called the mean spin operator [7]) dˆΣD dt= 0 . In the presence of an electromagnetic field, which is relevant for exploring the micros copic origin of the Landau-Lifshitz-Gilbert (LLG) equation, a satisfactory result ha s not yet given. KnowingthatthespinorsintheDiractheoryconsistoffourcompon ents, itisimportantto check whether the Dirac equation yields physically reasonable result s in the non-relativistic expansion case and to show that the Dirac equation reproduces th e two-component Pauli equation. We transform the Hamiltonian in such a way that all operat ors of the type αthat couple the large to the small components will be removed. This can be achieved by a Foldy Wouthuysen transformation [7–9] which is a non-relativistic expans ion of the Hamiltonian 2into series of the particle ′s Compton wave lengths λC≡h mc. Hickey and Moodera [10] have proposed that the spin-orbit interac tion, which arises from the non-relativistic expansion of the Dirac equation, may be respon sible for the intrinsic ferromagnetic line width. In their work, the term containing the curl of the electric field when coupled to Maxwell’s equations lead to a time-varying magnetic ind uction,and the theoretical methods employed involve previously developed formalis ms in which an effective non-Hermitian and time-dependent Hamiltonian is used. However, th e non-Hermiticity of the Hamiltonian imposes new rules which are modified with respect to th ose of standard quantum mechanics. This fact was not explicitly taken into account b y the authors of [10] and therefore, their derivation of the intrinsic damping process is u nfortunately incorrect. Moreover, there is another fundamental issue which emerges fro m this work [10] concerning how to properly perform the coupling between the classical Maxwell equations and the quantum evolution resulting from the non-relativistic limit of the Dirac equation. In what follows, we show how to overcome this difficulty by using the well-known correspondence principle. Intheref. [11], themain goalwastodemonstratethatt hereisawaytoderivethe LLG equation coming from a non Hermitian quantum mechanics and to s park a discussion about the connection between quantum and classical spin dynamics . Unfortunately, the quantum Heisenberg equation for a non-Hermitian Hamiltonian opera tor describing the damping process is not compatible with the time-evolution operators for non-Hermitian Hamiltonian operator. Fromthe relativistic Diracequation, performing a Foldy-Wouthuyse n transformationand using the Heisenberg equation of spin motion, Mondal et al [12–14] derive general relativistic expressions for the Gilbert damping, but the term involving the cros s-product between the magnetisationandthetime-derivativeofthemagneticfieldispurelyim aginary,andtherefore appears not to correspond to damping. In the seminal work made by Dirac on relativistic quantum mechanics, the corresponding Hamiltonian would be Hermitian. We stress that this property is very u seful to have in a physical system, however we argue that the same features can b e achieved when starting fromnon-HermitianHamiltonian systems. These features canalso b eobtainedfromtheories based on non-Hermitian Hamiltonians that have been considered in diff erent contexts. And we distinguish three separate regimes: i) The PT-symmetric regime where the eigenval- ues are real, ii) The spontaneously broken PTregime where the eigenvalues are complex 3conjugate pairs and iii) The regime with complex, unrelated, eigenvalu es in which the PT -broken regime. Our objective here is to derive the LLG equation based on a non-Her mitian Dirac Hamil- tonian when compared to the most common standard approaches [ 12–14]. Therefore, the Dirac equation in its fundamental representation is not unique to either Hermitian quantum mechanics orquantum field theory. By relaxing the assumption of Her- miticity andadoptinginstead the principles of P0T0-symmetry quantum mechanics outlined in the following paragraph, we will not make any modifications to the Dir ac equation. By P0T0-symmetry we mean reflection in space, with a simultaneous reversa l of time. The fundamental representation of the Dirac equation emerges comp letelyintact,identical in every aspect to the Dirac equation derived from Hermitian theory. Before constructing the analogous 4-d representation using the principles of P0T0quantum mechanics, let us briefly recall the notion of P0T0-symmetry. The Hermiticity of quantum Hamiltonians depends on the choice of the inner product of the states in the physical Hilbert space. This point was first point ed out by Bender et al [16, 17]. They showed that a wide class of Hamiltonians that respec tP0T0-symmetry can exhibit entirely real spectra. Since then P0T0-symmetry has been a subject of intense interest in the field of quantum mechanics. While any evidence of P0T0-symmetry has remained out of reach due to the hermitian natureofthequantummechanics theory,opticshave providedafe rtilegroundforobservation of this property- P0T0-symmetry-since this field mainly relies on the presence of gain and loss.Note that even though HandP0T0commute, they do not continuously have identical eigenvectors, as a result of the anti-linearity of the P0T0operator. If HandP0T0don’t have the same eigenvectors, we say that the P0T0−symmetry is broken. The parity operator P0 effects the momentum operator pand the position operator ras (P0:r→ −r,p→ −p). This parity transformation has the following effect on the various vector potentials P0A(r,t)(P0)−1=−A(r,t) andP0Φ(r,t)(P0)−1= Φ(r,t), expressing thus their scalar and vector nature. The anti-linear time reversal operator T0has the effect of changing the sign of the mo- mentum operator p,the pure imaginary complex quantity iand the time t(T0:r→r, p→ −p, i→ −i, t→ −t). Since A(r,t) is generated by currents, which reverse s signswhen the sense of time is reversed, it holds that T0A(r,t)(T0)−1=−A(r,t) 4andT0Φ(r,t)(T0)−1= Φ(r,t).The two reflection operators commute with each other: P0T0=T0P0. Therefore, it is natural to introduce a modified Hilbert space, which is now endowed with P0T0-inner product, for the P0T0-symmetric nonself-adjoint theories. In such a Hilbert space, the time evolution becomes unitary as the Hamiltonian is self- P0T0-adjoint and the eigenfunctions form a complete set of orthonormal functions . But the norms of the eigenfunctions have alternate signs even in the new Hilbert space en dowed with the P0T0- innerproducts. Infact, anytheoryhavinganunbroken P0T0-symmetryitexistsasymmetry of the Hamiltonian associated with the fact that there are equal nu mbers of positive-norm and negative-norm states [17]: /an}b∇acketle{tψm,ψn/an}b∇acket∇i}htP0T0=/integraldisplay dx/bracketleftbig P0T0ψn(x)/bracketrightbig ψm(x) =/integraldisplay dxψ∗ n(−x)ψm(x) = (−1)nδmn (1) The situation here is analogous to the problem that Dirac encounter ed in formulating the spinor wave equation in relativistic quantum theory [18] . This again raises an obstacle in probabilistic interpretation in spite of t he systembeingin anunbroken P0T0phase. Afterwards, a new symmetry C, inherent to all P0T0-symmetric non-Hermitian Hamiltonians, has been introduced [17]. Ccommutes with both HandP0T0 and fixes the problem of negative norms of the eigenfunctions when the inner products are taken with respect to CP0T0-adjoint. Does aP0T0-symmetric Hamiltonian Hspecify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answe r is that if Hhas an unbroken P0T0symmetry, then it has another symmetry represented by a linear o perator C.Thereforewecanconstruct atime-independent inner productwit hapositive-definite norm in terms of C. Another possibility to explain the reality of the spectrum is making use of the pseudo/quasi-Hermiticity transformations which do not alter the e igenvalue spectra. It was shown by Mostafazadeh [19] that P0T0-symmetric Hamiltonians are only aspecific class of the general families of thepseudo-Hermitian operators. A Hamiltonian is said to be η-pseudo-Hermitian if: H†=ηHη−1, (2) 5whereηisametric operator . Theeigenvaluesofpseudo-HermitianHamiltoniansareeither real or appear in complex conjugate pairs whilethe eigenfunctions satisfy bi-orthonormality relations in the conventional Hilbert space. Due to this reason, suc h Hamiltonians do not possessacomplete set of orthogonal eigenfunctions in the conventional Hilb ert space and hence the probabilistic interpretation and unitarity of time evolution have not been satisfied by these pseudo-Hermitian Hamiltonians. However, like the case of P0T0-symmetric non-Hermitian systems, thepresence of the additional operator ηin the pseudo-Hermitian theories allows usto define a new inner product in the fashion /an}b∇acketle{tφ|ψ/an}b∇acket∇i}htη=/an}b∇acketle{tηφ|ψ/an}b∇acket∇i}ht=/integraldisplay (ηφ(x))ψ(x)dx=/an}b∇acketle{tφ|ηψ/an}b∇acket∇i}ht. (3) Lateranovel concept ofthepseudoparity-time (pseudo -P0T0)symmetry wasintroduced in [15] to connect the non-Hermitian Hamiltonian Hto its Hermitian conjugate H† H†=/parenleftbig P0T0/parenrightbig H/parenleftbig P0T0/parenrightbig−1(4) where in the expression of the inner product (3), the metric ηis replaced by P0T0. We now turn our attention to the main topic of interest, the spatial reflec tion and the time-reversal invariance of the Dirac equation. The complete spatial reflection (p arity) transformation for spinors and the complete time-inversion operator are denoted asP=γ0P0andT= −iα1α3T0=iγ1γ3T0such that Pγ0P−1=γ0,PγiP−1=−γi Tγ0T−1=γ0,TαT−1=−α (5) where the Hermitian matrices β=γ0andαsatisfy the ‘Dirac algebra’ {αi,αj}= 2δij; {αi,β}= 0. Inwhat follows, we will denote by ˆHD(t) =i/parenleftbig cα.(ˆp+ieA(r,t))−eΦ(r,t)+mc2β/parenrightbig (6) the non-Hermitian DiracHamiltonian for a single electron in thepresence of a classical time- dependent external electromagnetic field defined by ( A(r,t),Φ(r,t)). The associated non- Hermitian Dirac equation for a single electron in thepresence of a classical time-dependent 6external electromagnetic field reads i/planckover2pi1∂ΨD(r,t) ∂t=ˆHD(t)ΨD(r,t) =i/parenleftBig ˆT+ˆV+mc2β/parenrightBig ΨD(r,t), (7) where ΨD(r,t) = col(u(r,t),v(r,t)),is bispinors verifying the pseudo-othogonality relation /angbracketleftbig ΨD|ΨD/angbracketrightbig PT=I,ˆV≡ −eΦ andˆT≡cα.ˆπ=cα.(ˆp+ieA(r,t)) is the kinetic energy which produces a coupling between small and large components of the Dirac wavefun ction ΨD, α,βandγ5(see Eq.(15)) are the Dirac matrices [20]. The PTsymmetry condition (PT)ˆHD(t)(PT)−1=ˆH†D(t), (8) connects the non-Hermitian Dirac Hamiltonian ˆHD(t) to its Hermitian conjugate ˆH†D(t).This observation leads us to introduce a novel concept of the pseu do-parity-time (pseudo-PT) symmetry, where ( PT) is interpreted as a metric. Thus as the case of pseudo- hermiticity, the bispinor ΨD(r,t) = col(u(r,t),v(r,t)),verifiesthe pseudo-othogonality relation/angbracketleftbig ΨD|ΨD/angbracketrightbig PT=I. Note that, there is another situation which differs from that described above ,also called pseudo- PTsymmetry which means that the system can have a real eigenvalues whether or not the original system is PT- symmetric [21, 22]. As we are dealing with the non-relativistic expansion of the Dirac equa tion, the following decomposition ΨD(r,t) = Π(r,t)×χD(us,t) [23] can be used, where χD(us,t) is the time- dependent bi-spinor representing the spin state oriented in the dir ection defined by usand Π(r,t) thescalarpart of the wave function. We argue that, in the non-relativistic limit, the operatorˆΣD is the one which must be interpreted as the spin operator in the Pauli theory and used to de fine the magnetization as M(r,t)≡µB/angbracketleftbigg χD|ˆΣD |χD/angbracketrightbigg PT=µB/angbracketleftbigg χD|ˆU−1ˆΣFWˆU|χD/angbracketrightbigg PT=µB/an}b∇acketle{tχFW|ˆΣFW|χFW/an}b∇acket∇i}htPTwhere χFWis the spin part of the Dirac bi-spinor wave function ΨDin a non-relativistic expansion, obtained by using the Foldy-Wouthuysen (FW) transformation [7] andˆUis the associated operator (see the definition in the following). It worth mentioning th at,according to the above definition ,expanding the spin operator (to a consistent order in h/mc) and using the Dirac representation of the wave function is equivalent to expand ingthe wave function (also to a consistent order in h/mcusing the FW transformation) and keep ingthe original spin operator in the Dirac representation. In this work we have chosen to expand the mean value operator. The classical magnetization, M(r,t), is obtained by using the correspondence 7principle. Indeed, we will show in what follows that the equation of mot ion of the mean spin operator for an electron interacting with a time-dependent ele ctromagnetic field leads to the LLG equation of motion revealing thus its microscopic origin. Ina seminal work, FoldyandWouthuysen (FW)solved theproblem of finding a canonical transformation thatallows to obtain a two-component theory in the low-energy limit (Pauli approximation) ,in the case of the Dirac equation coupled to an electromagnetic field [7 ]. Unfortunately, contrarytothefree-electroncase, thesolutio ncannotbeexpressed inaclosed form. However, FW showed how to obtain successive approximation s of this transformation as a power series expansion in powers of the Compton wave length of the particle λC≡ h mc. This procedure, generally restricted to the second-order in 1 /m, is presented in many textbookson relativistic quantum mechanics [8, 20, 24, 25] and has been exte nded to fifth order in powers of 1 /m[26]. In what follows, the symbol [ ˆC,ˆD] ({ˆC,ˆD}) denotes the commutator (anticommutator) of the operators ˆCandˆD. We shall also use the following notations: ˆX≡X(r,t) and ˆY≡Y(r,t). In the Hermitian Dirac representation ˆHhD(t) =/parenleftbig cα.(ˆp−eA(r,t))+eΦ(r,t)+mc2β/parenrightbig , (9) the Heisenberg equation of motion for the spin operator reads as f ollows dˆΣD dt=i /planckover2pi1/bracketleftBig ˆHhD,ˆΣD/bracketrightBig =−2c /planckover2pi1[α∧(ˆp−eA(r,t)]. (10) It is well established that the expectation value onto/vextendsingle/vextendsingleΨD/angbracketrightbig of the above equation does not lead to the LLG equation for the magnetization. However, as it w illbeshown in the following, the latter can be obtained by using the non unitary FW tran sformation and the new definitionof themagnetizationasanexpectation valueof means pinoperator. Thefirst- and second-order terms of the FW expansion in powers of 1 /mcorrespond, respectively, to the precessional motion of the magnetization around an effective m agnetic field and its damping. Since the Hamiltonian ˆHD(t) has a similar structure to the one in the Dirac case, by analogy with the latter, we use for ˆU(t) the form eˆS(t)whereˆSis a non self-adjoint 8operator. Therefore, the transformation ΨFW(r,t) =eˆS(t)ΨD(r,t)≡ˆU(t)ΨD(r,t) leads to a new Hamiltonian ˆHFW(t) =eˆS(t)/parenleftbigg ˆHD(t)−i/planckover2pi1∂ ∂t/parenrightbigg e−ˆS(t), (11) whereˆSis a non self-adjoint operator. More generally, any operator inthe FW representation ,that is not explicitly time dependent, ˆOFWwill be transformed in the Dirac representation as ˆOD(t) =ˆU−1(t)ˆOFWˆU(t). The most natural extension oftheEhrenfest equation tonon-He rmitian pseudo- PTsym- metric systems is by replacing a Hermitian ˆHhDwitha non-Hermitian one. The structure of the Ehrenfest equation does not change, having assumed that part of the action of T (i.e.T0) is to send t→ −t, i.e.;we show that it anticommute swith the operator ∂/∂t, consequently, the operator PTcommuteswithi∂/∂t.Which immediately leads us to deduce the Ehrenfest equation of motion for the diagonal matrix e lement of an operator ˆOD(t) d dt/angbracketleftBig ΨD|ˆOD|ΨD/angbracketrightBig PT=/angbracketleftBigg ΨD|i /planckover2pi1/bracketleftBig ˆHD,ˆOD/bracketrightBig +∂ˆOD ∂t|ΨD/angbracketrightBigg PT. (12) It’s straightforward to show that the equation of motion (12) lead s to d dt/angbracketleftBig ΨFW|ˆOFW|ΨFW/angbracketrightBig PT=/angbracketleftbigg ΨFW|i /planckover2pi1/bracketleftBig ˆHFW,ˆOFW/bracketrightBig |ΨFW/angbracketrightbigg PT. (13) By expanding ˆS=ˆS1m−1+ˆS2m−2+ˆS3m−3+...withˆS1=β 2c(α.ˆπ),ˆS2=−i/planckover2pi1e 4c3(α.E), ˆS3=−β 8c6/parenleftBig 4c3 3(α.ˆπ)(α.ˆπ)(α.ˆπ)+iec/planckover2pi12(α.∂tE)/parenrightBig andE=−∇Φ−∂A ∂t[26], the mean spin operator is computed using the inverse FW transformation of the s pin operator asˆΣD = e−(ˆS1/m+ˆS2/m2+ˆS3/m3)ˆΣDe(ˆS1/m+ˆS2/m2+ˆS3/m3)and may be expanded in power series of (1 /m) leading toˆΣD =ˆΣD 0+ˆΣD 1m−1+ˆΣD 2m−2+ˆΣD 3m−3+... ˆΣD 0=ˆΣD, ˆΣD 1=−iβ c(α׈π), ˆΣD 2=1 8c2/parenleftBig −4ie/planckover2pi1B+2e/planckover2pi1/parenleftBig ˆΣD×B/parenrightBig −4/parenleftBig ˆπ×/parenleftBig ˆΣD׈π/parenrightBig/parenrightBig/parenrightBig −e/planckover2pi1 2c3(α×E), ˆΣD 3=β 48c3/bracketleftBig (α.ˆπ),/bracketleftBig (α.ˆπ),/bracketleftBig (α.ˆπ),ˆΣD/bracketrightBig/bracketrightBig/bracketrightBig +β 6c3/bracketleftBig (α.ˆπ)(α.ˆπ)(α.ˆπ),ˆΣD/bracketrightBig −e/planckover2pi12β 4c5(α×(∂tE)), (14) 9whereB=∇ ×A.The free case which is investigated in [7] (may be obtained in closed form in this case) is recovered from the above formula by substitut ingE=B= 0 ,iβ→β andiˆπ→ˆpleading to ˆΣD =ˆΣD−iβ(α∧ˆp) Ep−/parenleftBig ˆp∧/parenleftBig ˆΣD∧ˆp/parenrightBig/parenrightBig Ep(Ep+mc) whereEp=/radicalbig m2c2+p2. The pseudo- PTequation of motion (12) for the mean spin operator is d dt/angbracketleftbig ΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig PT =/angbracketleftbig ΨD/vextendsingle/vextendsingleeβ m/parenleftBig ˆΣD∧B/parenrightBig −e/planckover2pi1 4m2c2/parenleftBig ˆΣD∧∂tB/parenrightBig −ie 2m2c2/parenleftBig ˆΣD∧(E∧ˆπ)/parenrightBig +1 4/planckover2pi1m2c/parenleftBig/bracketleftBig (α.π)ˆΣD(α.π),(α.π)/bracketrightBig −/bracketleftbig (α.π)(α.π)(α.π),ΣD/bracketrightbig/parenrightBig +ϑ(m−3)/vextendsingle/vextendsingleΨD/angbracketrightbig PT.(15) (In order) to check this result, we have applied to the above equat ion the direct FW trans- formation. It leads to d dt/angbracketleftbig ΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig PT=d dt/angbracketleftbig ΨFW/vextendsingle/vextendsinglee(ˆS1/m+ˆS2/m2+ˆS3/m3)ˆΣD e−(ˆS1/m+ˆS2/m2+ˆS3/m3)/vextendsingle/vextendsingleΨFW/angbracketrightbig PT ≡d dt/angbracketleftbig ΨFW/vextendsingle/vextendsingleˆΣ/vextendsingle/vextendsingleΨFW/angbracketrightbig PT=/angbracketleftbig ΨFW/vextendsingle/vextendsinglei /planckover2pi1/bracketleftbigg ˆHFW,ˆΣFW/bracketrightbigg/vextendsingle/vextendsingleΨD/angbracketrightbig PT(16) whereˆΣFW≡ˆΣand the expression of ˆHFWis given by ˆHFW=iβmc2−ieˆΦ+iβˆπ2 2m+iβˆπ4 8m3c2+βe/planckover2pi12 16m3c4{ˆπ,∂tE} −βe/planckover2pi1 2mˆΣ./bracketleftbigg B−β/planckover2pi1 4mc2/parenleftbigg (∇∧E)+2i /planckover2pi1E∧ˆπ/parenrightbigg +i/planckover2pi1 8m2c4(∂tE∧ˆπ)+ ˆπ∧∂tE)/bracketrightbigg −βe/planckover2pi1 8m3c2/braceleftBig ˆπ2,ˆΣ.B/bracerightBig −iβ/parenleftbigge/planckover2pi1 2m/parenrightbigg2B2 2mc2+ie/planckover2pi12 8m2c2∇.E+ϑ(m−4). (17) Now, the equation of motion for the mean spin operator (15) can be written in a simpler way being obtained as the pseudo- PTexpectation value a spin Dirac states |χD/an}b∇acket∇i}htassociated toˆHD, one gets d dt/an}b∇acketle{tχD|ˆΣD |χD/an}b∇acket∇i}htPT =eβ m/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧B−e/planckover2pi1 4m2c2/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧∂tB −e 2mc2/an}b∇acketle{tχD|ˆΣD|χD/an}b∇acket∇i}htPT∧/parenleftbigg E∧/an}b∇acketle{tχD|i mˆπ|χD/an}b∇acket∇i}htPT/parenrightbigg +ϑ(m−3). (18) 10The above expression has been obtained by using/angbracketleftbig ΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig PT=/an}b∇acketle{tχD|ˆΣD |χD/an}b∇acket∇i}htPT. It is of interest to mention that the non-diagonal terms which appear at thesecondline of (15) are due to the Zitterbewegung phenomenon [5]. They cancel o ut when they are pseudo averaged out in a Dirac states. Let us note that, it is more convenient to use the FW representatio n to find the evolution of spin, (indeed) from Eq.(7) whered dt/angbracketleftbig ΨD/vextendsingle/vextendsingleˆΣD/vextendsingle/vextendsingleΨD/angbracketrightbig PT≡d dt/angbracketleftbig ΨFW/vextendsingle/vextendsingleˆΣFW/vextendsingle/vextendsingleΨFW/angbracketrightbig PT, we get d dt/an}b∇acketle{tχFW|ˆΣ|χFW/an}b∇acket∇i}htPT=eβ m/an}b∇acketle{tχFW|ˆΣ∧/parenleftbigg B+1 2c2/bracketleftbigg E∧iβ mˆπ/bracketrightbigg/parenrightbigg −e/planckover2pi1 4m2c2/bracketleftBig ˆΣ∧(∇∧E)/bracketrightBig |χFW/an}b∇acket∇i}htPT (19) Concerning the damping process, as previously explained, the only t erm of importance in the expression (15) is −e/planckover2pi1 4m2c2/parenleftBig ˆΣD∧∂tB/parenrightBig which has been obtained from the commutator [(α×E),(α.ˆπ)] coming from/bracketleftBig ˆHD,ˆΣD/bracketrightBig and the partial derivative with respect to time of the mean spin angular momentum operator at second order in 1 /m[27] given in Eq. (14). In addition, classical Maxwell equations have been also employ ed. Similarly to the Breit Hamiltonian ,which is obtained from the classical Darwin Lagrangian (which also originates from Maxwell equations) by using the correspondence p rinciple (CP) [25] ,we here resort to the same procedure (in its inverse form, from quan tum to classical) for the Maxwell equations. According to this principle, the quantum counte rpartsˆf, ˆgof classical observables f,gsatisfy/angbracketleftBig/bracketleftBig ˆf,ˆg/bracketrightBig/angbracketrightBig =i/planckover2pi1{f,g}p,qwhere/angbracketleftBig/bracketleftBig ˆf,ˆg/bracketrightBig/angbracketrightBig is the expectation value of the commutator and the symbol {}p,qdenotes the Poisson bracket [28–31]. Let’s take for instance the Maxwell-Faraday equation, we have ∇∧E(r,t) =−∂B(r,t) ∂t(20) which can be rewritten as ǫijk{pi,Ej}p,qek=−∂B(r,t) ∂t, (21) and using the CP one gets ǫijk[ˆpi,Ej] i/planckover2pi1ek=−∂B(r,t) ∂t≡ −∂tB. (22) Consequently, by using our definition of the magnetization M (r,t)≡ µB/angbracketleftbigg χD|ˆΣD |χD/angbracketrightbigg PT≡µB/angbracketleftBig χFW|ˆΣ|χFW/angbracketrightBig PTand the CP, the equation of motion (18) 11may be rewritten for the electron part as dM(r,t) dt=e mM(r,t)∧B(r,t)−e 4m2c2M(r,t)∧∂tB(r,t) +e 2mc2M(r,t)∧(E(r,t)∧v)+ϑ(m−3). (23) The above equation constitutes the main result of this work. Moreover, if the electron is embedded in a magnetically polarizable med ium,defined by its magnetic polarizability χm,then∂M ∂tgenerates a time-dependent magnetic induction according to the relation ∂tB(r,t) =1 χm∂M ∂tand the equation (23) can be rewritten as dM(r,t) dt=−γM(r,t)∧Beff(r,t)−αG M/parenleftbigg M(r,t)∧∂M(r,t) ∂t/parenrightbigg (24) withγ=−e m>0 the gyromagnetic ratio for an isolated electron, Beff≡B−1 2c2v∧E andαG≡eM 4m2c2χm. The first term describes the precessional motion of the magnetiz ation vector around the direction of the effective magnetic field and the s econd term represents its damping ,characterized by the Gilbert’s constant αG. Let us stress that the first term in the right hand side of equation ( 24) can be retrieved from the non-relativistic expansion of the Bargmann-Michel-Telegd i’s equation [24, 32, 33] whichrepresents the relativistic equation of motion of a classical magnetic dipole momen t [34]. However, the damping term cannot be obtained from this classic al description due to its quantum origin. In summary, the mean spin angular momentum operator introduced for the first time by Foldy and Wouthuysen for the case of a free electron has been e xtended to the non -Hermitian or precisely to a pseudo PT-symmetric case of an electron interacting with a time-dependent electromagnetic field. The expectation equation of the motion of the latter leads to the Landau-Lifshitz-Gilbert equation revealing thus its microscopic origin. We therefore argue that the expectation value of the pseudo-me an spin operator with the new definition of PT-inner product must be used instead of the usual one to properly describe the dynamics of the spin magnetization. [1] T. L. Gilbert, IEEE Transactions on magnetics 40, 3443 (2 004). In this paper published in Classics in Magnetics, the key results of the Gilbert’s thes is are reproduced. 12[2] H. Kronm¨uller and M. F¨ahnle, ”Micromagnetism and the M icrostructure of Ferromagnetic Solids”, Cambridge (2003). [3] Arne Brataas, Yaroslav Tserkovnyak, and Gerrit E. W. Bau er, Phys. Rev. Lett. 101, 037207 (2008). [4] Jean-Marc L´ evy-Leblond, Commun. math. Phys. 6, 286-311 (1967). [5] J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wes ley (1967). [6] S. Blundell, Magnetism in Condensed Matter, Oxford Univ ersity Press, (2001). [7] L. Foldy and S. Wouthuysen, Phys. Rev. 78, 29 (1950). [8] W. Greiner, Relativistic quantum mechanics. - Wave equa tions, Springer, (2000). [9] J. D. Bjorken and S.D. Drell, Relativistic Quantum Mecha nics. McGraw-Hill Book Company, New York (1964). [10] M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102, 1376 01 (2009). [11] R. Wieser, Phys. Rev. Lett. 110, 147201 (2013). [12] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppeneer, P hys. Rev. B 96, 024425 (2017). [13] R. Mondal, Marco Berritta, Peter M. Oppeneer, J. Phys.: Condens. Matter 30, 165801 (2018). [14] R. Mondal, Peter M. Oppeneer, J. Phys.: Condens. Matter 32, 455802 (2020). [15] Naima Mana and Mustapha Maamache, International Journ al of Modern Physics A, 35, No.1, 2075001 (2020) [16] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5234 ( 1998). [17] C. M. Bender, Dorje C. Brody, and Hugh F. Jones, Phys. Rev . Lett.89, 270401 (2002). [18] P. A. M. Dirac, Proc. R. Soc. Lond. A 180,1 (1942). [19] A. Mostafazadeh, J. Math. Phys. 43, 205 (2002). [20] P. Strange, Relativistic Quantum Mechanics, Cambridg e University Press, (2005). [21] X. Luo, J. Huang, H. Zhong, X. Qin, Q. Xie, Y. S. Kivshar an d C. Lee, Phys. Rev. Lett. 110, 243902 (2013). [22] M. Maamache, S. Lamri and O. Cherbal, Annals Phys. 378, 1 50 (2017). [23] J. S. Roman, L. Roso and L. Plaja, J. Phys. B 37, 435 (2004). [24] C. Itzykson, J. -B. Zuber, Quantum Field Theory, McGraw -Hill (1985). [25] M. Reiher and A. Wolf, Relativistic Quantum Chemistry, Wiley-VCH (2009). [26] Y. Hinschberger and P.-A. Hervieux, Physics Letters A 3 76, 813 (2012). 13[27]ˆBof/parenleftBig ˆΣD׈B/parenrightBig in Eq. (14) also originates from a commutator [ˆ πi,ˆπj]. [28] R. L. Liboff, Foundations of Physics 17, 981 (1987). [29] L´ evy-Leblond, J.M. The pedagogical role and epistemological significance of gro up theory in quantum mechanics . Riv. Nuovo Cim. 4, 99–143 (1974) [30] Hove, L´ eon Van. “ Sur le probl` eme des relations entre les transformations un itaires de la m´ ecanique quantique et les transformations canoniques de la m´ ecanique classique .” (1951). [31] D. Sen, S. K. Das, A. N. Basu and S. Sengupta, Current Scie nce, Vol. 80, No.4,536-541 (2001). [32] V. Bargmann, L. Michel, and V. L. Telegdi, Phys. Rev. Let t. 2, 435 (1959). [33] J. D. Jackson, Classical Electrodynamics, John Wiley ( 1998). [34] In order to be compatible with our description which doe s not include QED effects, the Land´ e factor must be g= 2. Supplementary Materials In terms of the conjugate variable ( q,p) the classical spin− →Sis desribed by [1, 2] Sx=/radicalbig S2−p2cosq Sy=/radicalbig S2−p2sinq Sz=p(25) the Poisson brackets {Si,Sj}=εijkSk(i,j,karex,yorz) are analogous to the same rela- tionships one has with spin components and commutators in quantum mechanics. Suppose we have the following Hamiltonian H=− →B.− →S (26) which is formally identical to the Hamiltonian for a spin 1 /2 system in a uniform magnetic field. We can calculate the evolution of the vector components using the standard Hamil- tonian techniques and The motion of spin− →Son the sphere (phase space) with (conserved) radiusS=/vextendsingle/vextendsingle/vextendsingle− →S/vextendsingle/vextendsingle/vextendsinglegenerated by (26), can be obtained by regarding H(26) as classical hamil- tonian . It may be confirmed that Hamilton’s equation reproduce exa ctly what spin does in a magnetic field i.e,− →· S=− →B∧− →S. The two-level spin system can be written as a classical model if we em ploy the anticom- muting Grassmann variables [3–6]− →ζwhich are transformed to the spin operator after the 14quantization∧− →ζ=∧− →S/√ 2 . Unlike the classical spin defined in the equation ((25)) which does not tranformed into a spin operator after the quantization∧− →S/ne}ationslash=− →S. [1] M. V. Berry, in ”Fundamental Aspects of Quantum” (Edited by V. Gorini and A. Frigerio), Plenum, Nato ASI series vol. 144, 267-278 (1986)). [2] M. Maamache, exact solution and geometic Angle for the cl assical spin system, Phys. Scr. 54, 21 (1996). [3] R. Casalbuoni, On the quantization of systems with antic ommuting variables, Nuovo Cimento A 33, 115 (1976). [4] F.A. Berezin and M.S. Marinov, Particle Spin Dynamics as the Grassmann Variant of Classical Mechanics, Ann. Phys. (N.Y) 104, 336 (1977). [5] E. Gozzi and W. D. Thacker, Classical adiabatic holonomy in a Grassmannian system, Phys. Rev. D 35, 2388 (1987). [6] M. Maamache and O. Cherbal, Evolution of Grassmannian in variant-angle coherent states and nonadiabatic Hannay’s angle, Eur. Phys. J. D 6, 145 (1999). 15

2022-09-16

The pseudo-PT symmetric Dirac equation is proposed and analyzed by using a non-unitary Foldy-Wouthuysen transformations. A new spin operator PT symmetric expectation value (called the mean spin operator) for an electron interacting with a time-dependent electromagnetic field is obtained. We show that spin magnetization - which is the quantity usually measured experimentally - is not described by the standard spin operator but by this new mean spin operator to properly describe magnetization dynamics in ferromagnetic materials and the corresponding equation of motion is compatible with the phenomenological model of the Landau-Lifshitz-Gilbert equation (LLG).

Pseudo-PT symmetric Dirac equation : effect of a new mean spin angular momentum operator on Gilbert damping

2209.07908v1

Material developments and domain wall based nanosecond-scale switching process in perpendicularly magnetized STT-MRAM cells Thibaut Devolderand Joo-V on Kim Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Universit ´e Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France J. Swerts, S. Couet, S. Rao, W. Kim, S. Mertens, and G. Kar IMEC, Kapeldreef 75, B-3001 Leuven, Belgium V . Nikitin SAMSUNG Electronics Corporation, 601 McCarthy Blvd Milpitas, CA 95035, USA We investigate the Gilbert damping and the magnetization switching of perpendicularly magnetized FeCoB- based free layers embedded in magnetic tunnel junctions adequate for spin-torque operated magnetic memories. We first study the influence of the boron content in MgO / FeCoB /Ta systems alloys on their Gilbert damping pa- rameter after crystallization annealing. Increasing the boron content from 20 to 30% increases the crystallization temperature, thereby postponing the onset of elemental diffusion within the free layer. This reduction of the in- terdiffusion of the Ta atoms helps maintaining the Gilbert damping at a low level of 0.009 without any penalty on the anisotropy and the magneto-transport properties up to the 400C annealing required in CMOS back-end of line processing. In addition, we show that dual MgO free layers of composition MgO/FeCoB/Ta/FeCoB/MgO have a substantially lower damping than their MgO/FeCoB/Ta counterparts, reaching damping parameters as low as 0.0039 for a 3 ˚A thick Tantalum spacer. This confirms that the dominant channel of damping is the presence of Ta impurities within the FeCoB alloy. On optimized tunnel junctions, we then study the duration of the switching events induced by spin-transfer-torque. We focus on the sub-threshold thermally activated switch- ing in optimal applied field conditions. From the electrical signatures of the switching, we infer that once the nucleation has occurred, the reversal proceeds by a domain wall sweeping though the device at a few 10 m/s. The smaller the device, the faster its switching. We present an analytical model to account for our findings. The domain wall velocity is predicted to scale linearly with the current for devices much larger than the wall width. The wall velocity depends on the Bloch domain wall width, such that the devices with the lowest exchange stiffness will be the ones that host the domain walls with the slowest mobilities. I. INTRODUCTION Tunnel magnetoresistance (TMR) and spin transfer torque (STT) – the fact that spin-polarized currents manipu- late the magnetization of nanoscale magnets and in par- ticular magnetic tunnel junction (MTJ) nanopillars – are the basic phenomena underpinning an emerging technol- ogy called Spin-Transfer-Torque Magnetic Random Access Memory (STT-MRAM)1, which combines high endurance, low power requirement2,3, CMOS back-end-of-line (BEOL) compatibility4and potentially large capacity5. The core of an STT-MRAM stack is a magnetic tunnel junction composed6of an FeCoB/MgO/FeCoB central block. One of the FeCoB layer is pinned to a high anisotropy syn- thetic ferrimagnet to create a fixed reference layer (RL) sys- tem while the second FeCoB acts as a free layer (FL). Histor- ically, the FL is capped with (or deposited on) an amorphous metal such as Ta4,7and more recently capped with a second MgO layer to benefit from a second interface anisotropy7–9 in the so-called ’dual MgO’ configuration. So far, it is un- clear whether this benefit of anisotropy can be obtained with- out sacrificing the other important properties of the free layer, in particular the Gilbert damping. In this paper, we will first tailor the Boron content inside the FeCoB alloy to improve the properties of Ta / FeCoB / MgO ’single MgO’ free layers and their resilience to thermal annealing. The idea is to postpone the FeCoB crystalliza-tion till the very last stage of the BEOL annealing. Indeed maintaining the amorphous state of FeCoB allows to mini- mize the interdiffusion of materials –in our case: tantalum– within the stack. This interdiffusion is otherwise detrimental to the Gilbert damping. We then turn to dual MgO systems comprising a Ta spacer layer in the midst of the FL. This spacer is empirically needed to allow proper crystallization and to effectively get perpen- dicular magnetic anisotropy (PMA)8,10–14. Unfortunately, the presence of heavy elements inside the FeCoB free layer is ex- pected to alter its damping and to induce some loss of mag- netic moment usually referred as the formation of magneti- cally dead layers. We study to what extend the Ta spacer in the dual MgO free layers affects the damping and how this damping compares with the one that can be obtained with sin- gle MgO free layers. Once optimized, damping factors as low as 0.0039 can be obtained a dual MgO free layer. Besides the material issues, the success of STT-MRAM also relies on the capacity to engineer devices in accordance with industry roadmaps concerning speed and miniaturiza- tion. To achieve fast switching and design devices accordingly optimized, one needs to elucidate the physical mechanism by which the magnetization switches by STT. Several categories of switching modes – macrospin15, domain-wall based16, based on sub-volume nucleation17or based on the spin-wave amplification18– have been proposed, but single-shot time- resolved experimental characterization of the switching patharXiv:1703.03198v3 [cond-mat.mtrl-sci] 4 Sep 20172 are still scarce19–21. Here we study the nanosecond-scale spin- torque-induced switching in perpendicularly magnetized tun- nel junctions with sizes from 50 to 300 nm. Our time-resolved experiments argue for a reversal that happens by the motion of a single domain wall, which sweeps through the sample at a velocity set by the applied voltage. As a result, the switching duration is proportional to the device length. We model our finding assuming a single wall moving in a uni- form material as a result of spin torque. The wall moves with a time-averaged velocity that scales with the product of the wall width and the ferromagnetic resonance linewidth, such that the devices with the lowest nucleation current densities will be the ones that host the domain walls with the lowest mobilities. The paper is split in first a material science part, followed by a study of the magnetization reversal dynamics. After a de- scription of the samples and the caracterization methods, sec- tion II C describes how to choose the optimal Boron content in an FeCoB-based free layer for STT-MRAM applications. Section II D discusses the benefits of ’dual MgO’ free layers when compared to ’single MgO’ free layers. Moving to the magnetization switching section, the part III A gathers the de- scription of the main properties of the samples and the experi- mental methods used to characterize the STT-induced switch- ing speed. Section III B describes the electrical signatures of the switching mechanism at the nanosecond scale. The latter is modeled in section III C in an analytical framework meant to clarify the factors that govern the switching speed when the reversal involves domain wall motion. II. ADVANCED FREE LAYER DESIGNS A. Model systems under investigation Our objective is to study advanced free layer designs in full STT-MRAM stacks. The stacks were deposited by phys- ical vapor deposition in a Canon-Anelva EC7800 300 mm cluster tool. The MgO tunnel barriers were deposited by RF-magnetron sputtering. In dual MgO systems, the top MgO layer was fabricated by oxidation of a thin metallic Mg film. All stacks were post-deposition annealed in a TEL-MSL MRT5000 batch furnace in a 1 T perpendicular magnetic field for 30 minutes. Further annealing at 400C were done in a rapid thermal annealing furnace in a N 2atmosphere for a pe- riod of 10 minutes. We will focus on several kinds of free layers embod- ied in state-of-the art bottom-pinned Magnetic Tunnel Junc- tions (MTJ) with various reference systems comprising ei- ther [Co/Ni] and [Co/Pt] based hard layers22,23. Although we shall focus here on FLs deposited on [Co/Ni] based synthetic antiferromagnet (SAF) reference layers, we have conducted the free layer development also on [Co/Pt] based reference layers. While specific reference layer optimization leads to slightly different baseline TMR properties, we have found that the free layer performances were not impacted provided the SAF structure is stable with the concerned heat treatment (not shown).The first category of samples are the so-called ’single- MgO’ free layers. We shall focus on samples with a free layer consists of a 1.4 nm thick Fe 60Co20B20or a 1.6 nm thick Fe 52:5Co17:5B30layer sandwiched between the MgO tunnel oxide and a Ta (2 nm) metal cap. Note that these so-called ”boron 20%” and ”boron 30%” samples have dif- ferent boron contents but have the same number of Fe+Co atoms. A sacrificial4Mg layer is deposed before the Ta cap to avoid Ta and FeCoB mixing during the deposition, and avoid the otherwise resulting formation of a dead layer. The Mg thickness is calibrated so that the Mg is fully sputtered away upon cap deposition. This advanced capping method has proven to provide improved TMR ratios and lower RA prod- ucts thanks to an improved surface roughness and a higher magnetic moment4. The second category of free layers are the so-called ’dual MgO’ free layers in which the FeCoB layer is sandwiched by the MgO tunnel oxide and an MgO cap which concur to improve the magnetic anisotropy. The exact free layer com- positions are MgO (1.0 nm) / Fe 60Co20B20(1.1 nm) / spacer / Fe 60Co20B20(0.9 nm) / MgO (0.5 nm). We study shall two spacers: a Mg/Ta(3 ˚A) spacer and a Mg/Ta(4 ˚A) spacer, both comprising a sacrificial Mg layer. B. Experimental methods used for material quality assessment We studied our samples by current-in-plane tunneling (CIPT), vibrating sample magnetometry (VSM) and Vector Network Ferromagnetic resonance (VNA-FMR)24in out-of- plane applied fields. CIPT was performed to extract the tun- nel magneto-resistance (TMR) and the resistance-area product (RA) of the junction. VSM measurements of the free layer minor loops have been used to extract the areal moments. We then use VNA-FMR to identify selectively the properties of each subsystem. Our experimental method is explained in Fig. 1, which gathers some VNAFMR spectra recorded on optimized MTJs. The first panel records the permeability of a single MgO MTJ in the ffield-frequencygparameter space. We systematically investigated a sufficiently large parameter space to detect 4 different modes whose spectral characters can be used to index them22. Three of the modes belong to the reference system that comprises 3 magnetic blocks coupled by interlayer exchange coupling through Ru and Ta spacers as usually done22,23; the properties of these 3 modes are inde- pendent from the nature of the free layer. While we are not presently interested in analyzing the modes of the fixed sys- tem – thorough analyses can be found in ref.22,23– we empha- size that it is necessary to detect all modes to unambiguously identify the one belonging to the free layer, in order to study it separately. The free layer modes are the ones having V- shaped frequency versus field curves [Fig. 1(a)], whose slope changes at the free layer coercivity. in each sample, the free layer modes showed an asymmetric Lorentzian dispersion for the real part of the permeability and a symmetric Lorentzian dispersion for the imaginary part [see the examples Fig. 1(b, c)]. As we found no signature of the two-layer nature of the dual MgO free layers, we modeled each free layer as a sin-3 Single MgOfree layerModes of the reference layers Dual MgOTa spacerf2f=0.006f2f=0.016Contrastx 10Permeabilitymap ↵=12@f@f=0.0039 FIG. 1. (Color online). Examples of MTJ dynamical properties to illustrate the method of analysis. (a) Microwave permeability versus increasing out-of-plane field and frequency for an MTJ with a sin- gle MgO free layer after an annealing of 300C. Note that the scale of the permeability was increased by a factor of 10 above 58 GHz for a better contrast. The apparent vertical bars are the eigenmode frequency jumps at the different switching fields of the MTJ. (b) Real and imaginary parts of the experimental (symbols) and modeled (lines) permeability for an out-of-plane field of 1.54 T for the same MTJ. The model is for an effective linewidth
f=(2f) = 0:016, which includes both the Gilbert damping and a contribution from the sample inhomogeneity. (c) Same but for a dual MgO free layer based on a 3 ˚A Ta spacer, modeled with
f=(2f) = 0:006. (d) Cross symbols: FMR half frequency linewidth versus FMR frequency for a dual MgO free layer based on a 3 ˚A Ta spacer. The line is a guide to the eye corresponding to a Gilbert damping of 0.0039. glemacrospin, disregarding whether it was a single MgO or a dual MgO free layer. FMR frequency versus field fits [see one example in fig. 2(c)] were used to get the effective anisotropy fields HkMsof all free layers25. The curve slopes are 0, where 0= 230 kHz.m/A is the gyromagnetic factor multiplied by the vacuum permeability 0. It was consistent with a spec- troscopic splitting Land ´e factor ofg2:08. Damping analy- sis was conducted as follows: the free layer composition can yield noticeable differences in the FMR linewidths [see for in- stance Fig. 1(b) and (c)]. To understand these differences, we systematically separated the Gilbert damping contribution to the linewidth from the contribution of the sample’s inhomo- geneity using standard VNA-FMR modeling25. This is doneby plotting the half FMR linewidth
f=2versus FMR fre- quencyfFMR [see one example in Fig. 1(d)]. The Gilbert damping is the curve slope and the line broadening arising from the inhomogeneity of the effective field within the free layer is the zero frequency intercept1 2 0
fjf=0) of the curve. C. Boron content and Gilbert damping upon annealing of single MgO free layers Designing advanced free layer in STT-MRAM stacks re- quires to minimize the Gilbert damping of the used raw ma- terial. In Ta/FeCoB/MgO ’single MgO’ free layers made of amorphous FeCoB alloys or made of FeCoB that has been just crystallized, a damping of 0.008 to 0.011 can be found typically19,25. (Note that lower values can be obtained but for thicknesses and anisotropies that are not adequate for spin- torque application26). The damping of Ta/FeCoB/MgO sys- tems generally degrades substantially when further annealing the already crystallized state27. Let us emphasize than even in the best cases26, the damping of FeCoB based free layers are still very substantially above the values of 0.002 or slightly less than can be obtained on FeCo of Fe bcc perfect single crystals28,29. There are thus potentially opportunities to improve the damping of free layers by material engineering. We illustrate this in fig. 2 in which we show that a simple increase of the Boron content is efficient to maintain the damping unaffected, even upon annealing at 400C in a single MgO free layer. In- deed starting from Ta/FeCoB/MgO ’single MgO’ free layers sharing the same damping of 0.009 after annealing at 300C (not shown), an additional 100C yields= 0:015for the free layer with 20% of boron, while the boron 30% free lay- ers keep a damping of = 0:009[see fig. 2(d)]. Meanwhile the anisotropies of these two free layers remain perpendicu- lar [fig. 2(c)] with 0(HkMs)being 0.27 and 0.17 T, re- spectively, after annealing at 400C. Let us comment on this difference of damping. Two mechanisms can yield to extra damping: spin- pumping30and spin-flip impurity scattering of the conduc- tion electrons by a spin-orbit process31. Tantalum is known to be a poor spin-sink material as this early transition metal has practically no delectrons and therefore its spin-pumping contribution to the damping of an adjacent magnetic layer is weak30. We expect a spin pumping contribution to the damp- ing of Ta (2 nm) / FeCoB (1.4 nm) / MgO ’single MgO’ free layers that compares with for instance that measured by Mizukami et al. on Ta (3 nm) / Fe 20Ni80(3 nm) which was undetectable32since below 0.0001; we therefore expect that the spin-pumping contribution to the total free layer damping is too negligible to account for the differences observed be- tween a free layer and the corresponding perfect single crys- tals. The main remaining contribution to the damping is the magnon scattering by the paramagnetic impurities within the FeCoB material33. Indeed the Ta atoms within an FeCoB layer are paramagnetic impurities that contribute to the damping ac- cording to their concentration like any paramagnetic dopant; however the effect with Ta is particularly large34as Fe and Co4 /s50/s50 /s50/s52 /s50/s54/s48 /s49 /s49/s48/s50/s48/s51/s48 /s49/s48 /s50/s48 /s51/s48/s49/s48 /s50/s48 /s51/s48 /s48/s46/s51/s48/s46/s54 /s40/s100/s41/s66/s111/s114/s111/s110/s32/s51/s48/s37 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s40/s99/s41 /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 /s32/s70/s77/s82/s32/s70/s114/s101/s113/s46/s32/s40/s71/s72/s122/s41/s32/s66/s32/s40/s84/s41 /s72/s97/s108/s102/s32/s108/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s71/s72/s122/s41 /s32/s84 /s97/s110/s110/s101/s97/s108/s32/s61/s32/s52/s48/s48/s176/s67 FIG. 2. (Color online). Properties of single MgO free layers after annealing at 400C. (a) and (b): Real part (narrow lines) and imagi- nary part (bold lines) of the free layer permeability in a field of 0.7 T. The lines are macrospin fits. (c) Ferromagnetic resonance frequency versus field curves. (d) Half linewidth versus FMR frequencies. The lines have slopes of = 0:009(red, B30%) and = 0:015(green, B20%) atoms in direct contact with Ta atoms loose part of their mo- ment and get an extra paramagnetic character, an effect usu- ally referred as a ”magnetically dead layer”. Qualitatively, the Ta atoms in the inner structure of the free layer degrade its damping. As the cap of Ta / FeCoB / MgO ’single MgO’ free lay- ers contain many Ta atoms available for intermixing, a strong degradation of the damping can be obtained in single MgO systems when interdiffusion occurs. To prevent interdiffu- sion, we used the following strategy. Amorphous materials (including the glassy metals like FeCoB) are known to be ef- ficient diffusion barriers, as they exhibit atom mobilities that are much smaller than their crystalline counterparts. To avoid the diffusion of Ta atoms to the inner part of the FeCoB free layer, a straightforward way is to maintain the FeCoB in an amorphous state as long as possible during the annealing. In metal-metalloid glasses, the crystallization temperature in- creases with the metalloid content. In our FeCoB free lay- ers, we find crystallization temperatures of 200, 300, 340 and 375C for boron contents of respectively 10%, 20%, 25% and 30%. Increasing the boron content in FeCo alloys is a way to conveniently increase the crystallization temperature and thus preserve a low damping. However since to obtain large TMR requires the FeCoB to be crystalline35,36, one should en- gineer the boron content such that the crystallization tempera- ture matches with that used in the CMOS final BEOL anneal- ing of 400C. In practice, we have found that this situation is better approached with a boron content of 30% than 0% to 25%.D. Gilbert damping in single MgO and dual MgO free layers In our search to further improve the free layers for STT- MRAM applications, we have compared the damping of op- timized ’single MgO’ and optimized ’dual MgO’ free layers. For a fair comparison, we first compare samples made from FeCoB with the same boron content of 20% and the same 300C annealing treatement. From Fig. 1(b) and (c), there is a striking improvement of the FMR linewidths when pass- ing from a single MgO to a dual MgO free layer. To discuss this difference in linewidth, we have separated the Gilbert damping contribution to the linewidth from the contribution of the sample’s inhomogeneity. We find that dual MgO sys- tems have systematically a substantially lower damping than single MgO free layers which confirms the trends indepen- dently observed by other authors9. Damping values as low as low as 0.00390:005were obtained in Ta 3 ˚A-spacer dual MgO stacks [Fig. 1(d)] after 300C annealing. Samples with a thicker Ta spacer exhibit an increased damping (not shown). This trend –lower damping in dual MgO systems –is main- tained after 400C annealing; for that annealing temperature, the best damping are obtained for a slightly different internal configuration of the dual MgO free layer. Indeed a damping of 0.0048 was obtained (not shown) in MgO / Fe 52:5Co17:5B30 (1.4 nm) / Ta (0.2 nm) / Fe 52:5Co17:5B30(0.8 nm). This should be compared with that the corresponding single MgO free layer which had a damping of 0.009 for the same an- nealing condition [Fig. 2(d)]. This finding is consistent with the results obtained on the single MgO free layer if we as- sume that the Ta impurities within an FeCoB layer contribute to the damping according to their concentration. Somehow, the number of Tantalum atoms in the initial structure of the free layer sets an upper bound for the maximum degradation of the damping upon its interdiffusion that can occur during the annealing. Notably, the single MgO free layers contain much more Ta atoms (i.e. 2 nm compared to 0.2 to 0.4 nm) available for intermixing: not only the initial number of Ta impurities within the FeCoB layer directly after deposition is larger in the case of single MgO free layer, but in addition a much stronger degradation of the damping can be obtained in single MgO systems when interdiffusion occurs, in line with our experimental findings. This interpretation – the dominant source of damping is the Ta content – is further strengthened by the fact that the thickness of the Ta spacer strongly impacts the damping in dual MgO free layers. Let us now study the spin-torque induced switching process in nanopillars processed from optimized MTJs. III. SPIN-TORQUE INDUCED SWITCHING PROCESS A. Sample and methods for the switching experiments In this section we use two kinds of perpendicularly magne- tized MTJ: a ’single MgO’ and a ’dual MgO’ free layer whose properties are detailed respectively in ref.19and20. Note that the devices are made from stacks that do not include all the latest material improvement described in the previous sec-5 tions and underwent only moderate annealing processes of 300C. The ’single MgO’ free layer samples include a 1.4 nm FeCoB 20%free layer and a Co/Pt based reference synthetic antiferromagnet. Its most significant properties include19an areal moment of Mst1:54mA, a damping of 0.01, an ef- fective anisotropy field of 0.38 T, a TMR of 150% . The ’dual MgO’ devices are made from tunnel junctions with a 2.2 nm thick FeCoB-based free layer and a hard reference system also based on a well compensated [Co/Pt]-based synthetic antifer- romagnet. The perpendicular anisotropy of the (much thicker) free layer is ensured by a dual MgO encapsulation and an iron- rich composition. After annealing, the free layer has an areal moment of Mst1:8mA and an effective perpendicular anisotropy field 0.33 T. Before pattering, standard ferromag- netic resonance measurements indicated a Gilbert damping parameter of the free layer being = 0:008. Depending on the size of the patterned device, the tunnel magnetoresistance (TMR) is 220 to 250%. Both types of MTJs were etched into pillars of various size and shapes, including circles from sub-50 nm diameters to 250 nm and elongated rectangles with aspect ratio of 2 and foot- print up to 150300 nm. The MTJs are inserted in series between coplanar electrodes [Fig. 3(a)] using a device integra- tion scheme that minimizes the parasitic parallel capacitance so as to ensure an electrical bandwidth in the GHz range. The junction properties19,20are such that the quasi-static switching thresholds are typically 500 mV . Spin-wave spectroscopy ex- periments similar to ref.37indicated that the main difference between the two sample series lies in the FL intralayer ex- change stiffness. It is A= 89pJ/m in the 2.2 nm thick dual MgO free layers of the samples of ref.20and more usual (20pJ/m) in the 1.4 nm thick ’single MgO’ free layers of the samples of ref.19. For switching experiments, the sample were characterized in a set-up whose essential features are described in Fig. 3(a): a slow triangular voltage ramp is applied to the sample in se- ries with a 50 oscilloscope. As the device impedance is much larger than the input impedance of the oscilloscope, we can consider that the switching happens at an applied voltage that is constant during the switching. We capture the elec- trical signature of magnetization switching by measuring the current delivered to the input of the oscilloscope [Fig. 3(b)]. When averaging several switching events [as conducted in Fig. 3(b)], the stochasticity of the switching voltage induces some rounding of the electrical signature of the transition. However, the single shot switching events can also be cap- tured (Fig. 4-5). In that case, we define the time origins in the switching as the time at which a perceivable change of the resistance suddenly happens (see the convention in Fig. 5). This will be referred hereafter as the ”nucleation” instant. This measurement procedure – slow voltage ramp and time- resolved current – entails that the studied reversal regime is the sub-threshold thermally activated reversal switching. This sub-threshold thermally-activated switching regime is not di- rectly relevant to understand the switching dynamics in mem- ory devices in which the switching will be forced by short pulses of substantially higher voltage21. However elucidat- ing the sub-threshold switching dynamics is of direct inter- 50 ΩMTJVoltagebias Oscilloscope50 Ω(a)(b)FIG. 3. (Color online). (a) Sketch of the experimental set-up. Mea- surement procedure: the device is biased with a triangular kHz-rate voltage (green) and the current (red) is monitored by a fast oscillo- scope connected in series. (b) The switching transitions are seen as abrupt changes of the current (red) followed by a change of the cur- rent slope. The resistance (blue) can be computed from the voltage- to-current ratio when the current is sufficiently non-zero. In this fig- ure, the displayed currents and resistances are the averages over 1000 events for a 250 nm device with a dual MgO free layer of thickness 2.2 nm and a weak exchange stiffness. est for the quantitative understanding of read disturb errors that may happen at applied voltages much below the writing pulses. Note finally that sending directly the current to the os- cilloscope has a drawback: the current decreases as the MTJ area such that the signal-to-noise ratio of our measurement degrades substantially for small device areas (Fig. 5). As a result, the comfortable signal-to-noise ratio allows for a very precise determination of the onset of the reversal in large de- vices, but the precision degrades substantially to circa 500 ps for the smallest (40 nm) investigated devices. B. Switching results In samples whose (i) reference layers are sufficiently fixed to ensure the absence of back-hopping19and (ii) in which the stray field from the reference layer is rather uniform20, opti- mized compensation of the stray field of the reference layers leads to a STT-induced switching with a simple and abrupt electrical signature [Fig. 4(a)]. If examined with a better time resolution, the switching event [Fig. 4(b)] appears to induce a monotonic ramp-like evolution of the device conductance. For a given MTJ stack, the switching voltage is practically in- dependent from the device size and shape in our interval of investigated sizes (not shown). This finding is consistent with the consensual conclusion that the switching energy barrier is almost independent from the device area38,39for device ar- eas above 50 nm. In spite of this quasi-independence of the switching voltage and the device size, the switching duration was found to strongly depend on device size (Fig. 5); we have found that smaller devices switch faster, and the trend is that the switching duration correlates linearly with the longest di- mension of the device. This is shown in Fig. 5: 40 nm devices switch in typically 2 to 3 ns whereas devices that are 6 times6 33PAP FIG. 4. (Color online). Single-shot time-resolved absolute value of the current during a spin-torque induced switching for parallel to antiparallel switching for a circular device of diameter 250 nm made with a weak exchange stiffness, dual MgO 2.2 nm thick free layer. (a) Two microsecond long time trace, illustrating that the switching is complete, free of back-hopping phenomena, and occurs between two microwave quiet states. (b) 30 ns long time trace illustrating the regular monotonic change of the device conductance during the switching. larger switch in 10 to 15 ns. Such a reversal path can be interpreted this way: once a do- main is nucleated at one edge of the device, the domain wall sweeps irreversibly through the system at a velocity set by the applied voltage [sketch in Fig. 4(b)]. The average domain wall speed is then about 20 nm/ns for the low-exchange-free- layers of ref.20. The other devices (not shown but described in ref.19) based on a ’single MgO’ free layer with a more bulk- like exchange switch with a substantially higher apparent do- main wall velocity, reaching 40 m/s. C. Switching Model: domain wall-based dynamics To model the switching, we assume that there is a domain wall (DW) which lies at a position qand moves along the longest axis xof the device. The domain wall is assumed to be straight along the ydirection, as sketched in Fig. 4(b). We describe the wall in the so-called 1D model40: the wall is assumed to be a rigid object of fixed width presenting a tiltof its magnetization in the device plane; by convention = 0is for a wall magnetization along x, i.e. a N ´eel wall. -10-5051015202501002000510m odelduration (ns)switchingd evice diam. (nm) 90 nm 60 nm 150 nm4 0 nm80 nmCurrent (norm.)T ime after nucleation (ns)250 nm FIG. 5. (Color online). Single-shot time-resolved conductance traces for parallel to antiparallel switching events occurring at at -0.5 V for circular devices of various diameters. The curves are for the de- vices whose dual MgO free layer has a thickness of 2.2 nm and has a weak exchange stiffness. The curves have been vertically offset and vertically normalized to ease the comparison. The time origins and switching durations are chosen at the perceivable onset and end of the conductance change: they are defined by fitting the experimental conductance traces by 3 segments (see the sketch labelled ”model”). Inset: duration of the switching events versus free layer diameter (symbols) and linear fit thereof with an inverse slope of 20 m/s. The local current density at the domain wall position is writtenj. The wall is subjected to an out-of-plane field Hz assumed to vary slowly in space at the scale of the DW width. jis assumed to transfer p1spin per electron to the DW by a pure Slonczewski-like STT. We define =~ 2e 0 0MSt(1) as the spin-transfer efficiency in unit such that jis a fre- quency. With typical FeCoB parameters, i.e. magnetization Ms2[1:1;1:4]MA/m and free layer thickness t2[1:4;2:2] nm, we have 2[0:018;0:036] Hz / (A/m2) where the low- est value corresponds to the largest areal moment Mst. With switching current density of the order of 41010A/m2, this yieldsjdcbetween 0.72 and 1.4 GHz. Following ref.41, the wall position qand and wall tilt are7 linked by the two differential equations: _+ _q= 0Hz; (2) _q _=jdc+ 0HDW 2sin(2) (3) in whichis the width of a Bloch domain wall in an ultra- thin film, with 2= 2A=(0MsHeff k)whereAis the exchange stiffness. A wall parameter = 12 nm will be assumed for the normal exchange 1.4 nm free layer from various estimates including ref.37for the exchange stiffness and ref.22for the anisotropy of the free layer. The domain wall stiffness field42 HDWis the in-plane field that one would need to apply to have the wall transformed from a Bloch wall to a N ´eel wall. As it expresses the in-plane demagnetization field within the wall, it depends on the wall width and on the wall length when the finite size of the device constrains the wall dimensions. Using42, the domain wall stiffness field can be estimated to be at the most 20 mT in our devices. In circular devices, the domain wall has to elongate upon its propagation38such that the domain wall stiffness field HDWdepends in princi- ple on the DW position. It should be maximal when the wall is along the diameter of the free layer. However we will see thatHDWis not the main determinant of the dynamics. In- deed in the absence of stray field and current, the Walker field HWalker is proportional to the domain wall stiffness field times the damping parameter, i.e. HWalker =H DW=2. As the sam- ples required for STT switching are typically made of low damped materials with  < 0:01, the Walker field is very small and likely to be smaller than the stray fields emanating from either the reference layers or the applied field. This very small Walker field has implications: in practice as soon as there is some field of some applied current, any domain wall in the free layer is bound to move in the Walker regime and to make the back-and-forth oscillatory movements that are in- herent to this regime. The DW oscillates at a generally fast (GHz) frequency43such that only the time-averaged velocity matters to define how much it effectively advances. To see quantitatively the effect of a constant current on the domain wall dynamics, we assume that the sample is invari- ant along the domain wall propagation direction (x) (like in an hypothetical stripe-shaped sample). Solving numerically Eq. 2 and 3, we find that the Walker regime is maintained for jdc6= 0(not shown). Two points are worth noticing: The time-averaged domain wall velocity h_qivaries linearly with the applied current density. When in the Walker regime, the current effect can be understood from Eq. 3. Indeed the sin(2)term essentially averages out in a time integration as is periodic, and the term _is neligible, such that the time- averaged wall velocity reduces to: h_qij dc (4) For= 12 nm andjin the range of 1.4 GHz at the switching voltage for the bulk-like exchange stiffness sample with free layer thcikness 1.4 nm, the previous equation would predict atime-averaged domain wall velocity of 17 m/s (or nm/ns) dur- ing the switching. More compact domain walls are expected for the samples with a weaker exchange stiffness; the twice lowerj0:72GHz related to the larger thickness would reinforce this trend to a much a lower domain wall velocity (9 m/s for our material parameters estimates). This expectation compares qualitatively well with our experimental findings of slower walls in weakly exchanged materials (Fig. 5). We wish to emphasize that Eq. 4 can be misleading regard- ing the role of damping. Indeed a too quick look at Eq. 4 could let people wrongly conclude the domain wall velocity is es- sentially set by the areal moment Mstand that the wall veloc- ity under STT from a current perpendicular to the plane (CPP current) is independent from the damping factor (see Eq. 1). However this is not the case as the switching current jdcis a sweep-rate-dependent and temperature-determined fraction 2[1 2;1]of the zero temperature instability current jc0of a macrospin in the parallel state, which reads15,44: jc0=4e ~1 +p2 p0MstHeff k 2(5) wherep1is an effective spin polarization. Using Eq. 1, 4 and 5, the time-averaged wall velocity at the practical switching voltage is: h_qi 0Heff k (6) This expression indicates that the samples performing best in term of switching current (minimal damping and easy nu- cleation thanks to a small exchange) will host domain walls that are inherently slow when pushed by the CPP current in the Walker regime. The domain wall speed scales with the domain wall width, which may be the reason why the low exchange stiffness samples host domain walls that are experi- mentally slower. To summarize, once nucleated at the instability of the uni- formly magnetized state at jdc=jc0, the domain wall flows in a Walker regime through the device. The switching dura- tion varies thus simply with the inverse current: switch =L j dcL 1  0Heff k1 (7) Let us comment on this equation which is the main con- clusion of this section. The underlying simplifications are: (i) a rigid wall (ii) that does not sense the sample’s edges (iii) that moves at a speed equal to its average velocity in the Walker regime (iv) at a switching voltage that is independent from the sample geometry. Under these assumptions, the du- ration of the switching scales with the length Lof the sam- ple, as observed experimentally. It also scales with the inverse of the zero-field ferromagnetic resonance linewith 2 0Heff k. The practical switching voltage is below the zero temperature macrospin switching voltage by a factor , which gathers the effect of the thermal activation and of the sweeping rate of the applied voltage45.1=2for quasi-static experiments like reported here and !1for experiments in which the voltage rise timeVmax=_Vis short enough compared to the switching duration (Eq. 7).8 IV . SUMMARY AND CONCLUSION In summary, we have investigated the Gilbert damping of advanced free layer designs: they comprise FeCoB alloys with variable B contents from 20 to 30% and are organized in the single MgO or dual MgO free layer configuration fully em- bedded in functional STT-MRAM magnetic tunnel junctions. Increasing the boron content increases the cristallization tem- perature, thereby postponing the onset of elemental diffusion within the free layer. This reduction of the interdiffusion of the Ta atoms helps maintaining the Gilbert damping at a low level without any penalty on the anisotropy and the transport properties. Thereby, increasing the Boron content to at least 30% is beneficial for the thermal robustness of the MTJ up to the 400required in CMOS back-end of line processing. In addition, we have shown that dual MgO free layers have a substantially lower damping than their single MgO counter- parts, and that the damping increases as the thickness of the Ta spacer within dual MgO free layers. This indicates that the dominant source of extra damping is the presence of Ta impurities within the FeCoB alloy. Using optimized MTJs, we have studied the duration of the switching events as in- duced by spin-transfer-torque. Our experimental procedure – time-resolving the switching with a high bandwidth but dur- ing slow voltage sweep – ensures that we are investigating only sub-threshold thermally activated switching events. In optimal conditions, the switching induces a ramp-like mono- tonic evolution of the device conductance that we interpret as the sweeping of a domain wall through the device. The switching duration is roughly proportional to the device size: the smaller the device, the faster it switches. We studied twoMTJ stacks and found domain wall velocities from 20 to 40 m/s. A simple analytical model using a rigid wall approxima- tion can account for our main experimental findings. The do- main wall velocity is predicted to scale linearly with the cur- rent for device sizes much larger than the domain wall widths. The domain wall velocity depends on the material parame- ters, such that the samples with the thinnest domain walls will be the ones that host the domain walls with the lowest mo- bilities. Schematically, material optimization for low current STT-induced switching (i.e. in practice: fast nucleation be- cause of low exchange stiffness Aand low damping ) will come together with slow STT-induced domain wall motion at least in the range of device sizes in which the STT-induced re- versal proceeds through domain wall motion. If working with STT-MRAM memory cells made in the same range of device sizes, read disturb should be minimal (if not absent) provided that the voltage pulse used to read the free layer magnetiza- tion state has a duration much shorter than the time needed for a domain wall to sweep through the device at that voltage (Eq. 7). 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2017-03-09

We investigate the Gilbert damping and the magnetization switching of perpendicularly magnetized FeCoB-based free layers embedded in tunnel junctions adequate for spin-torque operated memories. We study the influence of the boron content in MgO / FeCoB /Ta systems alloys on their Gilbert damping after crystallization annealing. Increasing the boron content from 20 to 30\% increases the crystallization temperature, thereby postponing the onset of elemental diffusion within the free layer. This reduction of the interdiffusion of the Ta atoms helps maintaining the Gilbert damping at a low level of 0.009 without any penalty on the anisotropy and the magneto-transport properties up to the 400$^\circ$C annealing required in CMOS back-end of line processing. In addition, we show that dual MgO free layers of composition MgO/FeCoB/Ta/FeCoB/MgO have a substantially lower damping than their MgO/FeCoB/Ta counterparts, reaching damping parameters as low as 0.0039 for a 3 \r{A} thick Tantalum spacer. This confirms that the dominant channel of damping is the presence of Ta impurities within the FeCoB alloy. On optimized tunnel junctions, we then study the duration of the switching events induced by spin-transfer-torque. We focus on the sub-threshold thermally activated switching in optimal applied field conditions. From the electrical signatures of the switching, we infer that once the nucleation has occurred, the reversal proceeds by a domain wall sweeping though the device at a few 10 m/s. The smaller the device, the faster its switching. We present an analytical model to account for our findings. The domain wall velocity is predicted to scale linearly with the current for devices much larger than the wall width. The wall velocity depends on the Bloch domain wall width, such that the devices with the lowest exchange stiffness will be the ones that host the domain walls with the slowest mobilities.

Material developments and domain wall based nanosecond-scale switching process in perpendicularly magnetized STT-MRAM cells

1703.03198v3

1 Spin injection into silicon detected by broadband ferromagnetic resonance spectroscopy Ryo Ohshima,1 Stefan Klingler,2,3, Sergey Dushenko,1 Yuichiro Ando,1 Mathias Weiler,2,3 Hans Huebl,2,3,4 Teruya Shinjo,1 Sebastian T. B. Goennenwein,2,3,4 and Masashi Shiraishi1* 1Department of Electronic Science and Engineering, Kyoto Univ., 615 -8510 Kyoto, Japan. 2Walther -Meißner -Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany. 3Physik -Department, Technische Universität München, 85748 Garching, Germany 4Nanosystems Initiative Munich , 80799 München, Germany We studied the spin injection in a NiFe(Py)/Si system using broadband ferromagnetic resonance spectroscopy. The Gilbert damping parameter of the Py layer on top of the Si channel was determined as a function of the Si doping concentration and Py layer thickness . For fixed Py thickness w e observe d an increase of the Gilbert damping parameter with decreasing resistivity of the Si channel . For a fixed Si doping concentration we measured an increasing Gilbert damping parameter for decreasing Py layer thickness. No increase of the Gilbert damping param eter was found Py/Si samples with an insulating interlayer . We attribute our observations to an enhanced spin injection into the low -resistivity Si by spin pumping . 2 Spin injection into semiconductors was relentlessly studied in recent years in hope to harness their long spin relaxation time, and gate tunability to realize spin metal -oxide - semiconductor field -effect -transistors (MOSFET s). A central obstacle for a spin injection into semiconductors was the conductance mismatch1 between ferromagnetic metals (used for the spin injection) and semiconductor channels. In an electrical spin injection method —widely used from the early years of the non -local spin transport studies —tunnel barriers between the semiconductor and ferromagne t were formed to avoid the conductance mismatch problem2–5. Unfortunately, it complicated the production process of the devices , as high quality tunnel barriers are not easy to grow , and presence of impurities, defects and pinholes take s a heavy toll on th e spin injection efficiency and/or induces spurious effects. Meanwhile, the fabrication of electrical Si spin devices, like spin MOSFETs, with different resistivities is a time - consuming process, which so far prevented systematic studies of the spin inject ion properties (such as spin lifetime, spin injection efficiency etc.). However, such a systematic study is necessary for further progress towards practical applications of spin MOSFETs. In 2002, a dynamical spin injection method, known as spin pumping, was introduced to the scene of spintronics research6,7. While the method was initially used in the metallic multilayer sy stems, it was later implemented to inject spin current s into semiconductors. In contrast to electrical spin injection, spin pumping doe s not require the application of an electric current across the ferromagnet/semiconductor interface. Devices that operate using spin current s instead of charge current s can potentially reduce heat generation and power consumption problems of modern electro nics. From a technological point of view, spin pumping is also appealing because it does not require a tunnel barrier. Spin injection —using spin pumping —into semiconductors from an adjacent ferromagnetic metal was achieved despite the existence of conducti vity mismatch8-12. However, so far there was no systematic study of the spin pumping based spin injection in dependence on the resistivity of the Si channel. 3 In this letter, we focus on the study of spin injection by spin pumping in the NiFe(Py)/Si system with different resistivities of the Si channel using broadband ferromagnetic resonance (FMR) . The broadband FMR method allows for a precise determination of the Gilbert damping parameter 𝛼, which increases in the presence of spin pumping, and thus, spin injection . By tracking the change of the Gilbert damping parameter in various Py/Si system s, we determine d the spin pumping efficiency in the broad range of resistivities of the Si channel. For a first set of sample s 7nm-thick Py film s were deposited by electron beam evaporation on top of various Si substrates (1×1 cm2 in size) with resistivit ies in the range from 10-3 to 103 ･cm (see Table 1 for the list of the prepared samples ). The oxidized surface of the Si substrates was removed using 10% hydrofluoric acid (HF) prior to the Py evaporation. For a second set of samples Py films with thickness es 𝑑Py between 5nm and 80nm were deposited on P -doped SOI (silicon on insulator) with the same technique . As a control experiment , Py/AlO x and Py/ TiO x films were grown on Si, P -doped SOI and SiO 2 substrates , as spin pumping should be suppressed in systems with an insulati ng barrier13 (see Table 2) . Both Al (3 nm, thermal deposition) and Ti (2 nm, electron beam evaporation) were evaporated on the non - treated substrates and left in the air for one day for oxidation of the surface (for the Al layer , the process was repeated 3 times, with 1 nm of Al evaporated and oxidized at each step). After oxidation, we evaporated 7nm thick Py film s on the top of the tunnel barrier s. The properties of the prepared samples are summarized in Table s 1 and 2 . A sketch of the broadband ferromagnetic res onance setup is shown in Fig. 1 (a). T he samples were placed face down on the center conductor of a coplanar waveguide (CPW), which was located between the pole shoes of an electromagnet . A static magnetic field |𝜇0𝐻|≤2.5 T was applied perpendicular to the surface of the samples to avoid extra damping due to two - magnon scattering14. One end of the CPW was connected to a microwave source , where microwaves with frequenc y f < 40 GHz were generated . The other end of the CPW was 4 connected to a microwave diode and a lock -in amplifier to measure the rectified microwave voltage as a function of the applied magnetic field . All measurements were carried out at room temperature. The microwave current in the CPW generates an oscillating magnetic field around the center conductor which results in an oscillating torque on the s ample ’s magnetization. For 𝜇0𝐻=𝜇0𝐻FMR this torque results in a n absorption of microwave power . The resonance condition i s given by the out -of-plane Kittel equation15,16: ℎ𝑓 𝑔𝜇B=𝜇0𝐻FMR −𝜇0𝑀eff. (1) Here, ℎ is the Plan ck constant, 𝑔 is the Landé g-factor, 𝜇B is the Bohr magneton, 𝜇0 is the vacuum permeability, and 𝑀eff is the effective saturation magnetization of Py. We use the Gilbert damping model, which phenomenologically models the viscous damping of the magnetic resonance. The linear relat ion between the full width at half maximum 𝛥𝐻 of the resonance and the applied microwave frequency f is given by the Gilbert damping equation 17: 𝜇0𝛥𝐻=𝜇0𝛥𝐻0+2𝛼ℎ𝑓 𝑔𝜇B. (2) Here, 𝛥𝐻0 corresponds to frequency independent scattering processes and 𝛼 is the Gilbert damping parameter18,19: 𝛼=𝛼0+𝛼SP+𝛼EC. (3) Here, 𝛼0 is the intrinsic Gilbert damping , 𝛼SP=𝑔𝜇𝐵𝑔r↑↓4𝜋𝑀S𝑑Py ⁄ is the damping due to spin pumping20, 𝑔r↑↓ is the real part of the spin mixing conductance, and 𝛼EC=𝐶EC𝑑Py2 is the eddy -current damping . The parameter 𝐶EC describes efficiency of the eddy -current damping . To realize a net spin injection via spin pumping, the following conditions should be fulfilled in the system: (i ) carriers should be present in the underlying channel, (ii) the spin relaxation time in the channel should be small enough. The available carriers in the channel transfer spin angular momentum away from the spin injection interface, allowing propagation 5 of the spin current. On the other hand, the long spin relaxation time in the channel leads to a large spin accumulation at the interface and generates a diffusive spin backflow in the direction opposite to the spin pumping current20 (see Figs. 1(b) and 1(c )). Thus, the spin backflow effectively cancels out the spin pumping current for long spin relaxation time, and the spin pumping contribution to the Gilbert damping parameter should no longer be present in the system. In addition to spin pumping, charge currents can be induced in the Si channel and Py layer due to the Faraday ’s law and Py magnetization precession, which results in a Gilbert -like damping contribution . These processes are refer red to as radiative damping and eddy -current damping18. An e nhancement of the Gilbert damping parameter due to these processes is expected to be especially large for the Si channels with low resistivit ies and thick Py films , since energy dissipation through eddy currents scales linearly with the conductivity of the Si layer and quadratically with the Py layer thickness . Hence, both spin pumping and eddy current damping are expected to be most efficien t for low -resistivity Si. I n contrast to spin pumping , the radiative damping contribution doe s not require a direct electrical contact between Py and Si, and is hence unaffected by the tunnel barrie r. Figure 1(d ) show s a typical FMR spectr um of a 7nm thick Py film on a phosphorous P-doped Si on insulator (SOI) substrate , where t he microwave frequency was fixed at 30 GHz during the sweep of the magnetic field. A single FMR signal was observed (Fig. 1(d) red filled circles) , from which 𝜇0𝐻FMR and 𝜇0𝛥𝐻 were extracted by a fit of the magnetic ac susceptibility (Fig. 1(d) black line) 16,21 (see Supplemental Material for additional fitting examples). An excellent agreement of the fit with the measurement is achieved. Figures 2(a) and (b) show 𝐻FMR and 𝛥𝐻 versus the applied microwave frequency f for the Py/P-doped SOI, Py/SOI and Py/SiO 2 samples. From the f itting of the frequency dependence of HFMR with Eq.(1) , 𝑔 and 𝜇0𝑀eff of the Py /P-doped SOI ( Py/SOI) were 6 estimated to be 2.049 (2.051) and 0.732 T (0.724 T), and those of the Py /SiO 2 were estimated to be 2.038 and 0.935 T, respectively (Supplemental Material for fitting and data from other samples ). The difference in the 𝑔 and 𝑀eff between the Py /Si and the Py/ SiO 2 sample is attributed to the inter -diffusion of the Fe/Ni and Si at the interface , which is always present to some extent during the growth at room temperature22,23. The Gilbert damping 𝛼 of the Py /P- doped SOI , Py/SOI and Py/SiO 2 were estimated to be 1.25 ×10-2, 9.02 ×10-3 and 8.49 ×10-3, respectively , from the linewidth vs. frequency evolution . The intrinsic Gilbert damping parameter 𝛼0 is determined from the linewidth evolution of the Py/SiO 2 and Py/quartz samples to be 8.5×10−3 and 8.6×10−3, respectively, since no spin pumping contribution is expected in these insulating materials ( 𝛼=𝛼0). From this we can see an increasing Gilbert damping with decreasing resistivity. We additionally measured the samples with a n insulating tunnel barrier between the Si channel and the Py film. We found Gilbert damping parameters of 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, independent of the Si resistivity. The damping values are in agreement with the intrinsic damping extracted from the Py/SiO 2 sample , indicating that radiative damping is negligible in our samples. Figure 3 (a) summarizes the dependence of the Gilbert damping parameter 𝛼 on the resistivity of the Si channel (see Supplemental Material D for the g -factor, the effective saturation magnetization and the frequency independent term ), including the measured control samples. The dashed lines show the intrinsic contributions 𝛼0 to the Gilbert damping parameter 𝛼 measured from the Py/SiO 2 and Py/quartz samples (red dashed), Py/AlO x/SiO 2 (blue dashed) and Py/TiO x/SiO 2 (green dashed). All samples with Py on top of the conductive substrates without an additional tunnel barrier exhibited the Gilbert damping parameter 𝛼 larger than the intrinsic contribution 𝛼0. The experimentally measured Gilbert damping parameter decreases logarithmically 7 with the resistivity. This result is in agreement with condition (i) for the spin pumping. In the Si channels with a small resistivity more carriers were available to transfer the injected angular momentum, leading to an effective spin pumping. Additionally, both electron spin resonance25– 28 and non -local 4 -terminal Hanle precession29,30 experiments showed, that the spin lifetime in Si is increas ing with increasing resistivity . Our samples with low resistivities have a large doping concentration (see Table 1), leading to shorter spin relaxation time. In accordance with the spin pumping condition (ii), the decrease of the spin relaxation time should lead to the increase of the spi n pumping contribution, as now observe d experimentally. While the spin pumping shows a logarithmic dependence on the resistivity of the channel, we note that an increased Gilbert damping parameter is observed even for the Si channel with high resistivit ies. We comment on the Sb -doped sample, where the experimentally measured 𝛼SP was lower than one expected from the logarithmic trend of the other samples. We speculate that this might originate from the different doping profile, compared to the other samples . We note, that further studies are necessary to separate the influence of the number of carriers in the channel and the spin relaxation time on the spin pumping process. Finally, we show that the Py damping is increased in a broad range of Si resistivitie s and attribute this effect to the enhanced spin injection via spin pumping ( a discussion of the spin mixing conductance for various Si resistivities is given in the Supplemental M aterial A). Figure 3(b) shows the Py thickness dependence of the 𝛼 for Py/P -doped SOI samples. The solid line shows a fit of Eq.(3) to the measured data and a very good agreement of the spin pumping theory with our measurements is achieved. From the fit we estimate 𝛼0=6.1×10−3, 𝑔r↑↓=1.2×1019 m-2 and 𝐶EC=2.9×1011 m-2. Both the intrinsic damping and the real part of the spin mixing conductance are in good agreement with previous measurements30. The blue dashed line indicates 𝛼0+𝛼SP and the green dashed line shows 𝛼0+𝛼EC . Dominant influence of spin pumping to the total damping is observed in samples with small Py thickness , 8 while eddy current contribution is dominant in samples with thick Py layer . For the 7 nm -thick Py sample, we find 𝛼SP = 3.6 ×10-3 and 𝛼EC = 1.4 ×10-5. Thus, the eddy -current damping in our 7 nm Py samples is negligibly small and cannot explain the increase of the damping with decreasing resistivity. The Py thickness dependence of the Gilbert damping indicates spin pumping into the Si substrates. In conclusion , we studied spin pumping based spin injection from a Py layer into Si channels with various resistivit ies using broadband ferromagnetic resonance . We determine d the spin pumping contribution from the change of the Gilbert damping parameter. The observed logarithmic decrease of the Gilbert damping parameter with increasing resistivity of the Si channel is attribute to the decrease in the number of carriers in the channel, and the increase in the spin lifetime. De spite the reduction of the spin pumping contribution to the Gilbert damping parameter with the increasing resistivity of the Si channel , we observe spin pumping even for the channels with high resistivity . We furthermore observe an increase of the Gilbert damping parameter for decreasing Py thickness which is in agreement with the spin pumping theory. Our results show that spin pumping can be potentially used in a spin transistors, where low doping concentration in the channel is necessary for the gate control of the device. Supplement al Material See Supplementary M aterial for a discussion of the spin mixing conductance for various Si resistivities and additional fitting examples. ACKNOWLEGEMENTS This research was supported in part by a Gran t-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, Innovative Area “Nano Spin Conversion Science ” (No. 2 6103003), Scientific Research (S) 9 “Semico nductor Spincurrentronics ” (No. 16H0633) and JSPS KAKENHI Grant (No. 16J00485). R.O. acknowledges JSPS Research Fellowship. S.D. acknowledges support by JSPS Postdoctoral Fellowship and JSPS KAKENHI Grant No. 16F16064. 10 References 1 A. Fert and H. Jaffrès, Phys. Rev. B 64, 184420 (2001). 2 E.I. Rashba, Phys. Rev. B 62, R16267 (2000). 3 I. Appelbaum, B. Huang, and D.J. Monsma, Nature 447, 295 (2007). 4 O.M.J. van ’t Erve, A.T. Hanbicki, M. Holub, C.H. Li, C. Aw o-Affouda, P.E. Thompson, and B.T. Jonker, Appl. Phys. Lett. 91, 212109 (2007). 5 T. Sasaki, T. Oikawa, T. Suzuki, M. Shiraishi, Y. Suzuki, and K. Tagami, Appl. Phys. Express 2, 53003 (2009). 6 Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. B 66, 224403 (2002). 7 S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002). 8 K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Trypiniotis, C.H.W. Barnes, S. Maekawa, and E. Saitoh, Nat. Mater. 10, 655 (2011). 9 K. Ando and E. Saitoh, Nat. Commun. 3, 629 (2012). 10 E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, and M. Shiraishi, Phys. Rev. Lett. 110, 127201 (2013). 11 A. Yamamoto, Y. Ando, T. Shinjo, T. Uemura, and M. Shiraishi, Phys. Rev. B 91, 24417 (2015). 12 S. Dushenko, M. Koike, Y. Ando, T. Shinjo, M. Myronov, and M. Shiraishi, Phys. Rev. Lett. 114, 196602 (2015). 13 C.H. Du, H.L. Wang, Y. Pu, T.L. Meyer, P.M. Woodward, F.Y. Yang, and P.C. Hammel, Phys. Rev. Lett. 111, 247202 (2013). 14 K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. Farle, U. von Hörsten, H. Wende, W. Keune, J. Rocker, S.S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, and Z. Frait, Phys. Rev. B 76, 104416 (2007). 15 K. Ando and E. Saitoh, J. Appl. Phys. 108, 113925 (2010). 11 16 K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa, and E. Saitoh, J. Appl. Phys. 109, 103913 (2011). 17 H.T. Nembach, T. J. Silva, J.M. Shaw, M.L. Schneider, M. J. Carey, S. M aat, and J.R. Childress, Phys. Rev. B 84, 054424 (2011). 18 M.A.W. Schoen, J. M. Shaw, H.T. Nembach, M. Weiler and T.J. Silva, Phys. Rev. B 92, 184417 (2015). 19 H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Takahashi, Y. Kajiwara, K. Uchida, Y. Fujikwa and E. Saitoh, Phys. Rev. B 85, 144408 (2012). 20 Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, and B.I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 21 L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gross, M.S. Brandt, and S.T.B. Goennenwein, Phys. Rev. B 86, 134415 (2012). 22 J.M. Gallego, J.M. García, J. Alvarez, and R. Miranda, Phys. Rev. B 46, 13339 (1992). 23 N. Kuratani, Y. Murakami, O. Imai, A. Ebe, S. Nishiyama, and K. Ogata, Thin Solid Films 281–282, 352 (1996). 24 Y. Ochiai and E. Matsuura, Phys. Status Solidi A 38, 243 (1976). 25 J.H. Pifer, Phys. Rev. B 12, 4391 (1975). 26 V. Zarifis and T. Castner, Phys. Rev. B 57, 14600 (1998). 27 R. Jansen, Nat. Mater. 11, 400 (2012). 28 T. Suzuki, T. Sasaki, T. Oikawa, M. Shiraishi, Y. Suzuki, and K. Noguchi, Appl. Phys. Express 4, 23003 (2011). 29 T. Tahara, Y. Ando, M. Kameno, H. Koike, K. Tanaka, S. Miwa, Y. Suzuki, T. Sasaki, T. Oikawa, and M. Shiraishi, Phys. Rev. B 93, 214406 (2016). 30 Note, that the sample set with various Py thicknesses was grown in a different batch than the samples with v arious Si doping. Hence, small deviations in the damping and the spin 12 mixing conductance are due to slightly different growth conditions . Figure 1: (a) Experimental setup for the broadband FMR measurement. The samples were placed with the Py layer facing down on a coplanar waveguide. External magnetic field and microwave field from the waveguide induce the FMR of the Py and spins are injec ted into Si via spin pumping. Schematic images of spin injection and dephasing in Si that have (b) long and (c) short spin lifetimes. 𝜏1 and 𝜏2 are the spin lifetim e of Si in the case of (b) and (c), respectively. Spin injection efficiency becomes large in the case of (c) because of a reduction of the backflow of spins. (d ) The derivative of the FMR signal of Py at 30 GHz microwave frequency. I is the microwave absorption intensity. Figure 2: Frequency dependence of the (a) resonance field 𝐻FMR and (b) full width at half maximum 𝛥𝐻 of the FMR spectra obtained from Py on top of P -doped SOI, SOI and SiO 2. The solid lines show fitting using Eqs. (1) and (2) of 𝐻FMR and 𝛥𝐻, respectively. Figure 3: (a) Si resistivity dependence of the Gilbert damping parameter 𝛼. The damping of the samples with an insulating layer represents the intrinsic damping of the Py layer and is shown by the dashed line s. Red, blue and green coloration represents Py, Py/AlO x and Py/TiO x samples, respectively. The damping of the Py/ P-doped SOI is an averaged value extracted from the two Py/P-doped SOI samples fabricated at different times. (b) Py thickness dependence of 𝛼. The solid line shows a fit of Eq. (3) to the data . The b lue line shows 𝛼0+𝛼SP, whereas the green line shows 𝛼0+𝛼EC. 13 Fig. 1 R. Ohshima et al . Fig. 2 R. Ohshima et al. 14 Fig. 3 R. Ohshima et al. Table 1: Sample summary Name Dopant Doping density (cm-3) Structure Resistivity (･cm) Gilbert damping parameter Mixing conductance (m-2) Py/P - doped SOI P 6.5×1019 Py(7 nm)/Si(100 nm)/ SiO 2(200 nm)/Si 1.3×10−3 1.1×10−2 5.7×1018 Py/Sb - doped Si Sb 1×1019 Py(7 nm)/Si 5.0×10−3 9.3×10−3 2.3×1018 Py/N - doped Si N 1×1019 Py(7 nm)/Si 1.0×10−1 9.5×10−3 2.6×1018 Py/SOI N/A 1×1015 Py(7 nm)/Si(100 nm)/ SiO 2(200 nm)/Si 4.5 9.0×10−3 1.3×1018 Py/P - doped Si P 1×1013 Py(7 nm)/Si 1.0×103 8.7×10−3 5.1×1017 Py/SiO 2 - - Py(7 nm) /SiO 2(500 nm)/Si - 8.5×10−3 - Py/Quartz - - Py(7 nm) /Quartz - 8.6×10−3 - 15 Table 2: List of the samples for the control experiment Name Dopant Doping density (cm-3) Structure Resistivity (･cm) Gilbert damping parameter Py/AlO x/P- doped SOI P 6.5×1019 Py(7 nm)/AlO x(3 nm)/Si(100 nm)/ SiO 2(200 nm)/Si 1.3×10−3 8.6×10−3 Py/AlO x/P- doped Si P 1×1013 Py(7 nm)/AlO x(3 nm)/Si 1.0×103 8.5×10−3 Py/AlO x/ SiO 2 - - Py(7 nm)/ AlO x(3 nm)/ SiO 2(500 nm)/Si - 8.8×10−3 Py/TiO x/P- doped SOI P 6.5×1019 Py(7 nm)/TiO x(2 nm)/Si(100 nm)/ SiO 2(200 nm)/Si 1.3×10−3 7.5×10−3 Py/TiO x/P- doped Si P 1×1013 Py(7 nm)/TiO x(2 nm)/Si 1.0×103 7.9×10−3 Py/TiO x/ SiO 2 - - Py(7 nm)/TiO x(2 nm)/ SiO 2(500 nm)/Si - 7.8×10−3

2017-04-24

We studied the spin injection in a NiFe(Py)/Si system using broadband ferromagnetic resonance spectroscopy. The Gilbert damping parameter of the Py layer on top of the Si channel was determined as a function of the Si doping concentration and Py layer thickness. For fixed Py thickness we observed an increase of the Gilbert damping parameter with decreasing resistivity of the Si channel. For a fixed Si doping concentration we measured an increasing Gilbert damping parameter for decreasing Py layer thickness. No increase of the Gilbert damping parameter was found Py/Si samples with an insulating interlayer. We attribute our observations to an enhanced spin injection into the low-resistivity Si by spin pumping.

Spin injection into silicon detected by broadband ferromagnetic resonance spectroscopy

1704.07006v1

1 CoB/Ni-Based Multilayer Nanowire with High-Speed Domain Wall Motion under Low Current Control Duc-The Ngo*, Norihito Watanabe, and Hiroyuki Awanoj Information Storage Materials Laboratory, Toyota Technological Institute, Nagoya 468- 8511, Japan The spin-transfer torque motion of magnetic DWs in a CoB/Ni-based nanowire driven by a low current density of (1.12±0.8)×10 11 A m-2 has been observed indirectly by magnetotransport 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 11 A m-2, the DW velocity increases to 197±16 m/s before decreasing quickly in the high- current-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 saturation magnetization, Gilbert damping fact or, and density of pinning sites, making the CoB/Ni multilayer nanowire favora ble for practical applications. *Present address: Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576. j E-mail address: awano@toyota-ti.ac.jp. 2 1. Introduction When a spin-polarized electron current hits a magnetic moment, it exerts a torque on the moment, transfers its angular momentum to the moment, and thereby affects the precession motion and switching of the mo ment. This phenomenon was theoretically predicted by Berger [1] and Slonczewski [2], and subsequently was named the spin- transfer torque (STT). The motion of magne tic domain walls (DWs) caused by an electrical current in magnetic nanostructures is also a consequence of the SST in which the spin-polarized current switc hes the magnetic moments in the wall. This is nowadays widely applied in spintronic technology such as the DW logic gate [3,4] and racetrack memory [5, 6]. Over the last 15 years, most studies have focused on a NiFe patterned film, a typical soft magnetic material with in-plane magnetic anisotropy and nearly zero magnetocrystalline 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 relatively high current density (~10 12 A m−2) owing to a wide DW and a low spin-torque efficiency (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 heat released from electrical current would sometimes degrade the performance of such devices. Therefore, deceasing of current dens ity is one of the most important technical issues at the moment. 3 Perpendicularly magnetized thin films have recently been proposed to replace the in-plane NiFe film [8,9] to realize this goa l. In the perpendicular magnetic anisotropy films, formation of Bloch-type walls that are 1-2 orders of magnitudes thinner than Néel- type 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] have presented the decrease in threshold current to (2-5)×10 11 A m−2 using multilayer nanowires, 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 thickness in those multilayers was normally ~3-20 Ä and might be badly influenced by the 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. Among the researchers, Yamanouchi et al. [14] were successful in establishing the motion of magnetic DWs in a perpendicularly magnetized ferromagnetic semiconductor (Ga,Mn)As with a very low current density of about 10 9 A m−2. However, this material (and most ferromagnetic semiconductors) has a Curie temperature far below room temperature and therefore is not realistic for room-temperature devices. In this article, we present the enhancement of the motion of the magnetic DWs in the 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, and improved the stability of the multilayer by preventing the diffusion between the Co/Ni interfaces. 4 2. Experimental Methods A multilayer film of Pt 5 nm/[CoB 0.6 nm/Ni 1.1 nm]4/CoB 0.6 nm/Pt 1 nm was fabricated by radio-frequency (RF) magnetron sputtering using Ar gas. The base vacuum of the deposition chamber was 3×10−8 Torr whereas the Ar pressure was maintained at 5 mTorr during the deposition process. The com position of the CoB ta rget was chosen as Co80B20 (at.%). The film was grown on a naturally oxidized Si substrate. A nanowire with 300 nm width and 150 µm length was subsequently patterned by electron beam lithography and ion beam etching (Fig. 1). The nanowire was modified to have a planar Hall shape for magnetotransport measurements. A square pad was made at one end of the wire 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 pattern produced by photolithography was mounted to the wire for magnetotransport measurements. The magnetic properties of the film specimen were measured using an alternating gradient magnetometer (AGM). The magnetic DWs were nucleated in the square pad by an Oersted field generated from the 30 ns width, 6.5 MHz pulse current flowing in the Ti/Au electrode deposited on the pad (Fig. 1, electrodes A-B). The motion of DWs in the nanowire was then driven by a DC current (J DC, electrodes G-B in Fig. 1). Anomalous Hall effect [17,18] measurement (either electrodes C-D or E-F) was carried out to detect the propagation of DW in the wire. Hysteresis loop measurement on the continuous film specimen (data not shown) using the AGM confirmed that the film exhibited a strong perpendicular magnetic anisotropy with a saturation magnetization of M s=5.6×105 A/m 5 (about 15% lower than that of a Co/Ni-based film [13]) and a uniaxial anisotropy constant of K u=3.57×105 J/m3 (~8% higher than that of a Co/Ni film [12, 13]). 3. Results and Discussion The time-resolved Hall effect voltage signal obtained from the nanowire at a driven current (DC current) of 0.59 mA corresponding to a current density of jDC=1.12×1011 A m−2 and an external field of +5 mT is illustrat ed in Fig. 2(a) and represents the movement of DW along the wire [17,18]. Initially, the wire was magnetically saturated, then magnetization reversal was induced by the Oersted field generated from the pulse current with 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 the wire, toward the Hall bar. When the domain moved into the Hall bar, the presence of the domain in the cross bar induced a change in the Hall voltage signal, as seen in Fig. 2(a). The Hall signal switched to a low valu e when the domain (w ith the two walls) had passed the Hall bar. The progress of the Ha ll signal could be interpreted approximately on the basis of a simple schematic shown in Fig. 2(b). Because of a periodic pulse, the domains were nucleated and driven to the wire periodically and the Hall voltage signal appeared 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 sharp change in the magnetization, proving a fast propagation of a domain through the Hall 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 Oersted field released from the driven current was estimated to be about 30 mT, which 6 was much smaller than the coercive field of the sample (see the Hall effect hysteresis loop in the inset of Fig. 2). Therefore, the influence of the Oersted field on the motion of the wall along the wire (described in Fig. 2) could be minor, whereas the effect of the spin-polarized current is essentially considered. The current dependence of the variation of normalized Hall resistance, ΔRHall, is shown in Fig. 3. The normalized Hall resistance here was defined as the change in the Hall 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 density of 1.12×10 11 A m−2, whereas this value was low (0) below the threshold current density. 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 1.12×10 11 A m−2, confirming that it is possible to drive the DW motion in the 300 nm width CoB/Ni nanowire with a threshold current density of 1.12×1011 A m−2 by the STT mechanism. It is important that the threshold current density obtained here was reasonably lower than either ~(1-3)×10 12 A m−2 in the NiFe-based devices [6,7] or ~(2- 5)×1011 A m−2 in a similar multilayer Co/Ni wire [12,13], or lower than the current density in a spin-valve nanowire [19] reported recently. Moreover, it is shown in Fig. 2 that the pulse like signal of Hall voltage is periodic and coherent with the nucleation pulse, presuming the continuous propagation of a multidomain similar to a shift register writing process. From the pulse like Hall signal, the velo city of DW moving in the Hall bar could be derived [13] from Fig. 2(b): T 1 was the time when the front-edge wall of the domain started coming to the Hall bar and T 2 was the time that it passed the Hall bar (in L = 500 7 nm). Therefore, the velocity of the front-edge wall could be referred as L/ Δt1 (Δt1 = T 2- T1). On the other hand, the velocity of the rear-edge wall could be attained from the time interval Δt2 = T 4 – T 3. On the other hand, from the phase delay between the signals at the C-D and E-F Hall bars that reflected the time of flight of the wall between two Hall bars, the velocity of the wall in the straight wire (from C to E) was de termined. Figure 4 shows the wall velocity in the straight wire area as a function of external magnetic field measured at the threshold current density. The field dependence here is consistent with the following expression [20]: v(H) = µ H(H - H 0) + v(J), (1) where µ H(J) is the DW mobility, J is the current density, and H 0 is the “dynamic coercive force”. The term v(J) - µ HH0 can be referred as the velocity at zero field. Using this linear dependence, a zero-field wall velocity of 86±5 m/s was calculated at the critical current density (1.12×1011 A m−2) with a mobility of 2640±170 (m s−1 T−1). This matches well with the velocity measured directly at H = 0 (85±4 m/s). It is interesting to note that the field-free wa ll velocity here was much higher than that of the Co/Ni wire [12,13] or TbFeCo nanowire [18,21]. Therefore, this aspect is very promising for high-speed devices. As DW moved in the region of the Hall bars, the wall velocity (as defined above) was found to be sli ghtly lower than that in the straight wire area but only in the error scale of the measurement. The non-zero DW velocity and linear dependence of the wall velocity on the external field can be attributed to the motion driven by the nonadiabatic torque [20]. The velocities of the front-edge and rear-edge walls were perfectly identical to each other and remained invariable at positions of two Hall bars. These suggest that i) the 8 effect of the pinning on the motion of th e walls along the wire was predominantly governed by the material rather than the geometry of the Hall bars and ii) no distortion of the domain and the wall geometry as the domain length was conserved when they were located in the Hall bars. Usually, the distortion of the domain in the Hall bar, denoted by the small difference between the velocities of the front-edge wall (faster) and the rear- edge wall (slower), was only observed at a high applied field (over 90 mT), and can be imagined similarly to the distortion of a balloon, as reported elsewhere [22]. Regarding other interesting points, the time interval T 3 - T 1 [see Fig. 2(b)] expresses the period necessary for the whole domain to reach the rear side of the Hall bar, allowing 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 11 A m−2 and zero field. Under an external field, the domain size was slightly reduced to 630±25 nm 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. The dependence of wall velocity at zero field on controlled current density is shown in Fig. 5. The velocity firstly incr eased with current density from 85 m/s at the threshold current density to a maximum value of 197±16 m/s at a current density of 2.63×10 11 Am−2, then markedly dropped at higher current densities. The variation of wall velocity with current density in this case can be explained qualitatively by referring to the model given in refs.#7 and 24. The model proposed by Tatara et al. [24] predicted that the trend of the wall velocity variation (including a linear increase at low current and a 9 decrease with increasing current at high cu rrents) is a consequence of the nonadiabatic torque driving when the damping factor is low and the pinning effect is weak. In the low current regime, DW velocity was linearly dependent on current density, which is in accordance with the zero-field velocity de scribed in eq. (1) and somehow similar to a previous experimental observation [20]. At a high current density (above 2.63×1011 A m−2), wall velocity appeared to decrease, indicating that the nonadiabatic parameter β was not zero and not equal to the Gilbert damping fa ctor. This led to a deformation of the wall structure above the Walker breakdown current density [20]. This dependence and linear field-velocity function discussed in previous paragraphs indicated that the motion of the walls in our device was mainly governed by the nonadiabatic term. In an attempt to explain the decrease in the threshold current density, theoretical models [7,24] are employed, in which the threshold (or intrinsic critical) current density could be referred to as follows [7]: 21~s c BeMJgP , (2) where α and β are the Gilbert damping factor and nonadiabatic spin-transfer torque parameter, respectively; is the DW width; Ms is the saturation magnetization; P is the spin polarization of the material; g is the gyromagnetic ratio, e is the electron charge, and B is the Bohr magneton. As discussed in the previous paragraphs, the magnitude of saturation magnetization of the studied CoB/Ni multilayer film was decreased by ~15%, whereas uniaxial anisotropy was slightly enhanced which subsequently led to a thinner DW. Additionally, the substitution of B for Co is expected to decrease the Gilbert damping factor of the film. Hence, such decreases in saturation magnetization magnitude and wall 10 width could result in a decrease in the threshold critical current. Moreover, the addition of B atoms on the other hand weakens the pinning of the DW by decreasing the number of pinning sites [25] as mentioned in previous paragraphs. This effect also enhances the velocity of the walls. It should be noted that the Walker current at which the wall velocity dropped as seen in Fig. 5, is also expected to be a function of the intrinsic parameters of the materials [26,27]. Additionally, 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 closed-package degree of the lattice, thus preventing the diffusion between the layers and preserving 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 the lattice constant between Co/Ni and result s in a smooth Co/Ni interface. It should be noted that 20% addition of B to the Co layers (Co 80B20) only gives rise to ~12% of resistivity in comparison with pure Co laye rs. From the technical point of view, these benefits would to enhance the working stability of the devices. 4. Conclusions The motion of magnetic DW in the CoB/Ni multilayer nanowire with a very low current density of (1.12±0.8)×1011 A m-2 and a high DW velocity of 85±4 m/s has been successfully induced. DW velocity can be raised up to 197±16 m/s by increasing the current density to 2.63×1011 A m−2. The variation of wall velocity was consistent with the nonadiabatic STT mechanism. These advantages were attributed to the presence of CoB layers with a low Gilbert damping factor, a low saturation magnetization, and a low 11 density of pinning sites. 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Ohshima, N. Ishwata, Y. Nakatani, and T. Ono: Appl. Phys. Express 3 (2010) 073004. [14] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno: Nature 428 (2004) 539. [15] D. McGrouther, S. McVitie, J. N. Chap man, and A. Gentils: Appl. Phys. Lett. 91 (2007) 022506. [16] K. J. Kirk, J. N. Chapman, and C. D. W. Wilkinson: Appl. Phys. Lett. 71 (1997) 539. [17] J. Heinen, D. Hinzke, O. Boulle, G. Malinowski, H. J. M. Swagten, B. Koopmans, C. Ulysse, G. Faini, and M. Kläui: Appl. Phys. Lett. 99 (2011) 242501. [18] D.-T. Ngo, K. Ikeda, and H. Awano: J. Appl. Phys. 111 (2012) 083921. [19] L. Y. Jang, S. Yoon, K. Lee, S. Lee, C. Nam, and B.-K. Cho: Nanotechnology 20 (2009) 125401. [20] G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Erskine: Phys. Rev. Lett. 97 (2006) 057203. [21] D.-T. Ngo, K. Ikeda, and H. Awano: Appl. Phys. Express 4 (2011) 093002. [22] D. Ravelosona, D. Lacour, J. A. Katine, B. D. Terris, and C. Chappert: Phys. Rev. Lett. 95 (2005) 117203. [23] K.-J. Kim, K.-W. Moon, K.-S. Lee, and S.-B. Choe: Nanotechnology 22 (2011) 025702. [24] G. Tatara, H. Kohno, and J. Shibata: J. Phys. Soc. Jpn. 77 (2008) 031003. [25] R. Lavrijsen, G. Malinowski, J. H. Franken, J. T. Kohlhepp, H. J. M. Swagten, B. Koopmans, M. Czapkiewicz, and T. Stobiecki: Appl. Phys. Lett. 96 (2010) 022501. 14 [26] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonfim, A. Schuhl, and G. Gaudin: Nat. Mater. 10 (2011) 419. [27] O. Boulle, G. Malinowski, and M. Cläui: Mater. Sci. Eng. R 72 (2011) 159. 15 Figure captions Fig. 1 . Electron microscopy image of the CoB/Ni nanowire with Ti/Au electrodes for magnetotransport measurements. The plus sign denotes the direction of applied field. Fig. 2 . (a) Time-resolved Hall voltage signal measured at a driven current of 0.59 mA (j DC=1.12×1011 A.m-2) and external field of 5 mT; (b) Interpretation of the Hall-voltage pulse as domain motion progresses in the Hall bar. The inset shows the field dependence of Hall voltage. Fig. 3 . Hall resistance changes as a function of driven current density. The inset shows a time-resolved Hall voltage signal measure at the critical current density and zero field. Fig. 4 . External field dependence of the velocity of the wall measured at the threshold current density (j DC=1.12×1011 A.m-2). Fig. 5 . Variation of wall velocity as a function of driven current density at zero external field. 16 Figure 1 17 Figure 2 18 Figure 3 19 Figure 4 20 Figure 5

2011-10-24

The spin-transfer torque motion of magnetic domain walls (DWs) in a CoB/Ni-based nanowire driven by a low current density of (1.12\pm0.8)\times10^{11} A m^{-2} has been observed indirectly by magnetotransport measurements. A high DW velocity of 85\pm4 m/s at zero field was measured at the threshold current density. Upon increasing the current density to 2.6\times10^{11} A m^{-2}, the DW velocity increases to 197\pm16 m/s before decreasing quickly in the high-current-density regime attributed to nonadiabatic spin-transfer torque at a low damping factor and weak pinning. The addition of B atoms to the Co layers decreased the magnitude of saturation magnetization, Gilbert damping factor, and density of pinning sites, making the CoB/Ni multilayer nanowire favorable for practical applications.

CoB/Ni-Based Multilayer Nanowire with High-Speed Domain Wall Motion under Low Current Control

1110.5112v3

arXiv:1311.4070v1 [nucl-th] 16 Nov 2013Shear viscosity due to the Landau damping from quark-pion in teraction Sabyasachi Ghosh1,2, Anirban Lahiri2, Sarbani Majumder2, Rajarshi Ray2, Sanjay K. Ghosh2 1Instituto de Fisica Teorica, Universidade Estadual Paulis ta, Rua Dr. Bento Teobaldo Ferraz, 271, 01140-070 Sao Paulo, SP, Brazil and 2Center for Astroparticle Physics and Space Science, Bose In stitute, Block EN, Sector V, Salt Lake, Kolkata 700091, India We have calculated the shear viscosity coefficient ηof the strongly interacting matter in the relaxation time approximation, where a quasi particle desc ription of quarks with its dynamical mass is considered from NJL model. Due to the thermodynamic s cattering of quarks with pseudo scalar type condensate (i.e. pion), a non zero Landau dampin g will be acquired by the propagating quarks. This Landau damping may be obtained from the Landau c ut contribution of the in-medium self-energy of quark-pion loop, which is evaluated in the fr amework of real-time thermal field theory. From the basic idea of the QCD asymptotic freedom at high temperatures and densities, a weakly interact- ing quark gluon plasma (QGP) is naturally expected to be produced in the experiments of heavy ion colli- sion (HIC). However, the experimental data from RHIC, especially the measured elliptic flow indicates that nu- clear matter as a strongly interacting liquid instead of a weakly interacting gas. The recent hydrodynamical cal- culations [1, 2] as well as some calculations of kinetic transport theory [3, 4] conclude that the matter, pro- duced in HIC, must have very small shear viscosity. The shear viscosity of the fluid is generally quantified by the the coefficient ηand it physically interprets the ability to transfer momentum over a distance of mean free path. Hence the lower values of ηmeans the constituents of the matter interact strongly to transfer the momentum eas- ily. Whereasaweaklyinteractingsystemmust havelarge ηbecause in this case the momentum transfer between the constituents become strenuous. Several theoretical attempts [5–25] are taken to cal- culate theηof the strongly interacting matter at very high [5], intermediate [6, 7] and low temperature [8–16], where some special attentions are drawn on the small- ness of its original value with respect to its lower bound (η=s 4π, wheresisentropydensity), commonlyknownto as the KSS bound [26]. The most interesting fact, which hasbeen addedwiththe recenttheoreticalunderstanding ofηfor strongly interacting matter, is that the η/smay reachaminimum inthe vicinityofaphasetransition[19– 23] (see also [27]) like the liquid-gas phase transition of certain materials e.g. Nitrogen, Helium or Water. These investigationsdemand a better understanding to zoom in on the temperature ( T) dependence of ηof the strongly interactingmatternearthephasetransition. Inspiringby thismotivation, inthisbriefreportwehaveaddressedthe η(T) due to forward and backward scattering of quark- pion interaction. In the relaxation time approximation, the ηof the quark [23] and pion [15, 16] medium (for µ= 0) can be expressed as η=8β 5/integraldisplayd3/vectork (2π)3/vectork4 ω2 QnQ(1−nQ) ΓQ(k) (U=l−k)(l) (k) QQ (A) (B)(k) (U=k−l)(l) (k)QQ Q FIG. 1: The diagram of quark (A)and pion (B)self-energy for quark-pion and quark-anti quark loops respectively. +β 5/integraldisplayd3/vectork (2π)3/vectork4 ω2πnπ(1+nπ) Γπ(1) wherenQ=1 eβωQ+1andnπ=1 eβωπ−1are respectively Fermi-Dirac distribution of quark and Bose-Einstein dis- tribution of pion with ωQ=/radicalBig /vectork2+M2 Qandωπ=/radicalBig /vectork2+m2π. The Γ Qand Γ πare Landau damping of quark and pion respectively. Following the quasi parti- cle description of Nambu-Jona-Lasinio(NJL) model [28], the dynamical quark mass MQis considered and it is generated due to quark condensate /angbracketleftψfψf/angbracketright=−MQ−mQ 2G(2) wheremQis the current quark mass. In the medium, above relation become (for µ= 0) MQ=mQ+4NfNcG/integraldisplayd3/vectork (2π)3MQ ωQ(1−2nQ).(3) This relation shows that the constituent quark mass tends to be the current quark mass at very high tem- perature where the non-zero quark condensate becomes small. This Landau damping Γ Qand Γ πmay be estimated from the self-energy graphs of quark and pion at finite temperature for quark-pion and quark-anti quark loops respectively. These are respectively expressed as ΓQ=−ImΣR(k0=/radicalBig /vectork2+M2 Q,/vectork) (4) and Γπ=−1 mπImΠR(k0=/radicalBig /vectork2+m2π,/vectork) (5)2 where ΣRand ΠRare respectively retardedpart of quark and pion self-energy at finite temperature. Their dia- grammatic representations are shown in Fig.1 (A)and (B)respectively. Following the real-time formalism of thermal field theory, the retarded part of in-medium quark self energy for quark-pion loop is given by [30] ΣR(k0,/vectork) =/integraldisplayd3/vectorl (2π)31 4ωl QωUπ[(1−nl Q)LQ 1+nU πLQ 3 k0−ωl Q−ωUπ+iη +nl QLQ 1+nU πLQ 4 k0−ωl Q+ωUπ+iη+−nl QLQ 2−nU πLQ 3 k0+ωl Q−ωUπ+iη +nl QLQ 2+(−1−nU π)LQ 4 k0+ωl Q+ωUπ+iη]. (6) whereLQ i,i= 1,..4 denote the values of LQ(l0,/vectorl) for l0=ωl Q,−ωl Q,k0−ωU π,k0+ωU πrespectively with ωl Q=/radicalBig /vectork2+M2 QandωU π=/radicalBig (/vectork−/vectorl)2+m2π. Herenl Q(ωl Q) is Fermi-Dirac distribution function of quark whereas nU π(ωU π) denotes Bose-Einstein distribution function of π meson. During extracting the imaginary part of ΣR(k0,/vectork) we will get four delta functions associated with the four individual terms of Eq. (6), which generate four dif- ferent region in k0-axis where the ImΣR(k0,/vectork) will be non-zero. From the non-zero values of ImΣR(k0,/vectork) the region of discontinuities or branch cuts of ΣR(k0,/vectork) can be identified. The regions coming from the 1st and 4th terms of (6) are respectively ( k0=−∞to −/radicalBig /vectork2+(mπ+MQ)2) and (k0=/radicalBig /vectork2+(mπ+MQ)2 to∞). These are known as unitary cuts and different kind of forward and inverse decay processes are associ- ated with these cut contributions [29, 30]. Similarly the regions (k0=−/radicalBig /vectork2+(mπ−MQ)2to 0) and (k0= 0 to/radicalBig /vectork2+(mπ−MQ)2) are coming from 2nd and 3rd terms respectively. These purely medium dependent cuts are known as Landau cuts and different kind of forward and inverse scattering processes are physically interpreted by these cut contributions [29, 30]. So the 3rd term of ImΣR(k0,/vectork) at the on-shell mass ( k0=/radicalBig /vectork2+M2 Q,/vectork) of quark is responsible for the Landau damping Γ Qand it is given by [30] ΓQ=−ImΣR(k0=/radicalBig /vectork2+M2 Q,/vectork) = [/integraldisplayd3/vectorl (2π)3LQ 2 4ωl QωUπ (nl Q+nU π)δ(k0+ωl Q−ωU π)]k0=/radicalbig /vectork2+M2 Q.(7) Rearranging the statistical weight factor by (nl Q+nU π) =nl Q(1+nU π)+nU π(1−nl Q),(8) we can find thermalized πanduwith Bose enhanced probability (1+ nU π) and Pauli blocked probability (1 −nl Q) respectively. With the help of Eq. (8), the physi- cal significance of the Landau cut contribution may ex- pressed as follows. During the propagation of uquark, it may absorb the thermalized ufrom the heat bath and create a thermalized πin the bath (indicated by the sec- ond partof Eq.(8)). Again the thermalized πmaybe ab- sorbedbythe mediumandcreatethe thermalized ualong with a propagating u, which is slightly off-equilibrium with the medium (indicated by the first part of Eq. (8)). To calculate LQ ifrom quark-pion interaction, let us start with free Lagrangian of quarks and demanding the invariance properties of Lagrangian under chiral trans- formation, ψ′ f= exp(i/vector π·/vector τγ5 2Fπ)ψf (9) where the chiral angle is associated with the pion field /vector πandFπis pion decay constant. Expanding up to first order of pion field, we obtain the quark-pion interaction term [33, 34], LπQQ=−iMQ Fπψf/vector π·/vector τγ5ψf =−iMQγ5 Fπ/parenleftbig ud/parenrightbig/parenleftbigg π0√ 2π+√ 2π−π0/parenrightbigg/parenleftbigg u d/parenrightbigg .(10) As we are interested to calculate one-loop self-energy (ΣR) of any quark flavor, u(say) hence we have to con- sider two possible loops - uπ0anddπ+. Due to isospin symmetry consideration in Lagrangian, we can evaluate anyone of the loops, say uπ0loop and then we have to multiply it by a isospin factor IF= (1)2+(√ 2)2= 3. (11) From the interaction part, Lπ0uu=−igπQQuγ5π0u,withgπQQ=MQ Fπ(12) we can calculate LQ ab(l0,/vectorl) =−IFg2 πQQ(l /−ml)ab, where a,bare Dirac indices. For simplification we have taken the scalar part only i.e. LQ(l0,/vectorl) =IFg2 πQQml. We have taken the parameters mQ= 0.0056 GeV,MQ= 0.4 GeV (forT= 0), three momentum cut-off Λ = 0 .588 GeV and corresponding Tc= 0.222 GeV for µ= 0 [28]. Similar to Eq. (6), the pion self-energy ΠRfor quark- anti quark loop is also received similar kind of form only the quantities nU π,ωU π=/radicalBig (/vectork−/vectorl)2+m2π,U=k−l andLQ i’s are changed to −nU Q,ωU Q=/radicalBig (/vectorl−/vectork)2+m2 Q, U=−k+landLπ i’s respectively [31, 32]. As pion on- shell mass point ( k0=/radicalBig /vectork2+m2π,/vectork) will be inside the unitary cut region ( k0=/radicalBig /vectork2+(MQ+MQ)2to∞) of ΠR, therefore Γπ=−ImΠR(k0=/radicalBig /vectork2+m2π,/vectork) mπ=−1 mπ[/integraldisplayd3/vectorl (2π)3Lπ 1 4ωl QωU Q3 00.10.20.30.40.50.6Γ (GeV)with folding without folding πcomponent 0.15 0.2 0.25 0.3 0.35 T (GeV)0102030τ (fm)k=0 FIG. 2: Upper panel : The Tdependency of Γ Qwith (solid line) and without (dotted line) folding by Aπand Γ π(dashed line). Lower panel : The variation of corresponding collisi on timeτwith temperature. 010203040Aπ (GeV-2)T=0.300 GeV T=0.250 GeV 00.050.10.150.20.250.30.350.4 M (GeV)00.10.20.30.40.5ImΠ/mπ (GeV)k=0 FIG. 3: Lower panel shows Mdependency of the imaginary part of pion self-energy for Q¯Qloop, which is normalized after dividing by mπ. Upper panel shows invariant mass distribu- tion of pion spectral function due to its Q¯Qwidth. Dotted line indicates the position of pion pole. 0.05 0.1 0.15 0.2 0.25 0.3 T (GeV)00.010.020.030.04η (GeV3) πcomponent without folding with folding Ref. [23] Ref. [12] Ref. [14] FIG. 4: Temperature dependence of ηdue to Γ π(dashed line), Γ Qwithout (dotted line) and with (solid line) folding are separately shown. The results of Ref. [23](triangles) a re attached to compare with our results (solid line). [also the results of hadronic domain by Ref. [12](stars), Ref. [14] (o pen circles)].(1−nl Q−nU Q)δ(k0+ωl Q−ωU Q)]k0=√ /vectork2+m2π(13) whereLπ= 4IFg2 πQQ[M2 Q−l2−k·l] can be obtained from (12). In Fig. (2), we can see the temperature dependency of Landau damping Γ (upper panel) and collision time τ=1 Γ(lower panel) of quark (dotted line) and pion (dashed line) for their momentum /vectork= 0. Owing to the on-shell condition, the Γ Qand Γ πare received the non-zero values only in the temperature range where mπ>2MQ, which are clearly seen from dotted and dashed lines respectively. A correspondingnon-divergent collisional times are also achieved by them in the same temperature domain. Due to decay width of π→Q¯Q, it will more realistic to consider the pion resonance of finite width in Eq. (7). The pion spectral function due to Q¯Q width may be defined as Aπ(M) =1 πIm/bracketleftBigg 1 M2−m2π+iImΠRvac(k0,/vectork)/bracketrightBigg (14) where ImΠR vac(k0,/vectork) is vacuum part of ImΠR(k0,/vectork) and M=/radicalBig k2 0−/vectork2. The variation of ImΠR(M)/mπandAπ withMfor two different temperatures are respectively shown in lower and upper panel of Fig. (3). Replacing mπof ΓQin (7) byMand then convoluting or folding it byAπ(M), we have [31, 32] ΓQ(mπ) =1 Nπ/integraldisplay ΓQ(M)Aπ(M)dM2(15) whereNπ=/integraltext Aπ(M)dM2. One should notice that in the narrow width approximation i.e. for ImΠR vac→0, Eq. (15) is merged to (7). The Tdependency of Γ Qand its corresponding τafter folding are shown by solid line in the upper and lowerpanel of Fig. (2) respectively. Due to folding, Γ Qat lowTdomain (where mπ<2MQ) has acquired some non-zero values from its vanishing con- tributions and at the same time corresponding τrecover fromitsdivergenceuptothe approximatefreezeouttem- perature (T∼120−150GeV) of the strongly interacting matter. By using Γ Q(T,/vectork) from Eq. (7) and (15) in the quark component (first term) of Eq. (1), we get the results of shear viscosity as function of T, which are respectively described by dotted and solid line of Fig. (4). Being proportional to collisional time, the divergence of ηis removed after folding in those temperature region, where mπ<2MQ. The contribution of ηdue to Γ π(T,/vectork) from Eq. (13) is shown by dashed line in Fig. (4). After similar kind of folding as done in Eq. (15), an almost negligible (∼10−5 GeV3) contribution of ηfor pion component can be obtained which is not included in final results. Inlowtemperatureregion, ηisdecreasingwithincreas- ing ofTwhich is analogous to the behavior of liquid (From our daily life experience, we see that the cooking oil behaves like a less viscous medium when it is heated).4 Whereas in high temperature domain, ηbecome an in- creasing function of Tjust like a system of gas. The magnitude of ηin our approach is very close to the results of Sasaki and Redlich [23] (indicated by tri- angles) but underestimated with respect to the earlier estimation in NJL model by Zhuang et al. [22]. The LQCD calculation of η(η∼0.054−0.47 GeV3nearTc) by H. B. Meyer [7] is higher than all of these calculations. From the solid line in the lower panel of Fig. (2), we see thatτbelow theT∼160 MeV exceeds the typical value of time period ( ∼30−50 fm) during which a strongly interacting matter survive in the labs of heavy ion colli- sions. Therefore the estimation of ηin low temperature domain is quite higher than the standard calculations of ηof hadronic matter [12, 14, 16]. The earlier calculations of NJL model [22, 23] also displayed these discrepancy in the hadronic temperature domain. In summary we have investigated the shear viscosity of strongly interacting matter in the relaxation time ap-proximation, where quarks with its dynamical mass may have some non zero Landau damping because of its vari- ous forward and inverse scattering with pions. This Lan- dau damping can be obtained from the thermal field the- oretical calculation of quark self-energy for quark-pion loop. The temperature dependency of shear viscosity is coming from the thermal distribution functions, the tem- perature dependence of Landau damping as well as the constituent quark mass, supplied by the temperature de- pendent gap equation in the NJL model. Due to this gap equation, this constituent quark mass drops rapidly to- wards its current mass near Tcto restore the chiral sym- metry. A non-trivial influence of all these temperature dependency on η(T) is displayed in our results. Acknowledgment: S. G. thanks to Saurav Sarkar, Tamal K. Mukherjee, Soumitra Maity, Ramaprasad Adak, Kinkar Saha, Sudipa Upadhaya for some pieces of discussions which have some direct and indirect influ- ence on our present work. [1] P. Romatschke and U. Romatschke, Phys. Rev. Lett. 99, 172301 (2007); M. LuzumandP. Romatschke, Phys.Rev. C 78, 034915 (2008). [2] H. Song and U. W. Heinz, Phys. Lett. B 658, 279 (2008); Phys. Rev. C 78, 024902 (2008). [3] Z. Xu, C. Greiner, and H. Stocker, Phys. Rev. Lett. 101, 082302 (2008); Z. Xu and C. Greiner, Phys. Rev. C 79, 014904 (2009). [4] G. Ferini, M. Colonna, M. Di Toro, and V. Greco, Phys. Lett.B 670, 325 (2009). V. Greco, M. Colonna, M. Di Toro, and G. Ferini, Prog. Part. Nucl. Phys. 65, 562 (2009). [5] P. B. Arnold, G. D. Moore, and L. G. Yaffe, J. High Energy Phys. 11 (2000) 001; 05 (2003) 051. [6] A. Nakamura and S. Sakai, Phys. Rev. Lett. 94, 072305 (2005). [7] H. B. Meyer, Phys. Rev. D 76, 101701 (2007); Phys. Rev.D 82, 054504 (2010). [8] M. Prakash, M. Prakash, R. Venugopalan, and G. Welke, Phys. Rep. 227, 321 (1993). [9] A. Dobado and S.N. Santalla, Phys. Rev. D 65, 096011 (2002); A. Dobado and F. J. Llanes-Estrada, Phys. Rev. D 69, 116004 (2004). [10] J. W. Chen, Y. H. Li, Y. F. Liu, and E. Nakano, Phys. Rev. D 76, 114011 (2007); E. Nakano, arXiv:hep-ph/0612255. [11] K. Itakura, O. Morimatsu, and H. Otomo, Phys. Rev. D 77, 014014 (2008). [12] D. Fernandez-Fraile and A. Gomez Nicola, Eur. Phys. J. C 62, 37 (2009); Eur. Phys. J. A 31, 848 (2007); Int. J. Mod. Phys. E 16(2007) 3010. [13] A. Muronga, Phys. Rev. C 69, 044901 (2004). [14] R. Lang, N. Kaiser and W. Weise Eur. Phys. J. A 48, 109 (2012). [15] S. Gavin, Nucl. Phys. A, 435, 826 (1985). [16] S. Mitra, S. Ghosh, and S. Sarkar Phys. Rev. C 85, 064917 (2012); S. Sarkar, Advances in High Energy Physics, vol. 2013, Article ID 627137, (2013).[17] S. Plumari, A. Puglisi, F. Scardina, and V. Greco, Phys. Rev.C 86, 054902 (2012). [18] J. Peralta-Ramos, G. Krein, Int.J. Mod. Phys. Conf. Ser . 18 (2012) 204; Phys. Rev. C 84(2011) 044904. [19] L. P. Csernai, J. I. Kapusta, and L. D. McLerran, Phys. Rev. Lett. 97, 152303 (2006); J.I. Kapusta arXiv:0809.3746 [nucl-th]. [20] P. Chakraborty and J. I. Kapusta Phys. Rev C 83, 014906 (2011). [21] T. Hirano, M. Gyulassy Nucl. Phys. A 769(2006) 71. [22] P. Zhuang, J. Hufner, S. P. Klevansky, L. Neise Phys. Rev.D 51(1995) 3728; P. Rehberg, S. P. Klevansky, J. Hufner, Nucl. Phys. A 608(1996) 356. [23] C. Sasaki, K. Redlich, Nucl. Phys. A 832(2010) 62. [24] N. Demir and S.A. Bass Phys. Rev. Lett. 102, 172302 (2009). [25] R. Marty, E. Bratkovskaya, W. Cassing, J. Aichelin, H. Berrehrah, arXiv:1305.7180 [hep-ph]. [26] P. Kovtun, D. T. Son, and O. A. Starinets, Phys. Rev. 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2013-11-16

We have calculated the shear viscosity coefficient $\eta$ of the strongly interacting matter in the relaxation time approximation, where a quasi particle description of quarks with its dynamical mass is considered from NJL model. Due to the thermodynamic scattering of quarks with pseudo scalar type condensate (i.e. pion), a non zero Landau damping will be acquired by the propagating quarks. This Landau damping may be obtained from the Landau cut contribution of the in-medium self-energy of quark-pion loop, which is evaluated in the framework of real-time thermal field theory.

Shear viscosity due to the Landau damping from quark-pion interaction

1311.4070v1

Optimal damping ratios of multi-axial perfectly matched layers for elastic-wave modeling in general anisotropic media Kai Gaoa, Lianjie Huanga aGeophysics Group, Los Alamos National Laboratory, Los Alamos, NM 87545 Abstract The conventional Perfectly Matched Layer (PML) is unstable for certain kinds of anisotropic media. This in- stability is intrinsic and independent of PML formulation or implementation. The Multi-axial PML (MPML) removes such instability using a nonzero damping coecient in the direction parallel with the interface be- tween a PML and the investigated domain. The damping ratio of MPML is the ratio between the damping coecients along the directions parallel with and perpendicular to the interface between a PML and the investigated domain. No quantitative approach is available for obtaining these damping ratios for general anisotropic media. We develop a quantitative approach to determining optimal damping ratios to not only stabilize PMLs, but also minimize the articial re ections from MPMLs. Numerical tests based on nite- dierence method show that our new method can eectively provide a set of optimal MPML damping ratios for elastic-wave propagation in 2D and 3D general anisotropic media. Key words: Anisotropic medium, elastic-wave propagation, Multi-axial Perfectly Matched Layers (MPML), damping ratio. 1. Introduction Elastic-wave modeling usually needs to absorb outgoing waveelds at boundaries of an investigated domain. Two main categories of boundary absorbers have been developed: one is called the Absorbing Boundary Condition (ABC) (e.g., Clayton and Engquist, 1977; Reynolds, 1978; Liao et al., 1984; Cerjan et al., 1985; Higdon, 1986, 1987; Long and Liow, 1990; Peng and Toks oz, 1994), and the other is termed the Perfectly Matched Layer (PML) (e.g., Berenger, 1994; Hastings et al., 1996; Collino and Tsogka, 2001). In some literature, PML is considered as one of ABCs. However, there are fundamental dierences in the construction of PML and its variants compared with traditional ABCs, we therefore dierentiate them in names. For a brief summary, please refer to Hastings et al. (1996) and other relevant references. The PML approach was rst introduced by Berenger (1994) for electromagnetic-wave modeling, and has been widely used in elastic-wave modeling because of its simplicity and superior absorbing capability (e.g., Collino and Tsogka, 2001; Komatitsch and Tromp, 2003; Drossaert and Giannopoulos, 2007). Various improved PML methods for elastic-wave modeling have been developed, such as non-splitting convolutional PML (CPML) to enahce absorbing capability for grazing incident waves (Komatitsch and Martin, 2007; Preprint submitted to Geophysical Journal International May 24, 2022arXiv:1608.08326v3 [physics.geo-ph] 21 Dec 2016Martin and Komatitsch, 2009), and CPML with auxiliary dierential equation (ADE-PML) for modeling with a high-order time accuracy formulation (Zhang and Shen, 2010; Martin et al., 2010). However, a well- known problem of PML and its variants/improvements is that numerical modeling with PML is unstable in certain kinds of anisotropic media for long-time wave propagation. To address the instability problem of PML, B ecache et al. (2003) analyzed the PML for 2D anisotropic media and found that, if there exists points where the ithcomponent of group velocity vhas an opposite direction relative to the ithcomponent of wavenumber k, i.e.,viki<0 (no summation rules applied), then thexi-direction PML is unstable. The original version of this aforementioned condition was expressed with so-called \slowness vector" dened by B ecache et al. (2003), but it can be recast in such form according to the denition of the \slowness vector" in eq. (45) of (B ecache et al., 2003). This PML instability is intrinsic and independent of PML/CPML formulations adopted for waveeld modelings. To make PML stable, elasticity parameters of an anisotropic medium need to satisfy certain inequality relations (B ecache et al., 2003). These restrictions limit the applicability of PML for arbitrary anisotropic media. Meza-Fajardo and Papageorgiou (2008) presented an explanation for the instability of conventional PML. They recast the elastic wave equations in PML to an autonomous system and found that the PML instability is caused by the fact that the PML coecient matrix having one or more eigenvalues with positive imaginary parts. They showed that PML becomes stable when adding appropriate nonzero damping coecients to PML in the direction parallel with the PML/non-PML interface. The ratio between the PML damping coecients along the directions parallel with and perpendicular to the PML/non-PML interface is called the damping ratio. The resulting PML with nonzero damping ratios is termed the Multi-axial PML (MPML). A key step in the stability analysis of PML is to derive eigenvalue derivatives of the damped system coecient matrix. Meza-Fajardo and Papageorgiou (2008) derived expressions of the eigenvalue derivatives for anisotropic media. However, these expressions are valid only for two-dimensional isotropic media and anisotropic media with up to hexagonal/orthotropic anisotropy, that is, C116= 0,C336= 0,C556= 0, C15=C35= 0, andC13can either be zero or nonzero depending on medium properties. Furthermore, although Meza-Fajardo and Papageorgiou (2008) showed that the nonzero damping ratios can stabilize PML, they did not present a method to select the appropriate dampoing ratios. Adding these nonzero damping coecients makes the PML no longer \perfectly matched", and the larger are the ratios, the stronger the articial re ections become (Dmitriev and Lisitsa, 2011). This increase is linear. Therefore, it is necessary to nd a set of \optimal" damping ratios to not only ensure the stability of MPML, but to also eliminate articial re ections as much as possible. We develop a new method to determine the optimal MPML damping ratios for general anisotropic media. We show that, even for a two-dimensional anisotropic medium with nonzero C15andC35, the MPML stability analysis is complicated, and new equations must be derived to calculate both the eigenvalues of the undamped system and the eigenvalue derivatives of the damped system. The resulting expressions are functions of all nonzeroCijcomponents as well as the wavenumber k. For 3D general anisotropic media, we nd that 2such an analytic procedure becomes practically impossible because it requires denite analytic expressions of eigenvalues and eigenvalue derivatives. In the 3D case, the dimension of the asymmetric system coecient matrix is up to 27 27, and therefore a purely numerical approach should be employed. We present two algorithms with slightly dierent forms but essentially the same logic, to determine the optimal damping ratios for 2D and 3D MPMLs. With these algorithms, it is possible to stabilize PML for any kind of anisotropic media without using a trial-and-error method. Our new algorithms enable us to use MPML for nite-dierence modeling of elastic-wave propagation in 2D and 3D general anisotropic media where all elastic parameters Cijmay be nonzero. These algorithms are also applicable to other elastic-wave modeling methods such as spectral-element method (e.g., Komatitsch et al., 2000) and discontinuous Galerkin nite- element method (e.g., de la Puente et al., 2007). Our paper is organized as follows. In the Methodology section, we derive the equations for the eigen- value derivatives for 2D and 3D general anisotropic media. We also present two algorithms to obtain the optimal MPML damping ratios. To validate our algorithms, we give six numerical examples in the Results section, including three 2D anisotropic elastic-wave modeling examples and three 3D anisotropic elastic-wave modeling examples, and show that our algorithms can give appropriate damping ratios for both 2D and 3D modeling in general anisotropic media. 2. Methodology 2.1. Optimal damping ratios of 2D MPML In this section, we concentrate our analysis on the x1x3-plane. This analysis is also valid for the x1x2- andx2x3-planes. We assume that C15andC35are generally nonzero for an anisotropic medium. The 2D elastic-wave equations in the stees-velocity form are given by (e.g., Carcione, 2007), @v @t=+f; (1) @ @t=CTv; (2) where = (11;33;13)Tis the stress waveeld, v= (v1;v3)Tis the particle velocity waveeld, fis the external force, is the mass density, Cis the elasticity tensor in Voigt notation dened as C=0 BBB@C11C13C15 C13C33C35 C15C35C551 CCCA; (3) andis the dierential operator matrix dened as =0 @@ @x10@ @x3 0@ @x3@ @x11 A: (4) In the following analysis, we ignore the external force term fwithout loss of generality. 3Using the convention in Meza-Fajardo and Papageorgiou (2008) for isotropic and VTI/HTI/othotropic media, the undamped system of eqs. (1){(2) can be also written as @ @t2 4v1 v33 5=2 4@ @x111+@ @x313 @ @x113+@ @x3333 5; (5) @ @t2 666411 33 133 7775=2 6664C11C13C15C15 C13C33C35C35 C15C35C55C553 77752 6666664@ @x1v1 @ @x3v3 @ @x1v3 @ @x3v13 7777775: (6) Equivalently, the system of the above two equations can be written as @ @tv=D1@ @x1+D3@ @x3; (7) @ @t=C1@ @x1v+C3@ @x3v; (8) where v= (v1;v3)T; (9) = (11;33;13)T; (10) D1=12 41 0 0 0 0 13 5; (11) D3=12 40 0 1 0 1 03 5; (12) C1=2 6664C11C15 C13C35 C15C553 7775; (13) C3=2 6664C15C13 C35C33 C55C553 7775: (14) In the conventional 2D PML, each eld variable is split into two orthogonal components that are per- pendicular to and parallel with the interface between the PML and the investigated domain, and the system of wave equations in PML can be written as (Meza-Fajardo and Papageorgiou, 2008) @ @t=A ; (15) withA=A0+B, and A0=2 6666664033 033@ @x1C1@ @x1C1 033 033@ @x3C3@ @x3C3 @ @x1D1@ @x1D1022 022 @ @x3D3@ @x3D3022 0223 7777775; (16) 4B=2 6666664d1I3033 032 032 033d3I3032 032 023 023d1I2022 023 023 022d3I23 7777775; (17) where 0mnis themnzero matrix, Imis themmidentity matrix, and = ((1) 11;(1) 33;(1) 13;(3) 11;(3) 33;(3) 13;v(1) 1;v(1) 3;v(3) 1;v(3) 3)T(18) represents the split waveeld variables in PML. In the damping matrix B,d1andd3represent the PML damping coecients along the x1- andx3-axis, respectively. The PML damping coecients depend on the thickness of PML, the desired re ection coecient and the P-wave velocity at the PML/non-PML interface. Usually, they vary with the distance from a location inside the PML to the PML/non-PML interface according to the power of two or the power of three (e.g., Collino and Tsogka, 2001). Transforming system (15) into the wavenumber domain leads to @U @t=~AU; (19) where U=F[ ] is the Fourier transform of the split led variables ,~A=~A0+B, and ~A0now is ~A0=2 6666664033 033ik1C1ik1C1 033 033ik3C3ik3C3 ik1D1ik1D1022 022 ik3D3ik3D3022 0223 7777775; (20) wherek1andk3are respectively the x1- andx3-components of wavenumber vector k. As demonstrated by the stability theory of autonomous system in Meza-Fajardo and Papageorgiou (2008), for a stable PML in an elastic medium, either isotropic or anisotropic, all the eigenvalues of the system matrix ~Ashould have non-positive imaginary parts. In conventional PML, the outgoing waveeld is damped only along the direction perpendicular to the PML/non-PML interface, and the damping matrix Bfor PML in thex1andx3-directions can be respectively written as B1=2 6666664d1I3033 032 032 033 033 032 032 023 023d1I2022 023 023 022 0223 7777775; (21) B3=2 6666664033 033 032 032 033d3I3032 032 023 023 022 022 023 023 022d3I23 7777775; (22) 5resulting in an unstable PML. Meza-Fajardo and Papageorgiou (2008) analyzed the derivatives of eigenvalues of~Awith respect to the damping parameters d1andd3, and showed that if an appropriate damping ratio 1or3is added along the direction parallel with the PML/non-PML interface, i.e., B1(1) =2 6666664d1I3 033 032 032 0331d1I33 032 032 023 023d1I2 022 023 023 0221d1I223 7777775; (23) B3(3) =2 6666664d33I33 033 032 032 033d3I3 032 032 023 023d33I22 022 023 023 022d3I23 7777775; (24) then the PML becomes stable. The underlying principle of such stability comes from the fact that, after adding nonzero damping ratios 1and3, the eigenvalue derivatives of ~Ahave negative values along all kdirections and consequently, all the relevant eigenvalues of ~A0have negative imaginary parts (other eigenvalues are a pure zero), making autonomous system (15) stable. A key step for determing such damping ratios 1and3is to compute the eigenvalue derivatives of ~A. Meza-Fajardo and Papageorgiou (2008) adopted the following procedure: 1. Calculate eigenvalues eof undamped system coecient matrix ~A0, of which six are a pure zero, and the rest four are pure imaginary numbers as functions of elasticity coecient Cand wavenumber k; 2. Calculate the eigenvalue derivative of ~A=~A0+~B1(1) or ~A=~A0+~B3(3) with respect d1ord3at d1= 0 ord3= 0. These eigenvalue derivatives are functions of elasticity coecient C, wavenumber k and damping ratio 1or3; 3. Choose an appropriate value to g ensure that the values of the eigenvalue derivatives are negative in the range of (0;=2] (the direction of wavenumber k). Both eigenvalues of ~A0and eigenvalue derivatives of ~Aare calculated analytically in the above procedure. Specially, Step 2 involves implicit dierentiation operation and solving roots for high-order polynomials, and could not be accomplished numerically. We adopt the above procedure for obtaining optimal damping ratios for 2D general anisotropic media whereC15orC35may be nonzero. We rst derive relevant expressions for the eigenvalues of ~A0and the eigenvalue derivatives of ~A. Because our procedure is the same as that in Meza-Fajardo and Papageorgiou (2008), we only show the resulting equations. The four nonzero eigenvalues of ~A0(C;k) are e1(C;k) =ip 2q Pp Q; (25) e2(C;k) =ip 2q P+p Q; (26) 6P=[(C11+C55)k2 1+ 2(C15+C35)k1k3+ (C33+C55)k2 3]; (27) Q=2f[(C11+C55)k2 1+ 2(C15+C35)k1k3+ (C33+C55)k2 3]2 + 4[(C2 15C11C55)k4 1+ 2(C13C15C11C35)k3 1k3 + (C2 13C11C332C15C35+ 2C13C55)k2 1k2 3 + 2(C15C33+C13C35)k1k3 3+ (C2 35C33C55)]k4 3g; (28) whereCIJare components of the elasticity matrix and is the mass density. The two eigenvalues in e1or e2have the same length with dierent signs, and we take only the negative ones, i.e., e1(C;k) =ip 2q Pp Q; (29) e2(C;k) =ip 2q P+p Q: (30) The choice of the signs of e1ande2does not aect the following stability analysis and optimal damping ratios. The eigenvalue derivatives of ~Awith respect to the damping coecient d1atd1= 0 can be written as (l) 1(C;k;1;el) = (2C15C35k2 1k2 3+ 3C15C33k1k3 32C2 35k4 3 + 2C33C55k4 32C2 15k4 1+ 2C15C35k2 1k2 31 +C15C33k1k3 31C2 13k2 1k2 3(1 +1) C13k1k3(C15k2 1(1 + 31) +k3(2C55k1(1 +1) +C35k3(3 +1))) +e2 l(k3(3C15k1(1 +1) + 3C35k1(1 +1) +C33k3(2 +1)) +C55(k2 3(2 +1) +k2 1(1 + 21)))+ 2e4 l(1 +1)2 +C11k2 1(2C55k2 11+C33k2 3(1 +1) +C35k1k3(1 + 31) +e2 l(1 + 21))) =(2((C2 15C11C55)k4 1+ 2(C13C15C11C35)k3 1k3+ (C2 13C11C332C15C35 + 2C13C55)k2 1k2 3+ 2(C15C33+C13C35)k1k3 3+ (C2 35C33C55)k4 3) 3e2 l((C11+C55)k2 1+ 2(C15+C35)k1k3+ (C33+C55)k2 3)4e4 l2); (31) where subscript \ l" is for the ltheigvenvalue, and elstands for the ltheigenvalue of ~A0. The eigenvalue derivatives of ~Awith respect to the damping coecient d3atd3= 0 is given by (l) 3(C;k;3;el) = (2C2 15k4 1+k2 3(2(C2 35C33C55)k2 33 C2 13k2 1(1 +3)C13k1(2C55k1(1 +3) +C35k3(1 + 33))) +e2 l(k3(3C35k1(1 +3) +C33k3(1 + 23)) +C55(k2 1(2 +3) +k2 3(1 + 23)))+ 2e4 l(1 +3)2 +C15k1k3(C13k2 1(3 +3) +k3(2C35k1(1 +3) 7+C33k3(1 + 33)) + 3e2 l(1 +3)) +C11k2 1(2C55k2 1 +k3(C33k3(1 +3) +C35k1(3 +3)) +e2 l(2 +3))) =(2((C2 15C11C55)k4 1+ 2(C13C15C11C35)k3 1k3 + (C2 13C11C332C15C35+ 2C13C55)k2 1k2 3+ 2(C15C33+C13C35)k1k3 3 + (C2 35C33C55)k4 3)3e2 l((C11+C55)k2 1 + 2(C15+C35)k1k3+ (C33+C55)k2 3)4e4 l2): (32) There exists a subtle trade-o between the PML stability and articail boundary re ections for anisotropic media. On one hand, it is necessary to introduce nonzero damping ratios 1and3to stabilize PML. On the other hand, adding these nonzero damping ratios to PML makes PML no longer perfectly matched, and the larger are the damping ratios, the stronger the articial boundary re ections become (Dmitriev and Lisitsa, 2011). The original analysis of Meza-Fajardo and Papageorgiou (2008) only showed that a certain value of 1or3can ensure that (l) 1and(l) 3are negative in all kdirections. However, it did not provide a method to determine how large the damping ratios 1and3are adequate for an arbitrary anisotropic medium. We therefore develop a procedure to determine the optimal damping ratios 1and3to not only stabilize PMLs, but to also minimize resulting articial boundary re ections. We employ the following procedure described in Algorithm 1 to obtain the optimal damping ratios 1 and3of MPML for 2D general anisotropic media. Algorithm 1: Determine the optimal damping ratio i(i= 1;3) of MPML for 2D general anisotropic media input :i= 0,=0:005,
= 0:001. for2(0;]do 1) Calculate wavenumber k= (sin;cos) 2) Calculate eigenvalues el(l= 1;2) of ~A0(C;k) using eqs. (25) and (26) 3) Calculate eigenvalue derivatives (l) i(C;k;i;el) 4)i;max= max((1) i;(2) i) using eq. (31) or (32) ifi;max>then i=i+
 go to step 3 end end output:i We apply the above procedure to both the x1- andx3-directions to obtain the optimal values of 1and 3. 8Note that the eigenvalue derivatives along these two directions have dierent expressions, although the expressions for eigenvalues elare the same for both the x1- andx3-directions. Therefore, the optimal damping ratios in the x1- andx3-directions might be dierent from one another. In addition, the searching range for the eigenvalue derivative should be (0 ;] instead of (0 ;=2]. We verify these ndings in numerical examples in the next section. We call the aforementioned procedure based on analytic expressions of eigenvalues and eigenvalue deriva- tives the analytic approach. 2.2. Optimal damping ratios of 3D MPML Elastic-wave equations (1){(2) are also valid for 3D general anisotropic media, but with v= (v1;v2;v3)T; (33) = (11;22;33;23;13;12)T; (34) C=2 6666666666664C11C12C13C14C15C16 C12C22C23C24C25C26 C13C23C33C34C35C36 C14C24C34C44C45C46 C15C25C35C54C55C56 C16C26C36C64C56C663 7777777777775; (35) =2 6664@ @x10 0 0@ @x3@ @x2 0@ @x20@ @x30@ @x1 0 0@ @x3@ @x2@ @x103 7775: (36) Analogous to the 2D case, the 3D elastic-wave equations can be written using decomposed coecient matrices CiandDi(i= 1;2;3) as @ @tv=D1@ @x1+D2@ @x2+D3@ @x3; (37) @ @t=C1@ @x1v+C2@ @x2v+C3@ @x3v; (38) where C1=2 6666666666664C11C16C15 C12C26C25 C13C36C35 C14C46C45 C15C56C55 C16C66C563 7777777777775; (39) 9C2=2 6666666666664C16C12C14 C26C22C24 C36C23C34 C46C24C44 C56C25C54 C66C26C643 7777777777775; (40) C3=2 6666666666664C15C14C13 C25C24C23 C35C34C33 C45C44C34 C55C45C35 C65C46C363 7777777777775; (41) D1=2 66641 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 03 7775; (42) D2=2 66640 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 03 7775; (43) D3=2 66640 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 03 7775: (44) The system of wave equations in PMLs for the 3D case can be expressed in the form of an autonomous system in the wavenumber domain as @U @t=~AU; (45) where U=F[ ]; (46) = ((1) 11;(1) 22;(1) 33;(1) 23;(1) 13;(1) 12;(2) 11;(2) 22;(2) 33;(2) 23;(2) 13;(2) 12; (3) 11;(3) 22;(3) 33;(3) 23;(3) 13;(3) 12;v(1) 1;v(1) 2;v(1) 3;v(2) 1;v(2) 2;v(2) 3;v(3) 1;v(3) 2;v(3) 3)T; (47) ~A=~A0+B; (48) 10~A0=2 6666666666664066 066 066ik1C1ik1C1ik1C1 066 066 066ik2C2ik2C2ik2C2 066 066 066ik3C3ik3C3ik3C3 ik11D1ik11D1ik11D1033 033 033 ik21D2ik21D2ik21D2033 033 033 ik31D3ik31D3ik31D3033 033 0333 7777777777775; (49) B=2 6666666666664d1I6066 066 063 063 063 066d2I6066 063 063 063 066 066d3I6063 063 063 036 036 036d1I3033 033 036 036 036 033d2I3033 036 036 036 033 033d3I33 7777777777775; (50) anddiis the damping coecient along the xi-direction ( i= 1;2;3). To stabilize the PML in the xi-direction, we need to employ nonzero damping ratios along the other two directions perpendicular to xi. Therefore, for the x1-,x2- andx3-directions, we respectively set the damping matrix to be B1(1) =2 6666666666664d1I6 066 066 063 063 063 066d11I6 066 063 063 063 066 066d11I6063 063 063 036 036 036d1I3 033 033 036 036 036 033d11I3 033 036 036 036 033 033d11I33 7777777777775; (51) B2(2) =2 6666666666664d22I6066 066 063 063 063 066d2I6 066 063 063 063 066 066d22I6 063 063 063 036 036 036d22I3033 033 036 036 036 033d2I3 033 036 036 036 033 033d22I33 7777777777775; (52) B3(3) =2 6666666666664d33I6 066 066 063 063 063 066d33I6066 063 063 063 066 066d3I6 063 063 063 036 036 036d33I3 033 033 036 036 036 033d33I3033 036 036 036 033 033d3I33 7777777777775: (53) In the above 3D MPML damped matrices, we employ the same damping ratio along the two directions 11parallel with the PML/non-PML interface. For instance, for damping along the x1-direction, we use the same nonzero damping ratio 1for thex2- andx3-directions, so both the damping coecients along the x2- andx3-directions in 3D MPML are 1d1. Using dierent damping ratios along dierent directions may also stabilize PML, but searching optimal values of damping ratios becomes even more complicated. We develop a new approach to computing the eigenvalue derivatives of damped matrix ~Ausing eqs (51)- (53) for 3D general anisotropic media. Matrix ~A(as well as ~A0) has a dimension of 27 27, resulting in an order of 27 of the characteristic polynomial of ~Aor~A0. Therefore, it is very dicult, if not impossible, to derive analytic expressions for the eigenvalues and eigenvalue derivatives, particularly for media with all Cij6= 0. We therefore adopt a numerical approach to solving for the eigenvalues and eigenvalue derivatives. Using the denitions of eigenvalues and eigenvectors of matrix ~A, we have (~AI27)P=0; (54) whereis the eigenvalue of ~A, and the columns of Pare the eigenvectors of ~A. In addition, we have QT(~AI27) =0; (55) where the columns of QTare the left eigenvectors of ~A. Dierentiating equation (54) with respect to damping parameter digives @~A @di@ @diI27! P+ (~AI27)@P @di= 0: (56) Multiplying both sides of equation (56) with QTleads to QT @~A @di@ @diI27! P+QT(~AI27)@P @di= 0; (57) which implies QT@~A @diP=QT@ @diI27P: (58) Therefore, @ @di=QT@~A @diP QTI27P: (59) Because ~A=~A0+Bi(i) and ~A0is irrelevant to di, the eigenvalue derivative along xi-axis can be written as i(i) =QTRi(i)P QTP; (60) 12where R1(1) =2 6666666666664I6066 066 063 063 063 0661I6066 063 063 063 066 0661I6063 063 063 036 036 036I3033 033 036 036 036 0331I3033 036 036 036 033 0331I33 7777777777775; (61) R2(2) =2 66666666666642I6066 066 063 063 063 066I6066 063 063 063 066 0662I6063 063 063 036 036 0362I3033 033 036 036 036 033I3033 036 036 036 033 0332I33 7777777777775; (62) R3(3) =2 66666666666643I6066 066 063 063 063 0663I6066 063 063 063 066 066I6063 063 063 036 036 0363I3033 033 036 036 036 0333I3033 036 036 036 033 033I33 7777777777775: (63) These equations indicate that we only need to obtain the eigenvalues and the left and right eigenvectors of matrix ~A(C;k;i) for obtaining the optimal damping ratios of MPML in 3D general anisotropic media. This can be achieved using a linear algebra library such as LAPACK, and the procedure is summarized in Algorithm 2. 13Algorithm 2: Determine the optimal damping ratio i(i= 1;2;3) for MPML in 3D general anisotropic media input :i= 0,=0:005,
= 0:001. for2(0;]do for2(0;]do 1) Calculate wavenumber k= (cossin;sinsin;cos) 2) Calculate the left and right eigenvectors of ~A(C;k;i) using a numerical eigensolver 3) Calculate the eigenvalue derivatives (l) i(l= 1;2;3) according to eq. (60) 4)i;max= max((1) i;(2) i;(3) i) ifi;max>then i=i+
 go to step 2 end end end output:i In the above algorithm, it is not necessary to seek analytic forms of the left/right eigenvectors, which is generally impossible for matrix ~A. In our following numerical tests, we calculate the left/right eigenvectors with the Intel Math Kernel Library wrapper for LAPACK. The above numerical approach is obviously applicable to the 2D case with trivial modications. Therefore, for 2D MPML, one can use either the analytic approach or the numerical approach, yet for 3D MPML, one can use only the numerical approach. 3. Results We use three examples of 2D anisotropic media and three examples of 3D anisotropic media to validate the eectiveness of our new algorithms for calculating optimal damping ratios in 2D and 3D MPMLs. In the following, when presenting an elasticity matrix, we write only the upper triangle part of this matrix, but it should be clear that the elasticity matrix is essentially symmetric. We also assume that all the elasticity matrices have units of GPa, and all the media have mass density values of 1000 kg/m3for convenience. 3.1. MPML for 2D anisotropic media To validate our new algorithm for determining the optimal damping ratios in MPML for 2D general anisotropic media, we consider a transversely isotropic medium with a horizontal symmetry axis (HTI medium), a transversely isotropic medium with a tilted symmetry axis (TTI medium), and a transversely isotropic medium with a vertical symmetry axis (VTI medium) with serious qS triplication in both x1- and x3-directions. 14-6000 -4000 -2000 0 2000 4000 6000 x1component (m/s) -6000-4000-20000200040006000x3component (m/s) qP-wave qS-waveFigure 1: Wavefront curves in the 2D HTI medium with elasticity matrix (64). For the HTI medium example, we use a well-known example with elasticity matrix (B ecache et al., 2003; Meza-Fajardo and Papageorgiou, 2008): C=2 66644 7:5 0 20 0 23 7775: (64) Note that this medium is considered as an orthotropic medium in B ecache et al. (2003) and Meza-Fajardo and Papageorgiou (2008). However, it could also be considered as an HTI medium on the x1x3-plane. The only dierences are that C11=C22andC44=C55for a 3D HTI medium, while there exists no such equality restrictions for an orthotropic medium. The wavefront curves of qP- and qS-waves in Fig. 1 show the anisotropy characteristics of this HTI medium. We employ Algorithm 1 to determine the optimal damping ratios in MPML along the x1- and x3-directions, leading to 1= 0:108;  3= 0:259: (65) In Meza-Fajardo and Papageorgiou (2008), the suggested values of damping ratios are 1= 0:30 and 3= 0:25 for this HTI medium. Their suggested value for damping ratio 1is much larger than the optimal damping ratio given in eq. (65), while their suggested value for 3is similar to the optimal damping ratio. Figure 2 plots the values of eigenvalue derivatives of ~Aunder the optimal damping ratios in the x1- andx3-directions. In both panels, the blue curves represent the qP-wave eigenvalue derivatives, and the red curves are for the qS-wave eigenvalue derivatives. Clearly, the qS-wave gives rise to the large damping ratios along both axes. Note that we set the threshold =0:005, therefore, in both panels of Fig. 2, the maximum values of the eigenvalue derivatives are 0:005. We validate the eectiveness of our new MPML in numerical modeling of anisotropic elastic-wave pro- 150 30 60 90 120 150 Wavenumber polar angle (deg.) -1.5-1.2-0.9-0.6-0.30 qPwave 1 with 1=0.108 qSwave 1 with 1=0.108 (a) 0 30 60 90 120 150 Wavenumber polar angle (deg.) -1.5-1.2-0.9-0.6-0.30 qPwave 3 with 3=0.259 qSwave 3 with 3=0.259 (b) Figure 2: Eigenvalue derivatives of ~Aof MPML in the (a) x1- and (b)x3-directions under calculated optimal damping ratios in eq. (65) for the 2D HTI medium with elasticity matrix (64). gation. We use the rotated-staggered grid (RSG) nite-dierence method (Saenger et al., 2000) to solve the stress-velocity form elastic-wave equations (1){(2). The RSG nite-dierence method has 16th-order accuracy in space with optimal nite-dierence coecients (Liu, 2014). We compute the waveeld energy decay curves of our waveeld modelings to validate the eectiveness of MPML. In our numerical modeling, the model is dened in a 400 400 grid, and a PML of 30-node thickness are padded around the model domain. The grid size is 10 m in both the x1- andx3-directions. A vertical force vector source is located at the center of the computational domain, and a Ricker wavelet with a 10 Hz central frequency is used as the source time function. We simulate wave propagation for 20 s with a time interval of 1 ms, which is smaller than what is required to satisfy the stability condition (about 1.54 ms). Figure 3 shows the resulting waveeld energy curve under the optimal damping ratios in eq. (65) , together with three others under dierent eigenvalue derivative threshold values, or equivalently, dierent damping ratios. Figure 3 shows that within the 20 s of wave propagation, the MPML with our calculated damping ratios 1= 0:108 and3= 0:259 is stable. These damping ratios are obtained under threshold value =0:005, meaning that the damping ratios have to ensure the eigenvalues derivatives 1and3are not larger than 0:005 in the entire range of wavenumber direction. We test the behavior of MPML under threshold = 0:01, or equivalently, 1= 0:095 and3= 0:248, and show in Fig. 3 that the numerical modeling is stable. We further increase the threshold to be 0.05, and the MPML becomes unstable quickly after about 2 s. Finally, the conventional PML, which is equivalent to MPML under 1=3= 0, becomes unstable even earlier (before 1 s). These tests indicate that a small positive threshold may still result in a stable MPML. However, there is no simple method to determine how large this positive to ensure the numerical stability. In this HTI medium case, = 0:01 results in stability while= 0:05 results in instability. Because a negative threshold resulting in stable MPML is consistent with the stability theory presented by Meza-Fajardo and Papageorgiou (2008), we therefore should choose a 160 5 10 15 20 Time (s)1011 109 107 105 103 Wavefield energy |v|2=0.005,1=0.108,3=0.259 =0.01,1=0.095,3=0.248 =0.05,1=0.059,3=0.218 Conventional PMLFigure 3: Waveeld energy decay curves under dierent eigenvalue derivative thresholds for the 2D HTI medium with elasticity matrix (64) within 20 s. Conventional PML can be considered as a special case of MPML with 0 or equivalently 1=3= 0. negative threshold for the calculation of the optimal damping ratios to ensure that the resulting MPML is stable. This is also veried in the hereinafter numerical examples. Next, we rotate the aforementioned HTI medium with respect to the x2-axis clockwise by =6 to obtain a TTI medium represented by the following elasticity matrix: C=2 66647:8125 7:6875 3:35585 15:8125 3:57235 2:18753 7775; (66) with unit GPa. The rotation can be accomplished by rotation matrix (e.g., Slawinski, 2010). The wavefront curves in this TTI medium is shown in Fig. 4. Although this TTI medium is the rotation result of the HTI medium in the previous numerical example, it is not obvious how to change the damping ratios accordingly. We obtain the following optimal damping ratios of MPML under =0:005 using Algorithm 1: 1= 0:157;  3= 0:226: (67) The eigenvalue derivatives under this set of damping ratios are shown in Fig. 5. The eigenvalue derivative curves are no longer symmetric with respect to ==2 (or has a period of =2) as those for the 2D HTI medium (Fig. 2). Instead, they are periodic every angle, corresponding to the fact that there is always at least one symmetric axis for whatever kind of 2D anisotropic medium in the axis plane. These curves also indicate that, for 2D general anisotropic medium (TTI medium in this example), it is necessary to determine the values of eigenvalue derivatives within the range of (0 ;] instead of (0 ;=2]. Using only the range (0 ;=2] can lead to a totally incorrect optimal value of 1, since the maximum value of 1in the range (0 ;=2] is smaller than that in the range ( =2;] for this TTI medium. In other words, even though 1in (0;=2] 17-6000 -4000 -2000 0 2000 4000 6000 x1component (m/s) -6000-4000-20000200040006000x3component (m/s) qP-wave qS-waveFigure 4: Wavefront curves in the 2D HTI medium with elasticity matrix (66). 0 30 60 90 120 150 Wavenumber polar angle (deg.) -1.5-1.2-0.9-0.6-0.30 qPwave 1 with 1=0.157 qSwave 1 with 1=0.157 0 30 60 90 120 150 Wavenumber polar angle (deg.) -1.5-1.2-0.9-0.6-0.30 qPwave 3 with 3=0.226 qSwave 3 with 3=0.226 Figure 5: Eigenvalue derivatives of ~Aof MPML in the (a) x1- and (b)x3-directions under calculated optimal damping ratios in eq. (67) for the 2D TTI medium with elasticity matrix (66). indicates a stable MPML, the MPML may still be unstable since 1may be larger than zero in ( =2;]. Therefore, for anisotropic media with symmetric axis not aligned with a coordinate axis, it is necessary to consider the values of iin wavenumber direction 2(0;]. This statement is also true for 3D anisotropic media as shown in the hereinafter 3D numerical examples. Figure 6 displays the waveeld energy decay curves for this TTI medium under the optimal damping ratios, as well as under damping ratios calculated with positive eigenvalue derivative thresholds. In this example, the waveeld energy decays gradually within 20 s for both cases with =0:005 and= 0:05. The numerical modeling with = 0:15 becomes unstable. For comparison, in the previous HTI case, = 0:05 results in an unstable MPML. These results further demonstrate that a positive eigenvalue derivative threshold should not be chosen to calculate the damping ratios, although a small positive might result in stable MPML. In contrast, a negative can always ensure the stability of MPML. 180 5 10 15 20 Time (s)1013 1011 109 107 105 103 Wavefield energy |v|2=0.005,1=0.157,3=0.226 =0.05,1=0.110,3=0.183 =0.15,1=0.025,3=0.105 Conventional PMLFigure 6: Waveeld energy decay curves under dierent eigenvalue derivative thresholds in the 2D TTI medium with elasticity matrix (66) within 20 s. Our next numerical example uses a VTI medium dened by C=2 666410:4508 4:2623 0 7:5410 0 11:39343 7775: (68) The wavefront curves for this VTI medium are depicted in Fig. 7. The special feature of this VTI medium is that, in both the x1- andx3-directions, there exists serious qS-wave triplication phenomena. We obtain the following optimal damping ratios using Algorithm 1: 1= 0:215;  3= 0:225: (69) The corresponding eigenvalue derivatives in the x1- andx3-directions are displayed in Fig. 8. Again, it is the qS-wave that causes the damping ratios to be large to stabilize PML. Figure 9 depicts the waveeld energy decay curves under dierent eigenvalue derivative thresholds. Similar with that of 2D TTI medium example, a positive threshold 0.05 can still stabilize PML, yet a value of 0.15 makes the MPML unstable. We choose a negative value of to ensure a stable MPML. 3.2. MPML for 3D anisotropic media For 3D anisotropic media, we need to determine the optimal MPML damping ratios along all three coordinate directions. We use three dierent anisotropic media (a quasi-VTI medium, a quasi-TTI medium and a triclinic medium) to demonstrate the determination of optimal MPML damping ratios. 19-5000 -2500 0 2500 5000 x1component (m/s) -5000-2500025005000x3component (m/s) qP-wave qS-waveFigure 7: Wavefront curves in the 2D VTI medium with elasticity matrix (68). 0 30 60 90 120 150 Wavenumber polar angle (deg.) -1.5-1.2-0.9-0.6-0.30 qPwave 1 with 1=0.215 qSwave 1 with 1=0.215 0 30 60 90 120 150 Wavenumber polar angle (deg.) -1.5-1.2-0.9-0.6-0.30 qPwave 3 with 3=0.225 qSwave 3 with 3=0.225 Figure 8: Eigenvalue derivatives of ~Aof MPML in the (a) x1- and (b)x3-directions under calculated optimal damping ratios in eq. (69) for the 2D VTI medium with elasticity matrix (68). 200 5 10 15 20 Time (s)1020 1015 1010 105 100Wavefield energy |v|2=0.005,1=0.215,3=0.225 =0.05,1=0.172,3=0.182 =0.15,1=0.093,3=0.104 Conventional PMLFigure 9: Waveeld energy decay curves under dierent eigenvalue derivative thresholds in the 2D VTI medium with elasticity matrix (68) within 20 s. We rst use a 3D anisotropic medium represented by the elasticity matrix C=2 666666666666416:5 5 5 0 0 0 16:5 5 0 0 0 6:2 0 0 0 4:96 0 0 3:96 0 5:963 7777777777775: (70) This elasticity matrix is modied from the elasticity matrix of zinc (a VTI medium, or hexagonal anisotropic medium) to increase the complexity of the resulting wavefronts and the characteristics of the eigenvalue derivatives along all three directions. This modied elastic matrix still represents a physically feasible medium since it is easy to verify that it satises the following stability condition for anisotropic media (Slawinski, 2010): det2 6664C11


C1n ......... C1n


Cnn3 7775>0; (71) wheren= 1;2;


;6. We call this anisotropic medium the quasi-VTI medium. Figure 10 shows the wavefront curves of this quasi-VTI medium on three axis planes. For comparison, a standard 3D VTI medium can be expressed by its ve independent elasticity constants 21as (e.g., Slawinski, 2010) C=2 6666666666664C11C12C13 0 0 0 C11C13 0 0 0 C33 0 0 0 C44 0 0 C44 0 C11C12 23 7777777777775: (72) We calculate the optimal damping ratios for MPML using Algorithm 2, and obtain 1= 0:088;  2= 0:131;  3= 0:041: (73) We plot the eigenvalue derivatives in the polar angle range (0 ;] and azimuth angle range (0 ;] for three axis directions in Fig. 11. Since the three symmetric axes of this VTI medium are aligned with three coordinate axes, the three eigenvalue derivatives are symmetric with respect to both ==2 and==2 lines. We conduct numerical waveeld modeling to verify the stability of MPML under these optimal damping ratios. The model is dened on a 400 400400 grid with a grid size of 10 m in all three directions. The thickness of PML layer is 25 grids. A vertical force vector is located at the center of the computational domain, and the source time function is a Ricker wavelet with a central frequency of 10 Hz. The time step size is 1 ms, which is smaller than the stability-required time step of 1.69 ms. A total of 15,000 time steps, i.e., 15 s, are simulated, and the waveeld energy curve is shown in Fig. 12. The blue curve in Fig. 12 is for the case with the optimal damping ratios in eq. (73). We also carry out a waveeld modeling with the eigenvalue derivative threshold = 0:015 and= 0:025. The MPMLs under these thresholds are unstable according to the corresponding waveeld energy variation curves in Fig. 12. As in the 2D MPML case, we should always choose a negative to stabilize MPML for 3D anisotropic media. Our next numerical example uses a rotation version of the aforementioned quasi-VTI medium. We rotate the quasi-VTI medium (70) with respect to the x1-axis by 30 degrees, the x2-axis by 50 degrees, and the x3-axis by 25 degrees, and the resulting elasticity matrix for this quasi-TTI medium is given by C=2 666666666666415:7930 4:1757 4:9651 0:1582 0:65291:0343 12:5979 4:1844 2:0903 0:81861:7513 14:1587 1:9573 0:76430:1606 3:78790:8909 0:8065 5:0750 0:7668 4:34233 7777777777775: (74) The corresponding wavefront curves are shown in Fig. 13. Similar to the 3D quasi-VTI case, we obtain the following optimal damping ratios using Algorithm 2: 1= 0:089;  2= 0:051;  3= 0:080: (75) 22-5000 -2500 0 2500 5000 x1component (m/s) -5000-2500025005000x2component (m/s) qP-wave qS1-wave qS2-wave(a) -5000 -2500 0 2500 5000 x1component (m/s) -5000-2500025005000x3component (m/s) qP-wave qS1-wave qS2-wave (b) -5000 -2500 0 2500 5000 x2component (m/s) -5000-2500025005000x3component (m/s) qP-wave qS1-wave qS2-wave (c) Figure 10: Wavefront curves in the 3D quasi-VTI medium with elasticity matrix (70) on the (a) x1x2(b)x1x3and (c)x2x3 axis plane. qS1 and qS2 represents the two qS-waves. 2330 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1(a) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (b) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (c) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2 (d) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (e) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (f) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (g) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (h) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (i) Figure 11: Eigenvalue derivatives of ~Aof MPML in the (a)-(c) x1-, (d)-(f)x2-, and (g)-(i) x3-directions under calculated optimal damping ratios in eq. (73) for the 3D quasi-VTI medium with elasticity matrix (70). (a), (d) and (g) represent qP-wave, (b), (e) and (h) represent qS1-wave, (c), (f) and (i) represent qS2-wave. 240 5 10 15 Time (s)1013 1011 109 107 105 Wavefield energy |v|2 =0.005,1=0.089,2=0.131,3=0.041 =0.015,1=0.068,2=0.112,3=0.020 =0.025,1=0.058,2=0.102,3=0.009 Conventional PMLFigure 12: Waveeld energy decay curve in the 3D VTI medium with elasticity matrix (70). The eigenvalue derivatives under this set of damping ratios for three axis directions are shown in Fig. 14. Obviously, for the quasi-TTI medium where the symmetric axes are not aligned with coordinate axes, the eigenvalue derivatives of all three wave modes along any coordinate axis is no longer symmetric about any  orlines. Therefore, it is necessary to use the entire range of wavenumber polar angle and azimuth angle , i.e., (0;](0;], to determine the optimal damping ratios. Figure 15 depicts the waveeld energy decays under the optimal damping ratios in eq. (75) and damping ratios with thresholds = 0:05 and= 0:075. For the case where = 0:05, the waveeld energy does not diverge immediately after maximum energy value occurred. Instead, the curve indicates a very slow energy decay after about 1 s. In contrast, the optimal MPML with threshold =0:005 shows a \normal" energy decay. Therefore, although the MPML with = 0:05 does not show energy divergence within 15 s, it fails to eectively absorb the outgoing waveeld, and we consider this as a \quasi-divergence." Meanwhile, the MPML with = 0:075 shows an energy divergence after about 4 s. These results further demonstrate that the behavior of MPML with a positive eigenvalue derivative threshold is dierent and unpredictable for dierent kinds of anisotropic media. Figure 15 also shows that the conventional PML gives an unstable result. Our last 3D numerical example is based on a triclinic anisotropic medium represented by C=2 666666666666410 3:5 2:55 0:1 0:3 8 1:5 0:20:10:15 6 1 0 :4 0:24 5 0:35 0:525 41 33 7777777777775; (76) 25-5000 -2500 0 2500 5000 x1component (m/s) -5000-2500025005000x2component (m/s) qP-wave qS1-wave qS2-wave(a) -5000 -2500 0 2500 5000 x1component (m/s) -5000-2500025005000x3component (m/s) qP-wave qS1-wave qS2-wave (b) -5000 -2500 0 2500 5000 x2component (m/s) -5000-2500025005000x3component (m/s) qP-wave qS1-wave qS2-wave (c) Figure 13: Wavefront curves in the 3D quasi-TTI medium with elasticity matrix (74) on the (a) x1x2(b)x1x3and (c)x2x3 axis plane. qS1 and qS2 represents the two qS-waves. qS-wave wavefronts seem to be less complicated compared with those of the 3D quasi-VTI medium (70) only because the qS-wave triplications are now out of axis planes after rotation. 2630 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1(a) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (b) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (c) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (d) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (e) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (f) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (g) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (h) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1 (i) Figure 14: Eigenvalue derivatives of ~Aof MPML in the (a)-(c) x1-, (d)-(f)x2-, and (g)-(i) x3-directions under calculated optimal damping ratios in eq. (75) for the 3D quasi-TTI medium with elasticity matrix (74). (a), (d) and (g) represent qP-wave, (b), (e) and (h) represent qS1-wave, (c), (f) and (i) represent qS2-wave. 270 5 10 15 Time (s)1024 1019 1014 109 104 Wavefield energy |v|2 =0.005,1=0.089,2=0.051,3=0.080 =0.05,1=0.034,2=0,3=0.024 =0.075,1=0.009,2=0,3=0 Conventional PMLFigure 15: Waveeld energy decay curve in the 3D quasi-TTI medium with elasticity matrix (74). with unit GPa. The wavefront curves on three axis planes are shown in Fig. 16. We solve for the optimal damping ratios for this triclinic anisotropic medium using Algorithm 2, and obtain the following optimal damping ratios with =0:005: 1= 0:487;  2= 0:345;  3= 0:374: (77) The damping ratios for this anisotropic medium are unexpectedly very large compared with those for the heretofore 2D and 3D examples. We seek the reasons of these large damping ratios from the eigenvalue derivatives shown in Fig. 17, and nd that it is the qS2-wave that leads to such large damping ratios to achieve a stable MPML. In fact, for the damping ratios in eq. (77), the corresponding eigenvalue derivatives of qP- and qS1-waves are far smaller than zero, yet the eigenvalue derivative of qS2-wave merely smaller than zero (0:005 under our threshold setting), leading to a set of relatively large damping ratios for this 3D anisotropic medium. Our calculated optimal damping ratios result in a stable MPML, as indicated by the corresponding energy decay curve shown in Fig. 18. The waveeld energy decay curve with a threshold of = 0:1 displayed in Fig. 18 is surprisingly almost identical with that of =0:005. When using a threshold = 0:4, MPML become unstable, indicating that the positive threshold 0.4 is too large to make MPML stable. This veries again that, although a positive threshold might result in stable MPML, we should use a negative threshold to ensure a stable MPML for general anisotropic media. This is consistent with the stability condition described in Meza-Fajardo and Papageorgiou (2008), and is perhaps the only practical method to stabilize PML using nonzero damping ratios. 28-4000 -2000 0 2000 4000 x1component (m/s) -3000-1500015003000x2component (m/s) qP-wave qS1-wave qS2-wave(a) -4000 -2000 0 2000 4000 x1component (m/s) -3000-1500015003000x3component (m/s) qP-wave qS1-wave qS2-wave (b) -4000 -2000 0 2000 4000 x2component (m/s) -3000-1500015003000x3component (m/s) qP-wave qS1-wave qS2-wave (c) Figure 16: Wavefront curves in the 3D triclinic anisotropic medium with elasticity matrix (74) on the (a) x1x2(b)x1x3and (c)x2x3axis plane. qS1 and qS2 represents the two qS-waves. 2930 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5(a) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1.1-1-0.9-0.8-0.7-0.6-0.5-0.4 (b) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1.3-1.1-0.9-0.7-0.5-0.3-0.1 (c) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3 (d) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3 (e) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1.3-1.1-0.9-0.7-0.5-0.3-0.1 (f) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3 (g) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1-0.9-0.8-0.7-0.6-0.5-0.4-0.3 (h) 30 60 90 120 150 Wavenumber polar angle (deg.) 30 60 90 120 150Wavenumber azimuth angle (deg.) -1.3-1.1-0.9-0.7-0.5-0.3-0.1 (i) Figure 17: Eigenvalue derivatives of ~Aof MPML in the (a)-(c) x1-, (d)-(f)x2-, and (g)-(i) x3-directions under calculated optimal damping ratios in eq. (77) for the 3D triclinic anisotropic medium with elasticity matrix (76). (a), (d) and (g) represent qP-wave, (b), (e) and (h) represent qS1-wave, (c), (f) and (i) represent qS2-wave. 300 5 10 15 Time (s)1015 1012 109 106 103 Wavefield energy |v|2 =0.005,1=0.487,2=0.345,3=0.374 =0.1,1=0.420,2=0.258,3=0.291 =0.4,1=0.230,2=0.010,3=0.054 Conventional PMLFigure 18: Waveeld energy decay curve in the 3D triclinic anisotropic medium with elasticity matrix (76). The blue curve (=0:005) and red curve ( = 0:1) are almost identical. 4. Conclusions A denite analytic method for determining the optimal damping ratios of multi-axis perfectly matched layers (MPML) is generally impossible for 3D general anisotropic media with possible all nonzero elasticity parameters. We have developed a new method to eciently determine the optimal damping ratios of MPML for absorbing unwanted, outgoing propagating waves in 2D and 3D general anisotropic media. This numerical approach is very straightforward using the left and right eigenvectors of the damped system coecient matrix. We have used six numerical modeling examples of elastic-wave propagation in 2D and 3D anisotropic media to demonstrate that our new algorithm can eectively and correctly provide the optimal MPML damping ratios for even very complex, general anisotropic media. 5. Acknowledgments This work was supported by U.S. Department of Energy through contract DE-AC52-06NA25396 to Los Alamos National Laboratory (LANL). The computation was performed using super-computers of LANL's Institutional Computing Program. References B ecache, E., Fauqueux, S., Joly, P., 2003. Stability of perfectly matched layers, group velocities and anisotropic waves. Journal of Computational Physics 188 (2), 399 { 433. URL http://www.sciencedirect.com/science/article/pii/S0021999103001840 31Berenger, J.-P., 1994. A perfectly matched layer for the absorption of electromagnetic waves. Journal of Computational Physics 114 (2), 185 { 200. URL http://www.sciencedirect.com/science/article/pii/S0021999184711594 Carcione, J. M., 2007. Wave elds in real media: Wave propagation in anisotropic, anelastic, porous and electromagnetic media (Second Edition). Elsevier, Amsterdam, Netherlands. Cerjan, C., Koslo, D., Koslo, R., Reshef, M., 1985. A nonre ecting boundary condition for discrete acoustic and elastic wave equations. Geophysics 50 (4), 705{708. URL http://geophysics.geoscienceworld.org/content/50/4/705 Clayton, R., Engquist, B., 1977. Absorbing boundary conditions for acoustic and elastic wave equations. Bulletin of the Seismological Society of America 67 (6), 1529{1540. URL http://bssa.geoscienceworld.org/content/67/6/1529 Collino, F., Tsogka, C., 2001. Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66 (1), 294{307. URL http://dx.doi.org/10.1190/1.1444908 de la Puente, J., K aser, M., Dumbser, M., Igel, H., 2007. An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes IV. Anisotropy. Geophysical Journal International 169 (3), 1210{1228. URL http://dx.doi.org/10.1111/j.1365-246X.2007.03381.x Dmitriev, M. N., Lisitsa, V. V., 2011. Application of M-PML re ectionless boundary conditions to the numerical simulation of wave propagation in anisotropic media. Part I: Re ectivity. Numerical Analysis and Applications 4 (4), 271{280. URL http://dx.doi.org/10.1134/S199542391104001X Drossaert, F. H., Giannopoulos, A., 2007. A nonsplit complex frequency-shifted pml based on recursive integration for fdtd modeling of elastic waves. Geophysics 72 (2), T9{T17. URL http://dx.doi.org/10.1190/1.2424888 Hastings, F. D., Schneider, J. B., Broschat, S. L., 1996. Application of the perfectly matched layer (PML) ab- sorbing boundary condition to elastic wave propagation. The Journal of the Acoustical Society of America 100 (5). Higdon, R. L., Oct. 1986. Absorbing boundary conditions for dierence approximations to the multi- dimensional wave equation. Math. Comput. 47 (176), 437{459. URL http://dx.doi.org/10.2307/2008166 32Higdon, R. L., 1987. Numerical absorbing boundary conditions for the wave equation. Mathematics of Computation 49 (179), 65{90. URL http://www.jstor.org/stable/2008250 Komatitsch, D., Barnes, C., Tromp, J., 2000. Simulation of anisotropic wave propagation based upon a spectral element method. Geophysics 65 (4), 1251{1260. URL http://dx.doi.org/10.1190/1.1444816 Komatitsch, D., Martin, R., 2007. An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation. Geophysics 72 (5), SM155{SM167. URL http://dx.doi.org/10.1190/1.2757586 Komatitsch, D., Tromp, J., 2003. A perfectly matched layer absorbing boundary condition for the second- order seismic wave equation. Geophysical Journal International 154 (1), 146{153. URL http://dx.doi.org/10.1046/j.1365-246X.2003.01950.x Liao, Z.-F., Huang, K.-L., Yang, B.-P., Yuan, Y.-F., 1984. A transmitting boundary for transient wave analyses. Science China: Mathematics 27 (10), 1063. URL http://math.scichina.com:8081/sciAe/EN/abstract/article_379434.shtml Liu, Y., 2014. Optimal staggered-grid nite-dierence schemes based on least-squares for wave equation modelling. Geophysical Journal International 197, 1033{1047. URL http://gji.oxfordjournals.org/content/early/2014/02/20/gji.ggu032.abstract Long, L. T., Liow, J. S., 1990. A transparent boundary for nite-dierence wave simulation. Geophysics 55 (2), 201{208. URL http://geophysics.geoscienceworld.org/content/55/2/201 Martin, R., Komatitsch, D., 2009. An unsplit convolutional perfectly matched layer technique improved at grazing incidence for the viscoelastic wave equation. Geophysical Journal International 179 (1), 333{344. URL http://dx.doi.org/10.1111/j.1365-246X.2009.04278.x Martin, R., Komatitsch, D., Gedney, S. D., Bruthiaux, E., 2010. A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using auxiliary dierential equations (ADE-PML). Computer Modeling in Engineering & Sciences 56 (1), 17{40. Meza-Fajardo, K. C., Papageorgiou, A. S., 2008. A nonconvolutional, split-eld, perfectly matched layer for wave propagation in isotropic and anisotropic elastic media: Stability analysis. Bulletin of the Seismological Society of America 98 (4), 1811{1836. URL http://www.bssaonline.org/content/98/4/1811.abstract 33Peng, C., Toks oz, M. N., 1994. An optimal absorbing boundary condition for nite dierence modeling of acoustic and elastic wave propagation. The Journal of the Acoustical Society of America 95 (2), 733{745. URL http://scitation.aip.org/content/asa/journal/jasa/95/2/10.1121/1.408384 Reynolds, A. C., 1978. Boundary conditions for the numerical solution of wave propagation problems. Geo- physics 43 (6), 1099{1110. URL http://geophysics.geoscienceworld.org/content/43/6/1099 Saenger, E. H., Gold, N., Shapiro, S. A., 2000. Modeling the propagation of elastic waves using a modied nite-dierence grid. Wave Motion 31 (1), 77 { 92. URL http://www.sciencedirect.com/science/article/pii/S0165212599000232 Slawinski, M. A., 2010. Waves and Rays in Elastic Continua. World Scientic. URL http://www.worldscientific.com/worldscibooks/10.1142/7486#t=aboutBook Zhang, W., Shen, Y., 2010. Unsplit complex frequency-shifted PML implementation using auxiliary dier- ential equations for seismic wave modeling. Geophysics 75 (4), T141{T154. URL http://dx.doi.org/10.1190/1.3463431 34

2016-08-30

The conventional Perfectly Matched Layer (PML) is unstable for certain kinds of anisotropic media. This instability is intrinsic and independent of PML formulation or implementation. The Multi-axial PML (MPML) removes such instability using a nonzero damping coefficient in the direction parallel with the interface between a PML and the investigated domain. The damping ratio of MPML is the ratio between the damping coefficients along the directions parallel with and perpendicular to the interface between a PML and the investigated domain. No quantitative approach is available for obtaining these damping ratios for general anisotropic media. We develop a quantitative approach to determining optimal damping ratios to not only stabilize PMLs, but also minimize the artificial reflections from MPMLs. Numerical tests based on finite-difference method show that our new method can effectively provide a set of optimal MPML damping ratios for elastic-wave propagation in 2D and 3D general anisotropic media.

Optimal damping ratios of multi-axial perfectly matched layers for elastic-wave modeling in general anisotropic media

1608.08326v3

arXiv:1507.01124v1 [physics.flu-dyn] 4 Jul 2015Comments on turbulence theory by Qian and by Edwards and McComb R. V. R. Pandya∗ Department of Mechanical Engineering, University of Puerto Rico at Mayaguez, PR 00682, USA Abstract We reexamine Liouville equation based turbulence theories proposed by Qian [Phys. Fluids 26, 2098 (1983)] and Edwards and McComb [J. Phys. A: Math. Gen. 2, 157 (1969)], which are compatible with Kolmogorov spectrum. These theories obtai ned identical equation for spectral densityq(k)and different results for damping coefficient. Qian proposed v ariational approach and Edwards and McComb proposed maximal entropy principle to ob tain equation for the damping co- efficient. We show that assumptions used in these theories to o btain damping coefficient correspond to unphysical conditions. ∗rvrptur(AT)yahoo.com 1I. INTRODUCTION Edwards [2] proposed turbulence theory based on Liouville e quation for joint probability distribution function of Fourier modes uα(k,t)of velocity field governed by forced Navier- Stokes equation. Following Edwards, a few more turbulence t heories [3, 4, 10] were proposed to solve Liouville equation and were reviewed by Leslie [7] a nd McComb [8]. Edwards theory seeks Fokker-Planck model representation for Liouville eq uation and obtains closed set of equations for spectral density q(k)and damping coefficient (total viscosity) ω(k)for sta- tionary, isotropic turbulence. The theory failed to be cons istent with Kolmogorov spectrum [1, 5] and the failure was attributed to equation for ω(k)[8]. As a modification to Edwards’ theory, Edwards and McComb [3] proposed principle of maxima l entropy to derive an equa- tion forω(k)compatible with Kolmogorov spectrum. Within the Fokker-Pl anck framework of Edwards, Qian [10] proposed variational approach and obt ained different equation for damping coefficient consistent with Kolmogorov spectrum. In stead of using uα(k,t), Qian used real dynamical modal variables Xiand their governing equations which were suggested by Kraichnan [6] and later utilized by Herring [4] for his sel f-consistent turbulence theory. Herring’s theory also uses Liouville equation and obtains e quation for q(k)identical to equa- tion obtained by Edwards. In this paper, we reexamine Kolmog orov spectrum compatible theories proposed by Qian [10] and Edwards and McComb [3] to o btain damping coefficient. We show in the next two sections that assumptions made in thes e theories correspond to unphysical conditions. II. VARIATIONAL APPROACH BY QIAN For discussion purpose, hereafter we refer to Qian’s variat ional approach as Q83. We use (Q83; #) to represent equation number (#) in Q83 paper [10 ]. Qian based his theory on governing equation for real dynamical modal variables Xifor stationary, homogeneous, isotropic turbulence, written as dXi dt=−(νi−ν′ i)Xi+/summationdisplay j,mAijmXjXm,(Q83;8) (1) whereν′ iXirepresents external driving force. Einstein summation con vention of repeated in- dices is not utilized in Eq. (1) and in this section. It should be noted that the real dynamical 2modal variables and their equations were first suggested by K raichnan within the context of hydromagnetic turbulence [6]. Qian’s theory [10] seeks Lan gevin model representation for isotropic turbulence in the form d dtXi≃ −ηiXi+fi, ηi=ζi+(ν−ν′ i) (Q83;12) (2) by using /summationdisplay j,mAijmXjXm∼=−ζiXi+fi(Q83;11) (3) where−ζiXiis dynamical damping term and fiis white noise type forcing term. Qian pro- posed variational approach to obtain damping coefficient ηi. The approach yields equation forηiby minimizing a function I(ηi), written as I=/summationdisplay i/angbracketleftBigg /summationdisplay j,mAijmXjXm−(−ζiXi) 2/angbracketrightBigg (Q83;29) (4) and using ∂I ∂ηi= 0.(Q83;28) (5) Here/angbracketleft /angbracketrightrepresents ensemble average. Qian considered variation in Iunder constraint φi= constant , whereφiis related to /angbracketleftX2 i/angbracketrightby /angbracketleftBig X2 i/angbracketrightBig =φi/parenleftBigg 1−νi−ν′ i ηi/parenrightBigg .(Q83;25) (6) Also,φiis proportional to spectral density q(k)[10]. We now show that for I=I(ηi)and under the constraint of φi=constant , ∂I ∂ηi/negationslash= 0 (7) within the framework of Langevin model considered by Qian. C onsequently, the use of Eq. (5) to obtain ηiis in error. For stationary turbulence, solution of Eq. (2) s uggests 2ηi/angbracketleftBig X2 i/angbracketrightBig =Fi (8) in which correlation of white noise forcing term /angbracketleftfi(t)fi(t′)/angbracketright=Fiδ(t−t′) (9) 3is utilized. Here δ(t−t′)is Dirac delta function. Using Eqs. (3) , (4) and (9), functio nI(ηi) can be written as I=/summationdisplay i/angbracketleftBigg /summationdisplay j,mAijmXjXm−(−ζiXi) 2/angbracketrightBigg =/summationdisplay i/angbracketleftfifi/angbracketright=δ(0)/summationdisplay iFi. (10) Further, using Eqs (6), (8), (10) and for φi=constant , we can write ∂I ∂ηi=δ(0)/summationdisplay j∂Fj ∂ηi=δ(0)/summationdisplay j∂2ηj/angbracketleftBig X2 j/angbracketrightBig ∂ηi= 2δ(0)φi. (11) This Eq. (11) suggests that for all i ∂I ∂ηi/negationslash= 0 (12) asφi/negationslash= 0. In view of this, use of Eq. (5), i.e.∂I ∂ηi= 0, in Q83 to obtain ηiis in error and corresponds to unphysical condition φi= 0,∀ifor stationary turbulence. Now we suggest possible modification within the framework of Q83. Consider a function V, written as V=/summationdisplay j∂I ∂ηj=/summationdisplay j2δ(0)φj, (13) which satisfies an exact condition ∂V ∂ηi= 0. (14) This condition along with Eqs. (4) can be used, instead of Eq. (5), to obtain equation for ηi. III. MAXIMAL ENTROPY PRINCIPLE BY EDWARDS AND MCCOMB For discussion purpose, hereafter we refer to Edwards and Mc Comb’s theory as EM69. We use (M90; #) to represent equation number (#) in McComb’s b ook [8]. Edwards and McComb [3] considered stationary, homogeneous, isotropic , turbulence inside a cubic box of sideL. Their Liouville equation based theory uses equation for Fo urier modes uα(k,t)of the velocity field uα(x,t)governed by forced Navier-Stokes equation, written as /parenleftBigg∂ ∂t+νk2/parenrightBigg uα(k,t) =Mαβγ(k)/summationdisplay juβ(j,t)uγ(k−j,t)+fα(k,t),(M90;4.81) (15) 4wherefαrepresents external driving force, kis wavevector and k2=|k|2. The Einstein summation convention for repeated Greek indices is utilize d while writing Eq. (15) and in this section. Edwards and McComb [3] theory seeks Fokker- Planck model equation for Liouville equation. The model equation contains two model p arameters, namely r(k)and s(k), which account for contribution of nonlinear term in Eq. (15 ). The dynamical damping coefficient r(k)is related to damping coefficient ω(k)by ω(k) =νk2+r(k).(M90;6.80) (16) The coefficient s(k)accounts for correlation of white noise forcing in the Lange vin equa- tion for Fokker-Planck equation. Within the framework of Ed wards and McComb and for stationary turbulence, ω(k)ands(k)are related by 2ω(k)q(k) =d(k) (M90;6.84) (17) where d(k) =W(k)+s(k) (M90;6.79) (18) andW(k)accounts for correlation of forcing term fα(k,t). The spectral density q(k)is defined by /parenleftbigg2π L/parenrightbigg3 /angbracketleftuα(k)uβ(−k)/angbracketright=Dαβ(k)q(k),(M90;6.85) (19) whereDαβ(k) =δαβ−kαkβ |k|2. EM69 considered entropy function S=S[q(k),ω(k)]to obtain equation for ω(k)by maximizing S, corresponding to the condition δS δω(k)+/summationdisplay j/bracketleftBiggδS δq(j)/bracketrightBiggδq(j) δω(k)= 0.(M90;7.88) (20) Here δq(j) δω(k)=−d(k)δ(k−j) 2ω2(k)+1 2ω(j)δd(j) δω(k).(M90;7.89) (21) andδ(k−j) = 1 whenk=jotherwise δ(k−j) = 0 . Edwards and McComb realized the difficulty in obtaining the second term on the right-hand s ide (rhs) of Eq. (21). After neglecting the second term, an approximate equation δq(j) δω(k)=−d(k)δ(k−j) 2ω2(k)(22) 5was used for further calculation in EM69 [8]. This Eq. (22) su ggests thatδq(j) δω(k)= 0,∀j/negationslash=k. As a consequence, EM69 used following approximate equation δS δω(k)−/bracketleftBiggd(k) 2ω2(k)/bracketrightBiggδS δq(k)= 0 ( M90;7.90) (23) to obtain equation for ω(k). We now show that the neglect of the second term on the rhs of Eq. (21) corresponds to unphysical condition. Consequently, the use of Eq. (23) to o btainω(k)is in error. Since EM69 is proposed for stationary turbulence, /parenleftbigg2π L/parenrightbigg3/summationdisplay j1 2/angbracketleftuα(j)uα(−j)/angbracketright=/summationdisplay jq(j) =Constant. (24) and from which we can write exact equation /summationdisplay jδq(j) δω(k)=δq(k) δω(k)+/summationdisplay j,j/negationslash=kδq(j) δω(k)= 0. (25) Substituting approximate Eq. (22) of EM89 into Eq. (25) and u sing Eq. (17), we obtain d(k) 2ω2(k)=q(k) ω(k)= 0 (26) and which is not correct for all k. In view of this, approximation used in EM89 to obtain ω(k)corresponds to unphysical condition q(k) = 0 and does not comply with conservation of energy Eq. (24) for stationary turbulence where q(k)/negationslash= 0,∀k. It should be noted that, within the framework of EM69, the unp hysical behavior can be avoided if δS δω(k)= 0 (27) along with q(k) =constant, ∀kis used instead of Eq. (20). This means that S=S(ω(k)) and second term on the rhs of Eq. (20) is equal to zero and is neg lected. This kind of neglect by Qian in Q83 for function I(ηi)was considered mathematically incorrect by McComb [8]. In our view, Eq. (27) can be considered as valid equation whic h seeks to optimize Entropy when energy of turbulence remains constant as q(k) =constant, ∀k. IV. CONCLUDING REMARKS Within the Eulerian framework, a very few renormalized pert urbation theories of turbu- lence are consistent with Kolmogorov spectrum [7, 8]. In thi s paper, we have reexamined 6two such theories proposed by Qian [10] and Edwards and McCom b [3] and have revealed hidden unphysical conditions in these theories. We have sug gested possible modifications, Eqs. (13), (14) and (27), to these theories but have not explo red their usefulness in obtain- ing damping coefficient consistent with Kolmogorov spectrum . This will be explored in a broader context of our future work on turbulence theory deve lopment within the framework of Kraichnan’s direct interaction approximation [9]. [1] G. K. Batchelor. The Theory of Homogeneous Turbulence . Cambridge University Press, Cam- bridge, UK, 1959. [2] S. F. Edwards. The statistical dynamics of homogeneous t urbulence. J. Fluid Mech. , 18:239– 273, 1964. [3] S. F. Edwards and W. D. McComb. Statistical mechanics far from equilibrium. J. Phys. A , 2:157–171, 1969. [4] J. R. Herring. Self-Consistent-Field approach to turbu lence theory. Phys. Fluids , 8:2219–2225, 1965. [5] A. N. Kolmogorov. The local structure of turbulence in in compressible viscous fluid for large Reynolds numbers. Dokl. Akad. Nauk SSSR , 30(11):301–305, 1941. [6] R. H. Kraichnan. Irreversible statistical mechanics of incompressible hydromagnetic turbu- lence. Phys. Rev. , 109:1407–1422, 1958. [7] D. C. Leslie. Developments in the Theory of Turbulence . Clarendon Press, Oxford, 1973. [8] W. D. McComb. The Physics of Fluid Turbulence . Oxford University Press, New York, NY, 1990. [9] R. V. R. Pandya. Development of Eulerian theory of turbul ence within Kraichnan’s direct interaction approximation framework. arXiv:1407.1828 , pages 1–18, 2014. [10] J. Qian. Variational approach to the closure problem of turbulence theory. Phys. Fluids , 26:2098–2104, 1983. 7

2015-07-04

We reexamine Liouville equation based turbulence theories proposed by Qian {[}Phys. Fluids \textbf{26}, 2098 (1983){]} and Edwards and McComb {[}J. Phys. A: Math. Gen. \textbf{2}, 157 (1969){]}, which are compatible with Kolmogorov spectrum. These theories obtained identical equation for spectral density $q(k)$ and different results for damping coefficient. Qian proposed variational approach and Edwards and McComb proposed maximal entropy principle to obtain equation for the damping coefficient. We show that assumptions used in these theories to obtain damping coefficient correspond to unphysical conditions.

Comments on turbulence theory by Qian and by Edwards and McComb

1507.01124v1

arXiv:0811.4118v1 [cond-mat.mtrl-sci] 25 Nov 2008The quantum-mechanical basis of an extended Landau-Lifshitz-Gilbert equation for a current-carrying ferromagnetic wire D.M. Edwards1and O. Wessely1,2 1 Department of Mathematics, Imperial College, London SW7 2BZ, U nited Kingdom 2 Department of Mathematics, City University,London EC1V 0HB, Un ited Kingdom E-mail:d.edwards@imperial.ac.uk Abstract. An extended Landau-Lifshitz-Gilbert (LLG) equation is introduced to describe the dynamics of inhomogeneous magnetization in a current -carrying wire. The coefficients of all the terms in this equation are calculated quant um-mechanically for a simple model which includes impurity scattering. This is done by co mparing the energies and lifetimes of a spin wave calculated from the LLG equa tion and from the explicit model. Two terms are of particular importance since they describe non- adiabatic spin-transfer torque and damping processes which do no t rely on spin-orbit coupling. It is shown that these terms may have a significant influenc e on the velocity of a current-driven domain wall and they become dominant in the cas e of a narrow wall. PACS numbers:An extended Landau-Lifshitz-Gilbert equation 2 1. Introduction The effect of passing an electric current down a ferromagnetic wire is of great current interest. If the magnetization is inhomogeneous it experiences a sp in-transfer torque due to the current [1, 2, 3, 4]. The effect is described phenomenolo gically by adding terms to the standard LLG equation [5, 6]. The leading term in the spin -transfer torque is an adiabatic one arising from that component of the spin po larization of the current which is in the direction of the local magnetization. However , in considering the current-induced motion of a domain wall, Li and Zhang [3, 4] foun d that below a very large critical current the adiabatic term only deforms the wall and does not lead to continuous motion. To achieve this effect they introduced [7] a ph enomenological non-adiabatic term associated with the same spin non-conserving p rocesses responsible for Gilbert damping. Subsequently Kohno et al[8] derived a torque of the Zhang-Li formquantum-mechanically using amodel ofspin-dependent scatt ering fromimpurities. This may arise from spin-orbit coupling on the impurities. More recent ly Wessely et al[9] introduced two further non-adiabatic terms in the LLG equation in order to describe their numerical calculations of spin-transfer torques in a domain wall. These quantum-mechanical calculationsusingtheKeldyshformalismwerem adeintheballistic limit without impurities and with spin conserved. Other terms in the LLG equation, involving mixed space and time derivatives, have been considered by S obolevet al[12], Tserkovnyak et al[10], Skadsen et al[11] and Thorwart and Egger [13]. The object of this paper is to give a unified treatment of all these te rms in the LLG equation and to obtain explicit expressions for their coefficients by q uantum-mechanical calculations for a simple one-band model with and without impurity sca ttering. The strategy adopted is to consider a uniformly magnetized wire and to c alculate the effect of a current onthe energy andlifetime of a long wavelength spin wave propagating along the wire. It is shown in section 2 that coefficients of spin-transfer t orque terms in the LLG equation are directly related to qandq3terms in the energy and inverse lifetime of a spin wave of wave-vector q. The Gilbert damping parameter is the coefficient of the ωterm in the inverse lifetime, where ωis the spin-wave frequency. It corresponds to the damping of a q= 0 spin wave while higher order terms ωqandωq2relate to damping of spin waves with finite wave-vector q. The relation between the qterm in the spin wave energy and the adiabatic spin-transfer torque has been not iced previously [2, 14]. We find that the qterm in the spin wave lifetime relates to the Zhang-Li non-adiabatic spin transfer torque. Our result for the coefficient of the Zhang- Li term is essentially the same as that obtained by Kohno et al[8] and Duine et al[15] but our derivation appears simpler. The q3terms in the spin wave energy and lifetime are related to the additiona l non-adiabatic torques we introduced into the LLG equation [9], toge ther with an extra one arising from spin non-conserving scattering. Explicit expressio ns for the coefficients of these terms are obtained in section 3. In section 4 we discuss brie fly the importance of the additional terms in our extended LLG equation for current- driven motion of a domain wall. Some conclusions are summarized in section 5.An extended Landau-Lifshitz-Gilbert equation 3 2. The LLG equation and spin waves We write our extended LLG equation in the dimensionless form ∂s ∂t+αs×∂s ∂t+α1s×∂2s ∂z∂t−α′ 1s×/parenleftbigg s×∂2s ∂z∂t/parenrightbigg −α′ 2s×∂3s ∂z2∂t−α2s×/parenleftbigg s×∂3s ∂z2∂t/parenrightbigg =s×∂2s ∂z2−bexts×ez−a∂s ∂z−fs×∂s ∂z +a1/braceleftBigg s×/parenleftbigg s×∂3s ∂z3/parenrightbigg +/bracketleftBigg s·∂2s ∂z2−1 2/parenleftbigg∂s ∂z/parenrightbigg2/bracketrightBigg ∂s ∂z/bracerightBigg −f1s×/bracketleftbigg s×∂ ∂z/parenleftbigg s×∂2s ∂z2/parenrightbigg/bracketrightbigg +g1s×∂3s ∂z3. (1) Heres(z,t) is a unit vector in the direction of the local spin polarisation, time tis measured in units of ( γµ0ms)−1and the coordinate zalong the wire is in units of the exchange length lex= (2A/µ0m2 s)1/2. The quantities appearing here are the gyroscopic ratioγ= 2µB//planckover2pi1,thepermeabilityoffreespace µ0andtwopropertiesoftheferromagnetic material, namely the saturation magnetisation msand the exchange stiffness constant A.ezis a unit vector in the zdirection along the wire. The equation expresses the rate of change of spin angular momentum as the sum of various torq ue terms, of which theα1,α′ 1,a,f,a1,f1andg1terms are proportional to the electric current flowing. The second term in the equation is the standard Gilbert term, with da mping factor α, while the α′ 1andα′ 2terms introduce corrections for spin fluctuations of finite wave- vector. Skadsem et al[11] point out the existence of the α′ 2term but do not consider it further. It was earlier introduced by Sobolev et al[12] within a microscopic context based on the Heisenberg model. The α1andα2terms are found to renormalise the spin wave frequency, but for the model considered in section 3 we find t hatα1is identically zero. We shall argue that this result is model-independent. Tserko vnyaket al[10] and Thorwart and Egger [13] find non-zero values of α1which differ from each other by a factor 2; they attribute this to their use of Stoner-like and s−dmodels, respectively. ThorwartandEgger[13]alsofindthe α′ 1termandtheyinvestigatetheeffectof α1andα′ 1 terms on domain wall motion. Their results are difficult to assess beca use the constant |s|= 1 is not maintained during the motion. In eq.(1) we have omitted term s involving the second order time derivatives, whose existence was pointed ou t by Thorwart and Egger [13]; one of these is discussed briefly in section 3.2. The first term on the right-hand side of eq. (1) is due to exchange s tiffness and the next term arises from an external magnetic field Bextezwith dimensionless coefficient bext=Bext/µ0ms. The third term is the adiabatic spin transfer torque whose coefficie nt ais simple and well-known. In fact [3, 4] a=1 2/planckover2pi1JP eµ0m2 slex(2)An extended Landau-Lifshitz-Gilbert equation 4 whereJis the charge current density and eis the electron charge (a negative quantity). The spin polarisation factor P= (J↑−J↓)/(J↑+J↓), where J↑, J↓are the current densities for majority and minority spin in the ferromagnet ( J=J↑+J↓). Eq.(2) is valid for both ballistic and diffusive conduction . The fourth term on th e right-hand side of eq.(1) is the Zhang-Li torque which is often characterised [8 ] by a parameter β=f/a. The next term is the E1term of eq.(7) in ref. [9]. It is a non-adiabatic torque which is coplanar with s(z) ifs(z) lies everywhere in a plane. As shown in ref. [9] it is thezderivative of a spin current , which is characteristic of a torque occ urring from spin-conserving processes. In fact this term takes the form a1∂ ∂z/bracketleftBigg s×/parenleftbigg s×∂2s ∂z2/parenrightbigg −1 2s/parenleftbigg∂s ∂z/parenrightbigg2/bracketrightBigg . (3) Thef1term may be written in the form −f1/parenleftbigg s·∂s ∂z×∂2s ∂z2/parenrightbigg s+f1∂ ∂z/parenleftbigg s×∂2s ∂z2/parenrightbigg . (4) Ifs(z) lies in a plane, the case considered in ref. [9], the first term vanishes and we recover the F1term of eq.(9) in ref. [9]. Its derivative form indicate that it arises fr om spin-conserving processes so we conclude that the coefficient f1is of that origin. This is not true of the last term in eq.(1) and we associate the coefficient g1with spin non- conserving processes. For a spin wave solution of the LLG equation , where we work only to first order in deviations from a state of uniform magnetisatio n, the last three terms of eq.(1) may be replaced by the simpler ones −a1∂3s ∂z3+(f1+g1)s×∂3s ∂z3. (5) Apart fromadditional terms, eq.(1) looksslightly different fromeq.( 7) ofref. [9] because we use the spin polarisation unit vector srather than the magnetisation vector mand s=−m. Furthermore the dimensionless coefficients will take different nume rical values because we have used different dimensionless variables zandtto avoid introducing the domain wall width which was specific to ref. [9]. The torques due to an isotropy fields were also specific to the domain wall problem and have been omitted in e q.(1). We suppose that the wire is magnetised uniformly in the zdirection and consider a spin wave as a small transverse oscillation of the spin polarisation abo ut the equilibrium state or, when a current flows, the steady state. Thus we look fo r a solution of eq.(1) of the form s=/parenleftbig cei(qz−ωt),dei(qz−ωt),−1/parenrightbig (6) where the coefficients of the xandycomponents satisfy c≪1,d≪1. This represents a spin wave of wave-vector qand angular frequency ωpropagating along the zaxis. When (6) is substituted into eq.(1) the transverse components yie ld, to first order in c andd, the equations −iλc+µd= 0, µc+iλd= 0 (7)An extended Landau-Lifshitz-Gilbert equation 5 where λ=ω−aq+a1q3−α2ωq2+iα′ 1qω µ=−iαω+bext+q2+ifq+i(f1+g1)q3+α1ωq−iωq2α′ 2. (8) On eliminating canddfrom eq.(7) we obtain λ2=µ2. To obtain a positive real part for the spin wave frequency, we take λ=µ. Hence ω/parenleftbig 1−α1q−α2q2/parenrightbig =bext+aq+q2−a1q3 +i/bracketleftbig ω/parenleftbig −α−α′ 1q−α′ 2q2/parenrightbig +fq+(f1+g1)q3/bracketrightbig . (9) Thus the spin wave frequency is given by ω=ω1−iω2 (10) where ω1≃/parenleftbig 1−α1q−α2q2/parenrightbig−1/parenleftbig bext+aq+q2−a1q3/parenrightbig ω2≃/parenleftbig 1−α1q−α2q2/parenrightbig−1/bracketleftbig ω1/parenleftbig α+α′ 1q+α′ 2q2/parenrightbig −fq−(f1+g1)q3/bracketrightbig .(11) Here we have neglected terms of second order in α,α′ 1,α′ 2,f,f1andg1, the coefficients which appear in the spin wave damping. This form for the real and imag inary parts of the spin wave frequency is convenient for comparing with the qua ntum-mechanical results of the next section. In this way we shall obtain explicit expre ssions for all the coefficients in the phenomenological LLG equation. Coefficients of od d powers of qare proportional to the current flowing whereas terms in even powers ofqare present in the equilibrium state with zero current. 3. Spin wave energy and lifetimes in a simple model As a simple model of an itinerant electron ferromagnet we consider t he one-band Hubbard model H0=−t/summationdisplay ijσc† iσcjσ+U/summationdisplay ini↑ni↓−µBBext/summationdisplay i(ni↑−ni↓), (12) wherec† iσcreates an electron on site iwith spin σandniσ=c† iσciσ. We consider a simple cubic lattice and the intersite hopping described by the first term is r estricted to nearest neighbours. The second term describes an on-site interaction bet ween electrons with effective interaction parameter U; the last term is due to an external magnetic field. It is convenient to introduce a Bloch representation, with c† kσ=1√ N/summationdisplay iek·Ric† iσ, nkσ=c† kσckσ, (13) ǫk=−t/summationdisplay ieik·ρi=−2t(coskxa0+coskya0+coskza0). (14)An extended Landau-Lifshitz-Gilbert equation 6 The sum in eq.(13) is over all lattice cites Riwhereas in eq.(14) ρi= (±a0,0,0),(0,±a0,0),(0,0,±a0) are the nearest neighbour lattice sites. Then H0=/summationdisplay kσǫknkσ+U/summationdisplay ini↑ni↓−µBBext/summationdisplay k(nk↑−nk↓). (15) To discuss scattering of spin waves by dilute impurities we assume tha t the effect of the scattering from different impurity sites adds incoherently; hen ce we may consider initially a single scattering center at the origin, We therefore introdu ce at this site a perturbing potential u+vl·σ, wherel= (sinθcosφ,sinφsinθ,cosθ) is a unit vector whose direction will finally be averaged over. uis the part of the impurity potential which is indepndent of the spin σand the spin dependent potential vl·σis intended to simulateaspin-orbit L·σinteractionontheimpurity. Itbreaksspinrotationalsymmetry in the simplest possible way. Clearly spin-orbit coupling can only be trea ted correctly for a degenerate band such as a d-band, where on-site orbital angular momentum L occurs naturally. The present model is equivalent to that used by K ohnoet al[8] and Duineet al[15]. In Bloch representation the impurity potential becomes V=V1+V2 with V1=v↑1 N/summationdisplay k1k2c† k1↑ck2↑+v↓1 N/summationdisplay k1k2c† k1↓ck2↓ V2=ve−iφsinθ1 N/summationdisplay k1k2c† k1↑ck2↓+veiφsinθ1 N/summationdisplay k1k2c† k1↓ck2↑ (16) andv↑=u+vcosθ,v↓=u−vcosθ. To avoid confusion we note that the spin dependenceoftheimpuritypotentialwhichoccursinthemany-bod yHamiltonian H0+V is not due to exchange, as would arise in an approximate self consiste nt field treatment (e.g. Hartree-Fock) of the interaction Uin a ferromagnet. 3.1. Spin wave energy and wave function In this section we neglect the perturbation due to impurities and det ermine expressions for the energy and wave function of a long-wave length spin wave in t he presence of an electric current. The presence of impurities is recognised implicitly sin ce the electric current is characterised by a perturbed one-electron distributio n function fkσwhich might be obtained by solving a Boltzman equation with a collision term. We consider a spin wave of wave-vector qpropagating along the zaxis, which is the direction of current flow. Lengths and times used in this section and the next , except when specified, correspond to actual physical quantities, unlike the dim ensionless variables used in section 2. We first consider the spin wave with zero electric current and treat it, within the random phase approximation (RPA), as an excitation from the H artree-Fock (HF) ground state of the Hamiltonian (15). The HF one electron energies are given by Ekσ=ǫk+U/angbracketleftn−σ/angbracketright−µBσBext (17)An extended Landau-Lifshitz-Gilbert equation 7 whereσ= 1,−1 for↑and↓respectively, and /angbracketleftn−σ/angbracketrightis the number of −σspin electrons per site. In a self-consistent ferromagnetic state at T= 0,/angbracketleftnσ/angbracketright=N−1/summationtext kfkσand n=/summationtext σ/angbracketleftnσ/angbracketright, where, fkσ=θ(EF−Ekσ),nis the number of electrons per atom, and EFis the Fermi energy. Nis the number of lattice sites and θ(E) is the unit step function. The spin bands Ekσgiven by eq.(17) are shifted relative to each other by an energy ∆+2 µBBextwhere ∆ = U/angbracketleftn↑−n↓/angbracketrightis the exchange splitting. The ground state is given by |0/angbracketright=/producttext kσc† kσ|/angbracketrightwhere|/angbracketrightis the vacuum state and the product extends over all states kσsuch that fkσ= 1. Within the RPA, the wave function for a spin wave of wave-vector q, excited from the HF ground state, takes the form |q/angbracketright=Nq/summationdisplay kAkc† k+q↓ck↑|0/angbracketright (18) whereNqis a normalisation factor. The energy of this state may be written Eq=Egr+/planckover2pi1ωq=Egr+2µBBext+/planckover2pi1ω′ q (19) whereEgris the energy of the HF ground state and /planckover2pi1ωqis the spin wave excitation energy. On substituting (18) in the Schr¨ odinger equation ( H0−Eq)|q/angbracketright= 0 and multiplying on the left by /angbracketleft0|c† k′↑ck′−q↓, we find Ak′/parenleftbig ǫk′+q−ǫk′+∆−/planckover2pi1ω′ q/parenrightbig =U N/summationdisplay kAkfk↑(1−fk+q↓). (20) Hence we may take Ak= ∆/parenleftbig ǫk+q−ǫk+∆−/planckover2pi1ω′ q/parenrightbig−1(21) and, for small q,/planckover2pi1ω′ qsatisfies the equation 1 =U N/summationdisplay kfk↑−fk+q↓ ǫk+q−ǫk+∆−/planckover2pi1ω′q. (22) Thisistheequationforthepolesofthewell-knownRPAdynamicalsus ceptibility χ(q,ω) [16]. The spin wave pole is the one for which /planckover2pi1ω′ q→0 asq→0. To generalise the above considerations to a current-carrying sta te we proceed as follows. We re-interpret the state |0/angbracketrightsuch that /angbracketleft0|...|0/angbracketrightcorresponds to a suitable ensemble average with a modified one-electron distribution fkσ. When a current flows in thezdirection we may consider the ↑and↓spin Fermi surfaces as shifted by small displacement δ↑ˆkz,δ↓ˆkzwhereˆkzis a unit vector in the zdirection. Thus fkσ=θ(EF−Ek+δσˆkz,σ) ≃θ(EF−Ekσ)−δσδ(EF−Ekσ)∂ǫk ∂kz(23) and the charge current density carried by spin σelectrons is Jσ=e /planckover2pi1Na3 0/summationdisplay k∂ǫk ∂kzfkσ=−eδσ /planckover2pi1Na3 0/summationdisplay k/parenleftbigg∂ǫk ∂kz/parenrightbigg2 δ(EF−Ekσ) =−eδσ /planckover2pi1a3 0/angbracketleftBigg/parenleftbigg∂ǫk ∂kz/parenrightbigg2/angbracketrightBigg σρσ(EF) (24)An extended Landau-Lifshitz-Gilbert equation 8 where/angbracketleft(∂ǫk/∂kz)2/angbracketrightσis an average over the σspin Fermi surface and ρσ(EF) is the density of σspin states per atom at the Fermi energy. We shall also encounter the following related quantities; Kσ=1 N∆2a3 0/summationdisplay k∂ǫk ∂kz∂2ǫk ∂k2zfkσ =/planckover2pi1Jσ ∆2e/angbracketleftBigg/parenleftbigg∂ǫk ∂kz/parenrightbigg2∂2ǫk ∂k2z/angbracketrightBigg σ/slashbigg/angbracketleftBigg/parenleftbigg∂ǫk ∂kz/parenrightbigg2/angbracketrightBigg σ(25) Lσ=1 N∆3a3 0/summationdisplay k/parenleftbigg∂ǫk ∂kz/parenrightbigg3 fkσ =/planckover2pi1Jσ ∆3e/angbracketleftBigg/parenleftbigg∂ǫk ∂kz/parenrightbigg4/angbracketrightBigg σ/slashbigg/angbracketleftBigg/parenleftbigg∂ǫk ∂kz/parenrightbigg2/angbracketrightBigg σ. (26) To derive eqs.(25) and (26), δσhas been eliminated using eq.(24). To solve eqn.(22) for /planckover2pi1ω′ qwe expand the right-hand side of the equation in powers of (ǫk+q−ǫk−/planckover2pi1ω′ q)/∆ and make the further expansions ǫk+q−ǫk=q∂ǫk ∂kz+1 2q2∂2ǫk ∂k2z+1 6q3∂3ǫk ∂k3z... (27) /planckover2pi1ω′ q=Bq+Dq2+Eq3+... (28) in powers of q. We retain all terms up to q3except those involving B2; the coefficients BandEare proportional to the current and we keep only terms linear in the current. Hence we find a solution of eq.(22) in the form (28) with B=1 N↑−N↓/summationdisplay k(fk↑−fk↓)∂ǫk ∂kz=Na3 0 N↑−N↓/planckover2pi1 e(J↑−J↓) (29) D=1 N↑−N↓/bracketleftBigg 1 2/summationdisplay k(fk↑+fk↓)∂2ǫk ∂k2z−1 ∆/summationdisplay k(fk↑−fk↓)/parenleftbigg∂ǫk ∂kz/parenrightbigg2/bracketrightBigg (30) E=−a2 0B 6 +B (N↑−N↓)∆/bracketleftBigg/summationdisplay k(fk↑+fk↓)∂2ǫk ∂k2 z−3 ∆/summationdisplay k(fk↑−fk↓)/parenleftbigg∂ǫk ∂kz/parenrightbigg2/bracketrightBigg −Ua3 0/summationdisplay σ(Kσ−σLσ). (31) HereNσis the total number of σspin electrons so that Nσ=N/angbracketleftnσ/angbracketright. In the absence of spin-orbit coupling the expression for Bin terms of spin current is a general exact result even in the presence of disorder, as show n in Appendix A. The coefficient Dis the standard RPA spin-wave stiffness constant (e.g ref. [16]). W e note that, in the limit ∆ → ∞, E takes the simple form −a2 0B/6. On restoring the correct dimensions (as indicated after eq.(1)) to the expression forω1in eq.(11) we may determine the coefficients aanda1by comparing with theAn extended Landau-Lifshitz-Gilbert equation 9 equation /planckover2pi1ωq= 2µBBext+Bq+Dq2+Eq3. (32) From the coefficient of qwe have a+α1bext=B/(2µBµ0mslex). (33) aandBare both determined directly from the spin current JPindependently of a particular model (see appendix A) so that bextshould not enter their relationship. We conclude quite generally that α1= 0. In this case we find that on combining eqs.(33) and (29), and noting that ms=−µB(N↑−N↓)/Na3 0, eq.(2) is obtained as expected. In section 3.2 we show explicitly for the present model that α1= 0. This conflicts with the results of refs.[10] and [13]. From the coefficients of q2in eqs.(11) and (32) we find 1+α2bext=D/(4µBA/ms). Thus an external field slightly disturbs the standard relationA=Dms/4µB. However in the spirit of the LLG equation we take Aand ms, which enter the units of length and time used in eq.(1), to be consta nts of the ferromagnetic material in zero external field. The coefficients of q3in eqs.(11) and (32) yield the relation (taking α1= 0), −a1+α2a=E//parenleftbig 2µBµ0msl3 ex/parenrightbig . (34) We defer calculation of α2until section 3.2 and the result is given in eq.(44). Combining this with eqs.(34) and (31) we find 2µBµ0msl3 exa1=a2 0B 6−2BD ∆+Ua3 0/summationdisplay σ(Kσ−σLσ). (35) We have thus derived an explicit expression , for a simple model, for th e coefficient a1 of a non-adiabatic spin torque term which appears in the LLG equatio n (1). We have neglected the effect of disorder due to impurities . In the absence o f spin-orbit coupling the expression for the adiabatic torque coefficient a, given by eq.(2), is exact even in presence of impurities. In the next section we shall calculated furt her non-adiabatic torque terms, with coefficients f1andg1, as well as damping coefficients α,α′ 1andα′ 2. In the present model all these depend on impurity scattering for t heir existence. 3.2. Spin wave lifetime The solutions of eq.(22) are shown schematically in figure 1. They inclu de the spin wave dispersion curve and the continuum of Stoner excitations c† k+q↓ck↑|0/angbracketrightwith energies Ek+q↓−Ek↑. The Zeeman gap2 µBBextinthe spin wave energy at q= 0 doesnot appear because we have plotted /planckover2pi1ω′ qrather than /planckover2pi1ωq(see eq.(19)). Within the present RPA the spin wave in a pure metal has infinite lifetime outside the continuum and cannot decay into Stoner excitations owing to conservation of the momentum q. However, when the perturbation V1due to impurities is introduced (see eqn.(16)), crystal momentum is no longer conserved and such decay processes can occur. These ar e shown schematically by the dotted arrow in figure 1. If the bottom of the ↓spin band lies above the Fermi level there is a gap in the Stoner spectrum and for a low energy (sma llq) spin waveAn extended Landau-Lifshitz-Gilbert equation 10 Figure 1. Spin-flip excitations from the ferromagnetic ground state. The do tted arrow shows the mechanism of decay of a spin wave into Stoner excit ations which is enabled by the impurity potential V1. such processes cannot occur. However the spin-flip potential V2enables the spin wave to decay into single particle excitations c† k+qσckσ|0/angbracketrightabout each Fermi surface and these do not have an energy gap. The lifetime τ−1 qof a spin wave of wave-vector qis thus given simply by the “golden rule” in the form τ−1 q=2π /planckover2pi1Nimp(T1+T2) (36) whereNinpis the number of impurity sites and T1=/summationdisplay kp/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig 0/vextendsingle/vextendsingle/vextendsinglec† k↑cp↓V1/vextendsingle/vextendsingle/vextendsingleq/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2 fk↑(1−fp↓)δ(/planckover2pi1ωq−Ep↓+Ek↑) T2=/summationdisplay kpσ/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig 0/vextendsingle/vextendsingle/vextendsinglec† kσcpσV2/vextendsingle/vextendsingle/vextendsingleq/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2 fkσ(1−fpσ)δ(/planckover2pi1ωq−ǫp+ǫk). (37) We first consider T1and, using eqns.(16) and (18), we find /angbracketleftBig 0/vextendsingle/vextendsingle/vextendsinglec† k↑cp↓V1/vextendsingle/vextendsingle/vextendsingleq/angbracketrightBig =Nq Nfk↑(1−fp↓)[Akv↓(1−fp↓)−Ap−qv↑fp−q↑] =Nq Nfk↑(1−fp↓)(Akv↓−Ap−qv↑) (38) for small q. The last line follows from two considerations. Firstly, because of th eδ- function in eq.(37) we can consider the states k↑andp↓to be close to their respective Fermi surfaces. Secondly the ↓spin Fermi surface lies within the ↑Fermi surface and q is small. Hence T1=N2 q N2/summationdisplay kpfk↑(1−fp↓)δ(/planckover2pi1ωq−Ep↓+Ek↑)(Akv↓−Ap−qv↑)2.(39)An extended Landau-Lifshitz-Gilbert equation 11 To evaluate this expression in the case when a current flows we use t he distribution function fkσgiven by eq.(23). Thus, neglecting a term proportional to the squa re of the current, we have T1=N2 q N2/summationdisplay kpδ(/planckover2pi1ωq−Ep↓+Ek↑)(Akv↓−Ap−qv↑)2 ×/bracketleftbigg θ(EF−Ek↑)θ(Ep↓−EF)−δ↑θ(Ep↓−EF)δ(EF−Ek↑)∂ǫk ∂kz +δ↓θ(EF−Ek↑)δ(EF−Ep↓)∂ǫp ∂pz/bracketrightbigg . (40) We wish to expand this expression, and a similar one for T2, in powers of qtoO(q3) so that we can compare with the phenomenological expression (eq.(11 )) for the imaginary partofthespinwave frequency, which isgiven by τ−1 q/2. Itisstraight-forwardtoexpand the second factor in the above sum by using eqs.(21) and (28). We s hall show that the contribution to T1of the first term in square brackets in eq.(40) leads to a contributio n proportional to spin wave frequency ωq. Together with a similar contribution to T2it yields the Gilbert damping factor αas well as the coefficients α′ 1,α′ 2of the terms in eq.(11) which give the qdependence of the damping. The remaining terms in eq.(40) yield the spin-transfer torque coefficients f,f1andg1. The normalisation factor N2 qwhich appear in eq.(40) leads naturally to the factor (1−α1q−α2q2)−1which appears in eq.(11). From eq.(18) it is given by 1 =/angbracketleftq|q/angbracketright=N2 q N/summationdisplay k/parenleftbig A2 kfk↑−A2 k−qfk↓/parenrightbig . (41) By expanding A2 k−qin powers of q, and using eq.(23), we find to O(q2) that N−2 q= (N↑−N↓) ×/braceleftBigg 1+q2 ∆2(N↑−N↓)/summationdisplay k/parenleftbigg∂ǫk ∂kz/parenrightbigg2 [θ(EF−Ek↑)−θ(EF−Ek↓)]/bracerightBigg .(42) We deduce that α1= 0 (43) and α2=−1 l2 ex∆2(N↑−N↓)/summationdisplay k/parenleftbigg∂ǫk ∂kz/parenrightbigg2 [θ(EF−Ek↑)−θ(EF−Ek↓)].(44) The result α1= 0, which was predicted on general grounds in section 3.1 and in Appendix 1, arises here through the absence of a qterm, proportional to current, in the spin wave normalisation factor. In the derivation of eq.(42) th is occurs due to a cancellation involving the Bqterms in the spin energy, which appears in Ak. Without this cancellation we would have α1= 2B/lex∆ which is of the form obtained by Tserkovnyak et al[10] and Thorwart and Egger [13].An extended Landau-Lifshitz-Gilbert equation 12 We now return to the programme for calculating the LLG coefficients α,α′ 1,α′ 2,f,f1,g1which was outlined after eq.(40). We have seen that the qdependence ofN2 qcorresponds to the prefactor in eq.(11). Hence to determine the coefficients listed above we can take N2 q=N2 0= (N↑−N↓)−1inT1andT2when we expand terms in powers of qto substitute in eq.(36) and compare with eq.(11). We first consider the caseq= 0 in order to determine the Gilbert damping factor α. Thus only the first term in square brackets in eq.(40) contributes, since ∂ǫk/∂kzis an odd function kz, and T1(q= 0) =4v2cos2θ N↑−N↓ ×N−2/summationdisplay kpδ(/planckover2pi1ω0−Ep↓+Ek↑)θ(EF−Ek↑)θ(Ep↓−EF) (45) wherecos2θis an average over the angle appearing in the impurity potential V(eq.(16)) and we shall assume cosθ= 0. The summations in eq.(45) may be replaced by energy integrals involving the density of states of per atom ρσ(ǫ) of the states Ekσ. Then, to order (/planckover2pi1ω0)2, T1(q= 0) =/bracketleftBigg 4v2cos2θ N↑−N↓/bracketrightBigg/bracketleftbigg /planckover2pi1ω0ρ↑ρ↓+1 2(/planckover2pi1ω0)2/parenleftbig ρ↑ρ′ ↓−ρ′ ↑ρ↓/parenrightbig/bracketrightbigg (46) whereρσ(ǫ) and its derivative ρ′ σ(ǫ) are evaluated at ǫ=EF. Similarly T2(q= 0) =/bracketleftBigg v2sin2θ N↑−N↓/bracketrightBigg /planckover2pi1ω0/parenleftbig ρ2 ↑+ρ2 ↓/parenrightbig (47) and noω2 0terms appear. We have included the ω2 0term in eq.(46) merely because it corresponds to a term s×/parenleftBig s×∂2s ∂t2/parenrightBig in the LLG equation whose existence was noted by Thorwald and Egger [13]. We shall not pursue terms with second-ord er time derivatives any further. Since the imaginary part of the spin wave frequency is given by τ−1 q/2 it follows from eqs.(11), (36), (46) and (47) that α=πcv2 /angbracketleftn↑−n↓/angbracketright/bracketleftBig 4cos2θρ↑ρ↓+sin2θ/parenleftbig ρ2 ↑+ρ2 ↓/parenrightbig/bracketrightBig , (48) wherec=Nimp/Nis the concentration of impurities, in agreement with Khono et al[8] and Duine et al[15]. If the direction of the spin quantisation axis of the impurities is distributed randomly cos2θ= 1/3,sin2θ= 2/3 so that αis proportional to ( ρ↑+ρ↓)2. To investigate the qdependence of Gilbert damping, and thus evaluate α′ 1andα′ 2 in eq.(11), the second factor in the summation of eq.(40) must be ex panded in powers ofq. All the terms which contribute to the sum are of separable form g(k)h(p). The contribution to T1of interest here , proportional to ωq, again arises from the first term in square brackets in eq.(40), and similarly for T2. The summations required in eq.(40) are of the form /summationdisplay kpδ(/planckover2pi1ωq−Ep↓+Ek↑)θ(EF−Ek↑)θ(Ep↓−EF)g(k)h(p) =/angbracketleftg(k)/angbracketright↑/angbracketlefth(k)/angbracketright↓ρ↑ρ↓/planckover2pi1ωq (49)An extended Landau-Lifshitz-Gilbert equation 13 where/angbracketleftg(k)/angbracketrightσ=N−1/summationtext kg(k)δ(EF−Ekσ) is an average over the Fermi surface, as used previously in section 3.1. After some algebra we find α′ 1= 2Bα/∆lex (50) α′ 2=πc /angbracketleftn↑−n↓/angbracketrightl2 ex∆2/braceleftbigg ρ↑ρ↓/parenleftBig u2+5v2cos2θ/parenrightBig/summationdisplay σ/angbracketleftBigg/parenleftbigg∂ǫk ∂kz/parenrightbigg2/angbracketrightBigg σ −2ρ↑ρ↓∆v2cos2θ/summationdisplay σσ/angbracketleftbigg∂2ǫk ∂k2z/angbracketrightbigg σ −v2sin2θ/bracketleftBigg ∆/summationdisplay σσρ2 σ/angbracketleftbigg∂2ǫk ∂k2z/angbracketrightbigg σ−3/summationdisplay σρ2 σ/angbracketleftBigg/parenleftbigg∂ǫk ∂kz/parenrightbigg2/angbracketrightBigg σ/bracketrightBigg/bracerightbigg +2Dα ∆l2ex. (51) We note that, unlike αandα′ 1, the coefficient α′ 2is non-zero even when the spin- dependent part of the impurity potential, v, is zero. In this case the damping of a spin wave of frequency ωand small wave-vector qis proportional to ρ↑ρ↓u2ωq2. In zero external field ω∼q2so that the damping is of order q4. This damping due to spin-independent potential scattering by impurities was analysed in detail by Yamada and Shimizu [17]. One of the Fermi surface averages in eq.(51) is easily evaluated using eqs.(14) and (17). Thus/angbracketleftbigg∂2ǫk ∂k2 z/angbracketrightbigg σ=−a2 0 3/angbracketleftǫk/angbracketrightσ=−a2 0 3(Ef−U/angbracketleftn−σ/angbracketright+σµBBext). (52) In the spirit of the LLG equation we should take Bext= 0 in evaluating the coefficients α′ 2. We now turn to the evaluation of the non-adiabatic spin-transfer t orque coefficients f,f1andg1. These arise from the second and third terms in square brackets in eq.(40), and in a similar expression for T2. The summations involved in these terms differ from those in eq.(49) since one θ-function is replaced by a δ-function. This leads to the omission of the frequency factor /planckover2pi1ωq. The Fermi surface shifts δσare elininated in favour of currents Jσby using eq.(24). By comparing the coefficient of qin the expansion of eq.(36) with that in eq.(11) we find the coefficient of the Zhang-Li torque in the form f=πcv2 µ0m2s∆lex/planckover2pi1 e/bracketleftBig 2cos2θ(ρ↑J↓−ρ↓J↑)+sin2θ(ρ↓J↓−ρ↑J↑)/bracketrightBig .(53) This is in agreement with Khono et al[8] and Duine et al[15]. In the “isotropic” impurity case, with cos2θ= 1/3,sin2θ= 2/3, it follows from eqs.(53), (48) and (2) that β=f a=α2 U(ρ↑+ρ↓). (54) In the limit of a very weak itinerant forromagnet ρσ→ρ, the paramagnetic density of states, and Uρ→1 by the Stoner criterion. Thus in this limit β=α. Tserkovnyak etAn extended Landau-Lifshitz-Gilbert equation 14 al[10] reached a similar conclusion. For a parabolic band it is straightfor ward to show from Stoner theory that β/α >1 and may be as large as 1.5. As discussed in section 2 the coefficient f1is associated with spin-conserving processes, and hence involves the spin independent potential u. The coefficient g1 is associated with spin non-conserving processes and involves v. By comparing the coefficient of q3in the expansion of eq.(36) with that in eq.(11) we deduce that f1=πc 2µ0m2sl3exu2(K1+2L1+M1) (55) and g1=1 l2ex/parenleftbigg3D ∆−a2 0 6/parenrightbigg f +πcv2 2µ0m2sl3ex/bracketleftBig cos2θ(5K1+6L1−M1)+sin2θ(3K2+4L2)/bracketrightBig . (56) Here K1=K↓ρ↑+K↑ρ↓, K2=K↓ρ↓+K↑ρ↑ L1=L↓ρ↑−L↑ρ↓, L2=L↓ρ↓−L↑ρ↑ M1=/planckover2pi1 e∆3/summationdisplay σ/bracketleftBigg 2σ/angbracketleftBigg/parenleftbigg∂ǫk ∂kz/parenrightbigg2/angbracketrightBigg −σ+∆/angbracketleftbigg∂2ǫk ∂k2z/angbracketrightbigg −σ/bracketrightBigg Jσρ−σ. (57) This complete the derivation of expressions for all the LLG coefficien ts of eq.(1) within the present impurity model 4. The extended LLG equation applied to current-driven doma in wall motion In a previous paper [9] we introduced the a1andf1terms of the extended LLG equation (cf. eqns.(1), (3) and (4)) in order to describe numerically-calcula ted spin-transfer torques acting on a domain wall when it is traversed by an electric cur rent. In that work the origin of the small f1term for a pure ferromagnetic metal was specific to the domain wall problem; it was shown to be associated with those elec tronic states at the bulk Fermi surface which decay exponentially as they enter the wall. The analytic derivation of f1in section 3 (see eqn.(55)) is based on impurity scattering in the bulk ferromagnet and applies generally to any slowly-varying magnetizat ion configuration. For a ferromagnetic alloy such as permalloy both mechanisms should c ontribute in the domain wall situation but the impurity contribution would be expected to dominate. To describe a domain wall we must add to the right-hand side of eqn.(1 ) anisotropy terms of the form −(s·ey)s×ey+b−1(s·ez)s×ez, (58) whereeyis a unit vector perpendicular to the plane of the wire. The first term corresponds to easy-plane shape anisotropy for a wire whose widt h is large compared with its thickness and the second term arises from a uniaxial field Hualong the wire,An extended Landau-Lifshitz-Gilbert equation 15 so thatb=ms/Hu. The solution of eqn.(1), with the additional terms (58), for a stationary N´ eel wall in the plane of the wire, with zero external fie ld and zero current, is s= (sech(z/b1/2),0,−tanh(z/b1/2)). (59) As pointed out in ref. [9] there is no solution of the LLG equation of th e form s=F(z−vWt), corresponding to a uniformly moving domain wall, when the f1term is included. It is likely that the wall velocity oscillates about an average value, as predicted by Tatara and Kohno [18, 19] for purely adiabatic torque above the critical current density for domain wall motion. However, we may estimate t he average velocity vWusing the method of ref. [9]. The procedure is to substitute the ap proximate form s=F(z−vWt) in the extended LLG equation (1), with the terms (58) added, tak e the scalar product with F×F′and integrate with respect to zover the range ( −∞,∞). The boundary conditions appropriate to the wall are s→ ∓ezasz→ ±∞. Hence for bext= 0 we find the dimensionless wall velocity to be vW=f/integraltext∞ −∞(F×F′)2dz+f1/integraltext∞ −∞(F×F′′)2dz+g1/integraltext∞ −∞(F′′)2dz α/integraltext∞ −∞(F×F′)2dz+α′ 2/integraltext∞ −∞(F′′)2dz.(60) To estimate the integrals we take F(z) to have the form of the stationary wall s(z) (eqn.(59)) and, with the physical dimensions of velocity restored, the wall velocity is given approximately by vW=v0β α1+f1(3fb)−1 1+α′ 2(αb)−1(61) wherev0=µBPJ/(mse). We have neglected g1here because, like fandα, it depends on spin-orbit coupling but is a factor ( a0/lex)2smaller than f(cf. eqns.(53) and (56)). f1andα′ 2are important because they do not depend on spin-orbit coupling. It is interesting to compare vWwith the wall velocity observed in permalloy nanowires by Hayashi et al[20]. We first note that v0is the velocity which one obtains very simply from spin angular momentum conservation if the current -driven wall moves uniformly without any distortion such as tilting out of the easy plane a nd contraction [21]. This is never the case, even if f1= 0,α′ 2= 0, unless β=α. For a permalloy nanowire, with µ0ms= 1 T,v0= 110Pm/s forJ= 1.5·108A/cm2. Thus, from the standard theory with f1= 0,α′ 2= 0,vW= 110Pβ/αm/s for this current density. In fact Hayashi et al[20] measure a velocity of 110 m/s which implies β > αsince the spin polarization Pis certainly less than 1. They suggest that βcannot exceed αand that some additional mechanism other than spin-transfer torque is ope rating. However in the discussion following eqn.(54) we pointed out that in the model calculat ions it is possible to haveβ > α. Even if this is not the case in permalloy we can still have vW> v0if the last factor in eqn.(61) is greater than 1 when f1andα′ 2are non-zero. We can estimate termsinthisfactorusingtheobservationfromref. [20],that lW=lexb1/2= 23nm, where lWis the width of the wall. From eqns.(53) and (55) we find f1/(fb)∼(u/v)2(kFlW)−2, wherekFis a Fermi wave-vector. In permalloy we have Fe impurities in Ni so tha t inAn extended Landau-Lifshitz-Gilbert equation 16 the impurity potential u+v·σwe estimate u∼1 eV and v∼0.005 eV. The value for v is estimated by noting that the potential v·σis intended to model spin-orbit coupling of the form ξL·σwithξ/lessorsimilar0.1 eV and /angbracketleftLz/angbracketrightFe∼0.05,Lzbeing the component of orbital angular momentum in the direction of the magnetization [22]. Hence u/v∼200 and kFlW∼200 so that f1/(fb)∼1.α′ 2/(αb) is expected to be of similar magnitude. We conclude that the α′ 2andf1terms in the LLG equation (1) can be important in domain wall motion and should be included in micromagnetic simulations such as O OMMF [23]. For narrower domain walls these terms may be larger than the Gilb ert damping α and non-adiabatic spin- transfer torque fterms which are routinely included. Reliable estimates of their coefficients are urgently required using realistic m ultiband models of the ferromagnetic metal or alloy. 5. Conclusions The coefficients of all the terms in an extended LLG equation for a cu rrent-carrying ferromagnetic wire have been calculated for a simple model. Two of th ese (f1and α′ 2) are of particular interest since they do not rely on spin-orbit coup ling and may sometimes dominate the usual damping and non-adiabatic spin-tran sfer torque terms. One term ( α1) which has been introduced by previous authors is shown rigorously to be zero, independent of any particular model. Solutions of the exte nded LLG equation for domain wall motion have not yet been found but the average velo city of the wall is estimated. It is pointed out that the f1andα′ 2terms are very important for narrow walls and should be included in micrmagnetic simulations such as OOMMF. I t is shown that there is no theoretical reason why the wall velocity should not exceed the simplest spin-transfer estimate v0, as is found to be the case in experiments on permalloy by Hayashiet al[20] Acknowledgments We are grateful to the EPSRC for financial support through the S pin@RT consortium and to other members of this consortium for encouragement and s timulation. Appendix A. The simple single-band impurity model used in the main text is useful fo r obtaining explicit expressions for all the coefficients in the LLG equation (1). H ere we wish to show that some of these results are valid for a completely general s ystem. We suppose the ferromagnetic material is described by the many-body Hamilton ian H=H1+Hint+Hext (A.1) whereH1is a one-electron Hamiltonian of the form H1=Hk+Hso+V. (A.2)An extended Landau-Lifshitz-Gilbert equation 17 HereHkis the total electron kinetic energy, Hsois the spin-orbit interaction, Vis a potential term, Hintis the coulomb interaction between electrons and Hextis due to an external magnetic field Bextin thezdirection. Thus Hext=−2µBSz 0Bext (A.3) whereS0 zis thezcomponent of total spin. Both HsoandVcan contain disorder. Since we are interested in the energy and lifetime of a long-wavelength spin wave we consider the spin wave pole, for small q, of the dynamical susceptibility. χ(q,ω) =/integraldisplay dt/angbracketleft/angbracketleftS− q(t),S+ −q/angbracketright/angbracketrighte−iω−t(A.4) (ω−=ω−iǫ) whereS± q=Sx q±iSy qare Fourier components of the total transverse spin density. Here /angbracketleft/angbracketleftS− q(t),S+ −q/angbracketright/angbracketright=i /planckover2pi1/angbracketleft/bracketleftbig S− q(t),S+ −q/bracketrightbig /angbracketrightθ(t). (A.5) Ingeneral we shall take theaverage /angbracketleft/angbracketrightina steady statein which a charge current density Jis flowing in the qdirection. Following the general method of Edwards and Fisher [24] we use equations of motion to find that χ(q,ω) =−2/angbracketleftSz 0/angbracketright /planckover2pi1(ω−bext)+1 /planckover2pi12(ω−bext)2/braceleftbig χc(q,ω)−/angbracketleft/bracketleftbig C− q,S+ −q/bracketrightbig /angbracketright/bracerightbig (A.6) where/planckover2pi1bext= 2µBBext,C− q= [S− q,H1] and χc(q,ω) =/integraldisplay dt/angbracketleft/angbracketleftC− q(t),C+ −q/angbracketright/angbracketrighte−iωt. (A.7) For small qandω,χis dominated by the spin wave pole, so that χ(q,ω) =−2/angbracketleftSz 0/angbracketright /planckover2pi1(ω−bext−ωq)(A.8) wherebext+ωqis the spin wave frequency, in general complex corresponding to a fi nite lifetime. Following ref. [24] we compare (A.6) and (A.8) in the limit ωq≪ω−bextto obtain the general result ωq=−1 2/angbracketleftSz 0/angbracketright/planckover2pi1/braceleftbigg lim ω→bextχc(q,ω)−/angbracketleft/bracketleftbig C− q,S+ −q/bracketrightbig /angbracketright/bracerightbigg . (A.9) Edwards and Fisher [24] were concerned with Reωqwhereas Kambersky [25] derived the above expression for Imωqfor the case q= 0, and zero current flow. His interest was Gilbert damping in ferromagnetic resonance. Essentially the same re sult was obtained earlier in connection with electron spin resonance, by Mori and Kawa saki [26], see also Oshikawa and Affleck [27]. Since S− qcommutes with the potential term V, even in the presence of disorder, we have C− q=/bracketleftbig S− q,H1/bracketrightbig =/bracketleftbig S− q,Hk/bracketrightbig +/bracketleftbig S− q,Hso/bracketrightbig . (A.10) For simplicity we now neglect spin-orbit coupling so that C− q=/bracketleftbig S− q,Hk/bracketrightbig =/planckover2pi1qJ− q (A.11)An extended Landau-Lifshitz-Gilbert equation 18 where the last equation defines the spin current operator J− q. For a general system, with thenthelectron at position rnwith spin σnand momentum pn, S− q=/summationdisplay neiq·rnσ− n, Hk=/summationdisplay np2 n/2m. (A.12) Hence, from eqns.(A.11) and (A.12), /angbracketleft/bracketleftbig C− q,S+ −q/bracketrightbig /angbracketright=N/planckover2pi12q2 2m+2/planckover2pi1/summationdisplay n/angbracketleftσz nvn/angbracketright·q (A.13) whereNis the total number of electrons and vn=pn/mis the electron velocity, so thate/summationtext n/angbracketleftσz nvn/angbracketrightis the total spin current. Hence from eq.(A.9), we find ωq=/planckover2pi1q2 2/angbracketleftSz 0/angbracketright/bracketleftbiggN 2m−lim ω→bextχJ(0,ω)/bracketrightbigg +Bq /planckover2pi1(A.14) with B=/planckover2pi1µBPJ/em s. (A.15) This expression for Bhas been obtained by Bazaliy et al[2] and Fern´ andez-Rossier et al[14] for simple parabolic band, s−dand Hubbard models. The derivation here is completelygeneralforanyferromagnet,eveninthepresenceof disorderduetoimpurities or defects, as long as spin-orbit coupling is neglected. Eqs.(2) and ( A.14) are both valid for arbitrary bext, so that in eq. (33) we must have α1= 0. References [1] Berger L 1978 J.Appl.Phys. 492156 [2] Bazaliy Y B, Jones B A and Zhang S C 1998 Phys. Rev. B 57R3213 [3] Li Z and Zhang S 2004 Phys. Rev. B 70024417 [4] Li Z and Zhang S 2004 Phys. Rev. Lett. 92207203 [5] Landau L D, Lifshitz E M and Pitaevski L P 1980 Statistical Physics , part 2 (Oxford: Pergamon) [6] Gilbert T L 1955 Phys. Rev. 1001243 [7] Zhang S and Li Z 2004 Phys. Rev. Lett. 93127204 [8] Kohno H, Tatara G and Shibata J 2006 J. Phys. Soc. Japan 75113706 [9] Wessely O, Edwards D M and Mathon J 2008 Phys. Rev. B 77174425 [10] Tserkovnyak T, Skadsem H J, Brataas A and Bauer G E W 2006 Phys. Rev. B 74144405 [11] Skadsem H J, Tserkovnyak T, Brataas A and Bauer G E W 2007 Phys. Rev. B 75094416 [12] Sobolev V L, Klik I, Chang C R and Huang H L 1994 J. Appl. Phys. 755794 [13] Thorwart M and Egger R 2007 Phys. Rev. B 76214418 [14] Fern´ andez-Rossier J, Braun M, N´ u˜ nez A S and MacDonald A H 2004Phys. Rev. B 69174412 [15] Duine R A, N´ u˜ nez A S, Sinova J and Macdonald A H 2007 Phys. Rev. B 75214420 [16] Izuyama T, Kim D-J and Kubo R 1963 J. Phys. Soc. Japan 181025 [17] Yamada H and Shimizu M 1971 J. Phys. Soc. Japan 311344 [18] Tatara G and Kohno H 2004 Phys. Rev. Lett. 92086601 [19] Tatara G and Kohno H 2005 J. Electron. Microsc. 54i69 [20] Hayashi M, Thomas L, Rettner C, Moriya R, Bazaliy Y B and Parkin S S P 2007 Phys. Rev. Lett. 98037204 [21] Barnes S E and Maekawa S 2005 Phys. Rev. Lett. 95107204 [22] Daalderop G H O, Kelly P J and Schuurmans M F H 1990 Phys. Rev. B 4111919 [23] Donahue M and Porter D http://math.nist.gov/oommfAn extended Landau-Lifshitz-Gilbert equation 19 [24] Edwards D M and Fisher B 1971 J. Physique 32C1 697 [25] Kambersk´ y V 1976 Czech. J. Phys. B 261366 [26] Mori H and Kawasaki K 1962 Prog. Theor. Phys. 27529 [27] Oshikawa M and Affleck I 2002 Phys. Rev. B 65134410

2008-11-25

An extended Landau-Lifshitz-Gilbert (LLG) equation is introduced to describe the dynamics of inhomogeneous magnetization in a current-carrying wire. The coefficients of all the terms in this equation are calculated quantum-mechanically for a simple model which includes impurity scattering. This is done by comparing the energies and lifetimes of a spin wave calculated from the LLG equation and from the explicit model. Two terms are of particular importance since they describe non-adiabatic spin-transfer torque and damping processes which do not rely on spin-orbit coupling. It is shown that these terms may have a significant influence on the velocity of a current-driven domain wall and they become dominant in the case of a narrow wall.

The quantum-mechanical basis of an extended Landau-Lifshitz-Gilbert equation for a current-carrying ferromagnetic wire

0811.4118v1

arXiv:1206.4819v2 [cond-mat.mtrl-sci] 18 Aug 2013Fast domain wall propagation in uniaxial nanowires with tra nsverse fields Arseni Goussev1,2, Ross G. Lund3, JM Robbins3, Valeriy Slastikov3, Charles Sonnenberg3 1Department of Mathematics and Information Sciences, Northumbria University, Newcastle Upon Tyne, NE1 8ST, UK 2Max Planck Institute for the Physics of Complex Systems, N¨ othnitzer Straße 38, D-01187 Dresden, Germany 3School of Mathematics, University of Bristol, University W alk, Bristol BS8 1TW, United Kingdom (Dated: May 14, 2018) Under a magnetic field along its axis, domain wall motion in a u niaxial nanowire is much slower than in the fully anisotropic case, typically by several ord ers of magnitude (the square of the di- mensionless Gilbert damping parameter). However, with the addition of a magnetic field transverse to the wire, this behaviour is dramatically reversed; up to a critical field strength, analogous to the Walker breakdown field, domain walls in a uniaxial wire propa gate faster than in a fully anisotropic wire (without transverse field). Beyond this critical field s trength, precessional motion sets in, and the mean velocity decreases. Our results are based on leadin g-order analytic calculations of the velocity and critical field as well as numerical solutions of the Landau-Lifshitz-Gilbert equation. PACS numbers: 75.75.-c, 75.78.Fg Introduction The dynamics of magnetic domain walls in ferromag- netic nanowiresunder external magnetic fields [1–12] and spin-polarised currents [12–20] is a central problem in micromagnetics and spintronics, both as a basic physical phenomenonaswellasacornerstoneofmagneticmemory and logic technology [3, 16–18]. From the point of view of applications, it is desirable to maximise the domain wall velocity in order to optimise switching and response times. Partly because of fabrication techniques, attention has been focused on nanowires with large cross-sectional as- pect ratio, typically of rectangular cross-section. In this case, even if the bulk material is isotropic (e.g., permal- loy), the domain geometry induces a fully anisotropic magnetic permeability tensor, with easy axis along the wire and hard axis along its shortest dimension [21, 22]. Nanowires with uniaxial permeability, characteristic of moresymmetricalcross-sectionalgeometries(e.g., square or circular), have been less studied [23–25]. Here we in- vestigate domain wall (DW) motion in uniaxial wires in the presence of transverse fields. We show that the DW velocityinuniaxialwiresdependsstronglyonthelongitu- dinal applied field H1, increasing with H1up to a certain critical field and thereafter falling off as precessional mo- tion sets in. We employ a systematic asymptotic expan- sion scheme, which differs from alternative approaches based on approximate dynamics for the DW centre and orientation; a detailed account of this scheme, also in- cluding anisotropy and current-induced torques, will be given separately [29] . We employ a continuum description of the magnetisa- tion. For a thin nanowire, this is provided by the one- dimensionalLandau-Lifshitz-Gilbert(LLG)equation[22,26–28], which we write in the non-dimensionalised form ˙M=γM×H−αM×(M×H). (1) HereM(x,t) is a unit-vector field specifying the orien- tation of the magnetisation, which we shall also write in polar form M= (cosΘ ,sinΘcosΦ ,sinΘsinΦ). The effective magnetic field, H(m), is given by H=Am′′+K1m1ˆx−K2m2ˆy+Ha.(2) HereAis the exchange constant, K1is the easy-axis anisotropy, K2>0 is the hard-axis anisotropy, Hais the applied magnetic field (taken to be constant), γis the gyromagnetic ration, and αis the Gilbert damping parameter. For convenience we choose units for length, time and energy so that A=K1=γ= 1.Domains cor- respond to locally uniform configurations in which Mis aligned along one of the local minima, denoted m+and m−, of the potential energy U(m) =−1 2(m2 1−K2m2 2)−m·Ha.(3) Two distinct domains separated by a DW are described by the boundary conditions M(±∞,t) =m±. For purely longitudinal fields Ha=H1ˆxand forH1 belowtheWalkerbreakdownfield HW=αK2/2,theDW propagates as a travelling wave [1], the so-called Walker solution Θ( x,t) =θW(x−VWt), Φ(x,t) =φW, whereθW andφWare given by θW(ξ) = 2tan−1(e−ξ/γ),sin2φW=H1/HW.(4) The width of the DW, γ, is given by γ= (1 + K2cos2φW)−1/2, and the velocity is given by VW=−γ(α+1/α)H1. (5) ForH1> HW, the DW undergoes non-uniform preces- sion and translation, with mean velocity decreasing with2 H1[1, 5, 6, 9]. The effects of additional transverse fields have been examined recently [7, 11]. If the cross-sectional geometry is sufficiently symmet- rical (e.g., square or circular), the permeability tensor becomes uniaxial, so that K2= 0 [21, 22]. The dynamics in this case is strikingly different. The LLG equation has an exact solution, Θ( x,t) =θ0(x−VPt), Φ(x,t) =−H1t, in which the DW propagates with velocity VP=−αH1 (6) and precesses about the easy axis with angular velocity −H1[23, 24]. The precessing solution persists for all H1 – there is no breakdown field – but becomes unstable for H1/greaterorsimilar1/2 [25]. ForH1< HW, the ratio VW/VPisgiven by γ(α−2+1). For typical values of α(0.01 – 0.1), the uniaxial velocity VPis less than the fully anisotropic velocity VWby sev- eral orders of magnitude. As we show below, applying a transverse field H2>0 to a uniaxial wire dramatically changes its response to an applied longitudinal field H1. The transverse field, analogous to hard-axis anisotropy, inhibits precession and facilitates fast DW propagation. ForH1lessthanan H2-dependentcriticalfield H1c, given in the linear regime by (29) below, there appears a trav- elling wave, while for H1> H1c, there appears an oscil- lating solution, as in the Walker case. The DW velocity of travelling wave exceeds that of oscillating solution. Velocity of travelling wave We first obtain a general identity, of independent in- terest, which relates the velocity of a travelling wave M(x,t) =m(x−Vt) (assuming one exists) to the change in potential energy across the profile (for zero transverse field, this coincides with results of [1] and [10]). Noting that˙M=−Vm′, we take the squareof (1) and integrate over the length of the wire to obtain V2||m′||2= (1+α2)||m×H||2. (7) Here we use the notation ||u||2=/angbracketleftu,u/angbracketright,/angbracketleftu,v/angbracketright=/integraldisplay∞ −∞u·vdx(8) for theL2-norm and inner product of vector fields (anal- ogous notation for scalar fields is used below). Next, we take the inner product of (1) with Hto obtain V/angbracketleftm′,H/angbracketright=−α||m×H||2. (9) Noting that m′·H=/parenleftbig1 2m′·m′−U(m)/parenrightbig′, we combine (7) and (9) to obtain V=1 2(α+1/α)||m′||−2(U(m−)−U(m+)).(10)The identity (10) has a simple physical interpretation; the velocity is proportional to the potential energy dif- ference across the wire, and inversely proportional to the exchange energy of the profile. From now on, we consider the uniaxial case K2= 0 andappliedfieldwithlongitudinalandtransversecompo- nentsH1,H2>0 (by symmetry, we can assume H3= 0) with|Ha|<1. An immediate consequence of (10) is that, in the uniaxial case, the velocity must vanish as H1goes to zero. For when H1= 0, the local minima m±are related by reflection through the 23-plane, and U(m+) =U(m−). Small transverse field In order to understand travelling wave and oscillating solutions as well as the transition between them, we first carry out an asymptotic analysis in which both H1and H2are regarded as small, writing H1=ǫh1,H2=ǫh2 and rescaling time as τ=ǫt(a systematic treatment in- cludingcurrent-inducedtorqueswillbegivenin[29]). We seek a solution of the LLG equation (1) of the following asymptotic form: Θ(x,t) =θ0(x,τ)+ǫθ1(x,τ)+..., (11) Φ(x,t) =φ0(x,τ)+ǫφ1(x,τ)+... (12) It is straightforward to check that the boundary condi- tions, namely that mapproach distinct minima of Uas x→ ±∞, imply that m(±∞,τ) = (±1,ǫh2,0)+O(ǫ2).(13) The leading-order equations for Θ and Φ become θ0,xx−1 2(1+φ2 0,x)sin2θ0= 0, (14) /parenleftbig sin2θ0φ0,x/parenrightbig x= 0. (15) The only physical (finite-energy) solutions of (14) and (15) consistent with the boundary conditions (13) are of the form φ0(x,τ) =φ0(τ) (16) θ0(x,τ) = 2arctanexp( −(x−x∗(τ))),(17) whereφ0andx∗respectivelydescribetheDWorientation and centre, and are functions of τalone. It is convenient to introduce a travelling coordinate ξ=x−x∗(τ) and rewrite the ansatz (11)–(12) as Θ(x,t) =θ0(ξ,τ)+ǫθ1(ξ,τ)+..., (18) Φ(x,t) =φ0(ξ,τ)+ǫφ1(ξ,τ)+... (19) To obtain equations for φ0(τ) andx∗(τ) we must pro- ceed to the next order. It is convenient to introduce new3 variables at order ǫwhich, in light of the boundary con- ditions (13), vanish at x=±∞, as follows: Θ1:=θ1−h2cosφ0cosθ0, (20) u:=φ1sinθ0+h2sinφ0. (21) These satisfy the linear inhomogeneous equations LΘ1=f, (22) Lu=g. (23) HereLis the self-adjoint Schr¨ odinger operator given by L=−∂2 ∂ξ2+W(ξ), (24) where W=θ′′′ 0 θ′ 0= 1−2sech2ξ, (25) andf(ξ,τ) andg(ξ,τ) are given by f= (1+α2)−1sinθ0(−α˙x∗−˙φ0)−h1sinθ0, g= (1+α2)−1sinθ0(˙x∗−α˙φ0)+2h2sin2θ0sinφ0. (26) TheDWposition x∗andorientation φ0aredetermined from the solvability conditions for (22) – (23). According to the Fredholm alternative, given a self-adjoint opera- torLonL2(R), a necessary condition for the equation LΘ1=fto have a solution Θ 1is thatfbe orthogonal to the kernel of L. If this is the case, a sufficient condition is thatthespectrumof Lisisolatedawayfrom0. From(24) and (25) it is clear that θ′ 0belongs to the kernel of L, and since the eigenvalues of a one-dimensional Schr¨ odinger operator are nondegenerate, it follows that θ′ 0spans the kernel of L. Moreover, since W(ξ)→1 asξ→ ±∞, it followsthat the spectrum of Lis discrete near0. (In fact, Wis a special case of the exactly solvable P¨ oschl-Teller potential, but we won’t make use of this fact.) Requiring fandgin (22) and (23) to be orthogonal to θ′ 0and not- ing that /angbracketleftθ′ 0,θ′ 0/angbracketright= 2,/angbracketleftθ′ 0,sinθ0/angbracketright=−2,/angbracketleftθ′ 0,1/angbracketright=−π, and /angbracketleftθ′ 0,cosθ0/angbracketright= 0, we obtain the following system of ODEs forφ0andx∗: ˙φ0=−h1−απ 2h2sinφ0, (27) ˙x∗=−αh1+π 2h2sinφ0. (28) Travelling wave solutions appear provided (27) has fixed points; this occurs for h1below a critical field h1,c given by h1,c=απh2 2, (29)The velocity and orientation of the travelling wave are given by ˙x∗=−/parenleftbigg α+1 α/parenrightbigg h1, (30) sinφ0=−h1 h1,c. (31) There are two possible solutions for φ0∈[0,2π), only one of which is stable. Oscillating solutions appear for h1> h1c, and are given by h1tan1 2φ0=−h1,c−/radicalBig h2 1−h2 1,ctan/parenleftBig 1 2/radicalBig h2 1−h2 1,cτ/parenrightBig (32) with the period T= 2π//radicalBig h2 1−h2 1,c. The mean preces- sional and translational velocities are obtained by aver- aging over a period, with result /angbracketleftBig ˙φ0/angbracketrightBig =−sgn(h1)/radicalBig h2 1−h2 1,c, (33) /angbracketleft˙x∗/angbracketright=−/parenleftbigg α+1 α/parenrightbigg h1+1 αsgn(h1)/radicalBig h2 1−h2 1,c.(34) Note that for h1=h1,c, (34) coincides with the travelling wave velocity (30), whereas for h1≫h1,c, (34) reduces to the velocity of the precessing solution given by (6). The behaviour is similar in many respects to the Walker case (i.e., K2/negationslash= 0 and H2= 0). Here, the trans- verse field rather than hard-axis anisotropy serves to ar- rest the precession of the DW (provided the longitudinal field is not too strong). There are differences as well; in the transverse-field case there is just one stable trav- elling wave, whereas in the Walker case there are two. Also, in the transverse-field case the asymptotic value of the magnetisation has a transverse component, whereas in the Walker case it has none. Moderate transverse field We can extend the travelling wave analysis to the regime where H2is no longer regarded as small. We continue to regard H1as small, writing H1=ǫh1and V=ǫv, and expand the travelling wave ansatz Θ( x,t) = θ(x−Vt), Φ(x,t) =φ(x−Vt) to first order in ǫ, writing θ=θ0+ǫθ1,φ=φ0+ǫφ1. Substituting into the LLG equation, we obtain the O(ǫ0) equations θ′ 0= (H2−sinθ0), φ0= 0, (35) with boundary conditions sin θ0±=H2,θ0+> π/2 and θ0−< π/2. Thus, for H2=O(ǫ0), azimuthal symmetry isbrokenatleadingorder,andthestaticprofileisparallel to the transversefield (the alternativesolution with φ0= πis unstable). The solution of (35) is given by tanθ0 2=κ H2tanh/bracketleftbigg tanh−1/parenleftbiggH2−1 κ/parenrightbigg −κ 2ξ/bracketrightbigg +1 H2, (36)4 whereκ=/radicalbig 1−H2 2. At order ǫwe obtain the linear inhomogeneous equa- tions Lθ1=α 1+α2vθ′ 0−h1sinθ0, (37) Mφ1=1 1+α2v(cosθ0)′, (38) where L=−d2 dξ2+θ′′′ 0 θ′ 0, M=−d dξsin2θ0d dξ+H2sinθ0.(39) Hereθ0is given by (36), and θ1,φ1are requiredto vanish asξ→ ±∞. As above, the Fredholm alternative implies that the right-hand side of (37) must be orthogonal to θ′ 0 in order for a solution to exist. Calculation yields V=−/parenleftbigg α+1 α/parenrightbigg (1−(H2/κ)cos−1H2)−1H1.(40) ForH2= 0, this coincides with (30); thus, (40) gives H2-nonlinear corrections to the velocity. Moreover, it is straightforward to show that (40) is consistent with the general identity (10). Finally, one can also show that Mhas trivial kernel with spectrum bounded away from zero, so that (38) is automatically solvable. ItisinterestingtocomparetheDWvelocitywithtrans- verse field to the Walker case. From (5) and (40), VW/V=γ(1−(H2/κ)cos−1H2)<1.(41) Thus, to leading orderin H1, the DWvelocity in a uniax- ial wire with transverse field exceeds the Walker velocity. Numerical results below establish that this continues to hold asH1approaches the critical field H1c. Numerical results Toverifyouranalyticalresults, wesolvetheLLGequa- tion (1) using a finite-difference scheme on a domain −L≤x≤LwhereL= 100 (the DW has width of order 1). Neumann boundary conditions, m′= 0, are maintained at the endpoints. The damping parameter α is taken to be 0 .1 throughout. As initial condition we take the stationary profile, with θ0given by (36) and φ0= 0. After an initial transient period, during which the asymptotic values of matx→ ±Lconverge to m±, a stable solution emerges, in which the DW propagates with a characteristic mean velocity V. (For convenience, wehavetaken H1<0, sothat Vis positive.) In Figure1, numerically computed values of Vare plotted as a func- tion of|H1|for three fixed values of the transverse field: H2= 0.2,H2= 0.1, and the limiting case H2= 0, where the dynamics is given by the precessing solution. There is good quantitative agreement with the analytic results for small transverse fields, (30), for |H1|< H1,c, and(34), for |H1|> H1,c, In Figure 2, the analytic expres- sions for the velocity for small and moderate transverse fields are compared to numerical results for H2= 0.2 and |H1| ≪H1c. The moderate-field expression (40), which depends nonlinearly in H2, gives excellent agreement for smalldrivingfields. Fornonzero H2, the velocityexhibits a peak at a critical field |H1c|, which depends on H2. 0 0.1 0.2 0.3 0.400.050.10.150.20.250.30.35 |H1|V H2= 0.2 H2= 0.1 H2= 0 FIG. 1: Average DW velocity Vas a function of the driving field|H1|for three values of the transverse field H2. The an- alytic formulas (solid curves) (30), for |H1|< H1,c, and (34), for|H1|> H1,c, are plotted against numerically computed values (open circles). For H2= 0, the analytic formula is exact. 00.005 0.01 0.015 0.02 0.02500.050.10.150.20.250.3 |H1|V Small −H2theory Moderate −H2theory Numerics FIG. 2: DW velocity Vas a function of the driving field |H1|forH2= 0.2. The expressions for small-transverse field (30) (red curve) and moderate-transverse field (40) (light b lue curve) are plotted against numerically computed values (op en circles). Figure 3 shows the dependence of the critical field |H1,c|5 onH2, in close agreement with the analytic result (29). 0 0.05 0.1 0.15 0.200.0050.010.0150.020.0250.030.035 H2|H1,c| FIG. 3: The critical driving field |H1,c|as a function of the transverse field H2. A linear fit (blue curve) through the nu- merically computed data (blue diamonds) is plotted alongsi de the analytical result (29) (red curve). FIG. 4: The magnetization distribution, θ(x,t) andφ(x,t), for two values of the driving field: H1=−0.01 in figures (a) and (b), and H1=−0.05 in figures (c) and (d). The transverse field is taken as H2= 0.1 throughout. As in the Walker case, the properties of the propagat- ing solution are qualitatively different for driving fields |H1|below and above the critical field. This is confirmed in Figure 4, which shows contour plots of the magne- tization in the ( x,t)-plane. Figs. 4(a) and 4(b), where H1=−0.01, exemplify the case |H1|<|H1c|. The mag- netisation evolves as a fixed profile translating rigidly with velocity V. For|H1|>|H1c|, as exemplified by Figs. 4(c) and 4(d), where H1=−0.05. the magnetiza- tion profile exhibits a non-uniform precession as it prop-agates along the nanowire, with mean velocity in good agreement with (34). Summary We haveestablished, both analyticallyin leading-order asymptotics and numerically, the existence of travelling waveandoscillatingsolutionsoftheLLGequationinuni- axial wires in applied fields with longitudinal and trans- verse components. We have obtained analytic expres- sions for the velocity, (30) and (40), and for the critical longitudinal field, (29), above which the travelling wave solution ceases to exist. We have also obtained the mean precessional and linear velocities (33) and (34) for oscil- lating solutions. The analytic results are confirmed by numerics. [1] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). [2] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, T. Shinjo, Phys. Rev. Lett. 92077205 (2004). [3] D.A. Allwood, G. Xiong, C.C. Faulkner, D. Atkinson, D. Petit and R.P. Cowburn, Science 309, 1688 (2005). [4] R.P. Cowburn, Nature (London) 448, 544 (2007). [5] G.S.D Beach, C. Nistor, C. Knutson, M. Tsoi, and J.L. Erskine, Nature Mater. 4, 741(2005). [6] J. Yang, C. Nistor, G.S.D. Beach, and J.L. Erskine, Phys. Rev. B 77, 014413 (2008). [7] M.T. Bryan, T. Schrefl, D. Atkinson, D.A. Allwood, J. Appl. Phys. 103, 073906 (2008). [8] O.A. Tretiakov, D. Clarke, Gia-Wei Chern, Ya. B. Baza- liy and O. Tchernyshyov, Phys. Rev. Lett. 100127204 (2008). [9] X.R. Wang, P. Yan, J. Lu, Europhys. Lett. 86, 67001 (2009). [10] X.R. Wang, P. Yan , J. Lu, C. He, Ann. Phys. 324, 1815– 1820 (2009). [11] J. Lu and X.R. Wang, J. Appl.Phys. 107, 083915 (2010). [12] A. Mougin, M. Cormier, J.P. Adam, P.J. Metaxas and J. Ferr´ e, Europhys. Lett. 78, 57007 (2007). [13] Z. Li and S. Zhang, Phys. Rev. Lett. 92207203 (2004). [14] A. Thiaville, Y. Nakatani,, J. Miltat and Y. Suzuki, Eu- rophys. Lett. 69, 990 (2005). [15] G.S.D Beach, C. Knutson, C. Nistor, M. Tsoi, and J.L. Erskine, Phys. Rev. Lett. 97057203 (2006) [16] S. S. P. Parkin, M. Hayashi and L. Thomas, Science 320, 190 (2008). [17] M. Hayashi, L. Thomas, R. Moriya, C. Rettner and S. S. P. Parkin, Science 320, 209 (2008). [18] L. Thomas, R. Moriya, C. Rettner, S. and S. P. Parkin, Science330, 1810 (2010). [19] O.A. Tretiakov and Ar. Abanov, Phys. Rev. Lett. 105157201 (2010). [20] O.A. Tretiakov, Y. Liu and Ar. Abanov, Phys. Rev. Lett. 108247201 (2012). [21] V. Slastikov and C. Sonnenberg, IMA J. Appl. Math. 77 no. 2, 220 (2012)6 [22] A. Hubert and R. Sch¨ afer, Magnetic Domains: The Analysis of Magnetic Microstructures (Springer, Berlin, 1998). [23] Z. Z. Sun and J. Schliemann, Phys. Rev. Lett. 104, 037206 (2010). [24] A. Goussev, J.M. Robbins, V. Slastikov, Phys. Rev. Lett . 104, 147202 (2010). [25] Y. Gou, A. Goussev, J. M. Robbins, V. Slastikov, Phys. Rev. B84, 104445 (2011)[26] L. D. Landau and E. M. Lifshitz, Phys. Zeitsch. Sowietu- nion8, 153 (1935). [27] T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans. Mag.40, 3443 (2004). [28] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep.194, 117 (1990). [29] A. Goussev, R. Lund, J.M. Robbins, C. Sonnenberg, V. Slastikov, in preparation.

2012-06-21

Under a magnetic field along its axis, domain wall motion in a uniaxial nanowire is much slower than in the fully anisotropic case, typically by several orders of magnitude (the square of the dimensionless Gilbert damping parameter). However, with the addition of a magnetic field transverse to the wire, this behaviour is dramatically reversed; up to a critical field strength, analogous to the Walker breakdown field, domain walls in a uniaxial wire propagate faster than in a fully anisotropic wire (without transverse field). Beyond this critical field strength, precessional motion sets in, and the mean velocity decreases. Our results are based on leading-order analytic calculations of the velocity and critical field as well as numerical solutions of the Landau-Lifshitz-Gilbert equation.

Fast domain wall propagation in uniaxial nanowires with transverse fields

1206.4819v2

Nonlinear features of the superconductor{ferromagnet{superconductor '0Josephson junction in ferromagnetic resonance region Aliasghar Janalizadeh1, Ilhom R. Rahmonov2;3;4, Sara A. Abdelmoneim5, Yury M. Shukrinov2;3;4, and Mohammad R. Kolahchi1 1Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45137-66731, Zanjan, Iran 2BLTP, JINR, Dubna, Moscow Region, 141980, Russia 3Dubna State University, Dubna, 141980, Russia 4Moscow Institute of Physics and Technology, Dolgoprudny, 141700, Moscow Region, Russia 5Physics department, Menoya University, Faculty of Science, 32511, Shebin Elkom,Egypt (Dated: October 4, 2022) We demonstrate the manifestations of the nonlinear features in magnetic dynamics and IV- characteristics of the '0Josephson junction in the ferromagnetic resonance region. We show that at small values of system parameters, namely, damping, spin-orbit interaction, and Josephson to magnetic energy ratio, the magnetic dynamics is reduced to the dynamics of the scalar Dung os- cillator, driven by the Josephson oscillations. The role of increasing superconducting current in the resonance region is claried. Shifting of the ferromagnetic resonant frequency and the reversal of its damping dependence due to nonlinearity are demonstrated by the full Landau-Lifshitz-Gilbert- Josephson system of equations, and in its dierent approximations. Finally, we demonstrate the negative dierential resistance in the IV{characteristics, and its correlation with the foldover eect. I. I. INTRODUCTION The coupling of superconducting phase dierence with magnetic moment of ferromagnet in the '0junction leads to a number of unique features important for supercon- ducting spintronics, and modern information technology [1{5]. It allows to control the magnetization preces- sion by superconducting current and aects the current{ voltage (IV) characteristics by magnetic dynamics in the ferromagnet, in particular, to create a DC component in the superconducting current [6{8]. A remarkable mani- festation of such coupling is the possibility to stimulate a magnetization reversal in the ferromagnetic layer by applying current pulse through the '0-junction [3, 9{13]. There are two features of our Josephson junction that come into play in our study. One is the broken inver- sion symmetry in the weak link of the Josephson junc- tion, when the link is magnetic, which introduces an ex- tra phase in the current|-phase relation, preventing it from being antisymmetric. Such Josephson junctions are named'0junctions [1], and examples exist such as MnSi and FeGe. Second is the nonlinear property of the system that makes for an anomalous resonance behavior [14]. We couple such a Josephson junction to the model that describes the magnetodynamics in thin lms or heterostructure, to form the Landau-Lifshitz-Gilbert- Josephson model (LLGJ)[14{16]. It is shown that for a particular set of parameters, the coupled equations reduce to the dynamics of a Dung oscillator [14]. The cubic nonlinearity in this oscillator has applications in describing several eects in other models too [17]. One being the resonance eects in the antiferromagnetic bimeron in response to an alternating current, which has applications in the detection of weak signals [15, 18, 19]. The Gilbert damping term is added phenomenologi- cally to the Landau|-Lifshitz model, to reproduce the damping of the precessing magnetic moment. Gilbertdamping is important in modeling other resonance fea- tures too, as its temperature dependence aects them [20, 21], and in return in the superconducting correla- tions that aect it [22]. The magnetization precession in the ultra thin Co20Fe60B20layer stimulated by mi- crowave voltage under a large angle, needs modeling by Dung oscillator too. This gets help from the so called foldover features, again due to nonlinearity [16, 23, 24]. The consequences of the nonlinear nature of the cou- pled set of LLGJ system of equations in the weak cou- pling regime was demonstrated recently in Ref. [14]. We showed in this regime, where the Josephson energy is small compared to the magnetic energy, the '0Joseph- son junction is equivalently described by a scalar non- linear Dung equation. An anomalous dependence of the ferromagnetic resonant frequency (FMR) with the increase of the Gilbert damping was found. We showed that the damped precession of the magnetic moment is dynamically driven by the Josephson supercurrent, and the resonance behavior is given by the Dung spring. The obtained results were based on the numerical simu- lations. The role of dc superconducting current, and the state with negative dierential resistance (NDR) in IV- characteristic were not claried. Also, the eects of the Josephson to magnetic energy ratio and the spin-orbit coupling (SOC) were not investigated at that time. In the present paper, we study the nonlinear aspects of the magnetic dynamics and IV-characteristics of the '0Josephson junction in the ferromagnetic resonance re- gion. We compare description of the anomalous damp- ing dependence (ADD) exhibited by full LLGJ system of equations with approximated equations and demon- strate the Dung oscillator features in the small param- eter regime. Eects of the Josephson to magnetic energy ratio, and the spin-orbit coupling on the ADD, referred to earlier as the -eect [14] are demonstrated. By de- riving the formula which couples the dc superconduct-arXiv:2210.00366v1 [cond-mat.supr-con] 1 Oct 20222 ing current and maximal amplitude of magnetization we discuss the correlation of superconducting current and the negative dierential resistance in the resonance re- gion. Finally, we discuss the experimentally important features by emphasizing the details of the magnetization dynamics and the IV-characteristics of the '0junction. We have shown that in the limit of small system pa- rameters; that is, the Josephson to magnetic energy ra- tioG, the damping , and the spin-orbit coupling r, the dynamics is given by the Dung spring [14]. We focus on the shift in resonance and the eects of nonlinear in- teractions. We give semi-analytic models to explain our results in various limits. The paper is organized as follows. In Section II we outline the theoretical model and discuss the methods of calculations. The ferromagnetic resonance and ef- fects of system parameters on the anomalous damping dependence are considered in Subsection A of Section III. In Subsection B we present analytical description of the dynamics and IV-characteristics of the '0junction at small system parameters. Manifestation of the nega- tive dierential resistance in IV-characteristics through the foldover eect is discussed. We compare the de- scription of the anomalous damping dependence by full LLGJ system of equation with approximated equation, and show how the Dung oscillator captures the non- linearities in the small parameter regime in Subsection C. We present results on the critical damping and de- rive the formula which couples the dc superconducting current and maximal amplitude of magnetization in the ferromagnetic layer. Finally, in Section IV we concludes the paper. II. II. MODELS AND METHOD The following section is closely related to our work in [13]. The '0junction [6, 12, 25] that we study is shown in Fig.1. The current-phase relation in varphi 0junction has the form Is=Icsin (''0), where'0=rMy=M0, Mydenotes the component of magnetic moment in ^ ydi- rection,M0is the modulus of the magnetization. The physics of'0Josephson juncton is determined by system of equations which consists of Landau-Lifshits-Gilbert (LLG), resistively capacitively shunted junction (RCSJ) model expression with current-phase relation ( Is) de- scribed above, and Josephson relation between phase dif- ference and voltage. The dynamics of the magnetic moment Mis described by the LLG equation [26] dM dt= MHeff+ M0 MdM dt ; (1) where Mis the magnetization vector, is the gyromag- netic relation, Heffis the eective magnetic eld, is Gilbert damping parameter, M0=jMj. Figure 1. Schematic view of SFS '0Josephson junction. The external current applied along x direction, ferromagnetic easy axis is along z direction. In order to nd the expression for the eective mag- netic eld we have used the model developed in Ref.[6], where it is assumed that the gradient of the spin-orbit potential is along the easy axis of magnetization taken to be along ^z. In this case the total energy of the system can be written as Etot=0 2'I+Es(';' 0) +EM('0); (2) where'is the phase dierence between the supercon- ductors across the junction, Iis the external current, Es(';' 0) =EJ[1cos (''0)], andEJ=  0Ic=2 is the Josephson energy. Here  0is the ux quantum, Icis the critical current, r=lso=Fl= 4hL=~F,L is the length of Flayer,his the exchange eld of the Flayer,EM=KVM2 z=(2M2 0), the parameter so=F characterizes a relative strength of spin-orbit interaction, Kis the anisotropic constant, and Vis the volume of the ferromagnetic ( F) layer. The eective eld for LLG equation is determined by He=1 V@Etot @M = F  Grsin 'rMy M0 by+Mz M0bz (3) where F= K=M 0is frequency of ferromagnetic reso- nance andG=EJ=(KV) determines the ratio of Joseph- son energy to magnetic one. In order to describe the full dynamics '0junction the LLG equations should be supplemented by the equation for phase dierence ', i.e. equation of RCSJ model for bias current and Josephson relation for voltage. Accord- ing to the extended RCSJ model, which takes into ac- count derivative of '0phase shift, the current owing through the system in underdamped case is determined by I=~C 2ed2' dt2+~ 2eRd' dtr M0dMy dt (4) +Icsin 'r M0My : whereIis the bias current, CandRare the capacitance and resistance of Josephson junction respectively. The3 Josephson relation for voltage is given by : ~ 2ed' dt=V: (5) We note that in the framework of RCSJ{model the displacement current is proportional to the rst deriva- tive of voltage (or second derivative of phase dierence). From the other hand, the magnetization dynamics plays role of the external force and rst order derivative of '0 is a source of external current for JJ. This was demon- strated in Ref.[25, 27] where the authors included the rst derivative of '0as the source of the electromotive force. Voltage is determined by the phase dierence, and does not depend on '0. From this point of view, in the frame- work of RCSJ model the external current source cannot modify the expression for displacement current. That's why we do not include the second derivative of varphi 0 in our model. Using (1), (3), (4) and (5) we can write the system of equations, in normalised variables, which describes the dynamics of '0junction _mx=!F 1 +2fmymz+Grm zsin('rmy) [mxm2 z+Grm xmysin('rmy)]g; _my=!F 1 +2fmxmz [mym2 zGr(m2 z+m2 x) sin('rmy)]g; _mz=!F 1 +2fGrm xsin('rmy) [Grm ymzsin('rmy)mz(m2 x+m2 y)]g; _V=1 c[IV+r_mysin('rmy)]; _'=V(6) wheremx;y;z =Mx;y;z=M0and satisfy the constraintP i=x;y;zm2 i(t) = 1,c= 2eIcCR2=~is McCumber pa- rameter. In order to use the same time scale in the LLG and RCSJ equations in this system of equations we have normalized time to the !1 c, where!c=2eIcR ~, and!F= F=!cis the normalized frequency of ferro- magnetic resonance F= K=M 0. Bias current is nor- malized to the critical current Icand voltage V{ to the Vc=IcR. The system of equations (6), is solved numer- ically using the fourth-order Runge-Kutta method(see Ref.[14]). III. III. RESULTS AND DISCUSSION A. A. Eect of system parameters on the anomalous damping dependence ADD of the FMR frequency with increasing was dis- cussed in Ref. [14]. It was found that the resonance curves demonstrate features of Dung oscillator, re- ecting the nonlinear nature of Landau-Lifshitz-Gilbert- 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8mymax Valpha=0.01 alpha=0.02 alpha=0.03 alpha=0.04 alpha=0.05 alpha=0.06 alpha=0.07 alpha=0.08 alpha=0.09 alpha=0.1 0 0.1 0.5Figure 2. Maximal amplitude of magnetization mycomponent at each values of bias current and voltage along IV-characteristics of the '0junction in the ferromag- netic resonance region for various . Inset enlarges the main maximum. Parameters: c= 25,G=0.05,r=0.05, !F= 0:5. Josephson (LLGJ) system of equations. There is a criti- cal damping value at which anomalous dependence comes into play. This critical value depends on the system pa- rameters. Here we present the details of such transforma- tion from usual to anomalous dependence with variation in spin-orbit coupling and ratio of Josephson to magnetic energy. To investigate the eect of damping, we calculate the maximal amplitude of magnetization component my taken at each value of the bias current based on the LLGJ system of equations (6). In Fig.2 we show the voltage dependence of maximal amplitude mmax yin the ferromagnetic resonance region at dierent damping pa- rameter and small values of Josephson to magnetic en- ergy ratio G=0.05 and spin-orbit coupling r= 0:05. We found that the ferromagnetic resonance curves demon- strate the dierent forms. An increase in damping shows a nonuniform change in the resonant frequency: it is ap- proaching the !Finstead of moving away with increase in. We stress that this happens at small Gandr. We consider that such behavior can be explained by the non- linear nature of the LLGJ system of equations. There is a manifestation of subharmonics of the FMR in Fig.2 at != 1=2;1=3;1=4. We usually expect the resonance peak to move away from resonance as the increases. Figure 2 shows that this normal eect is accompanied with an anomalous be- haviour as can be seen in the inset to this gure, where the resonance peak approaches !Fasincreases [14]. The manifestation of FMR in IV-characteristics of the '0junction at three values of damping parameter is demonstrated in Fig. 3. The strong deviation of the IV-curve is observing at = 0:01, which is characteristic4 Figure 3. Part of the IV characteristic of the '0junction atG= 0:05;r= 0:05 and dierent values of Gilbert damp- ing. The numbers show value. Inset shows the total IV- characteristic and arrow indicates the resonance region value for many magnetic materials. This fact indicates that ADD can be observed experimentally by measuring IV-characteristics in wide interval of the damping param- eter. Interesting features of ADD appear by a variation of spin-orbit coupling. As it was demonstrated in Ref.[28], an increase in SOC leads to the essential change in IV- characteristics and magnetization precession in the fer- romagnetic resonance region. The nonlinearity is going stronger and the state with negative dierential resis- tance appears at large SOC. Figure 4(a) demonstrates results of numerical simu- lations ofmmax ydependence on at dierent values of SOC parameter r. It shows two specic features of ADD. First, with an increase in r, the critical value of Vpeakis decreasing (the curve moves away from !F). The sec- ond important feature is an increasing of critwhich is demonstrated in this gure by arrows. Another model parameter which aects the phe- nomenon discussed in the present paper is the ratio G of Josephson to magnetic energies. Figure 4(b) demon- strates the results of numerical simulations of mmax yde- pendence on at dierent values of G. Similar to the eect of r, increasing Galso causes the value ofcritto increase. By changing the volume of the ferromagnetic layer, the ferromagnetic energy and con- sequently the value of G can be changed [6]. For small G, i.e. a situation where the magnetic energy is much larger than the Josephson energy, the magnetic layer re- ceives less energy, and its amplitude decreases in the y direction, and also the maximum value of the oscillation frequency is closer to the magnetic frequency, !F. Vpeakα 0.46 0.47 0.48 0.49 0.500.050.10.150.2 αcrit 1 23 Vpeakα 0.46 0.47 0.48 0.49 0.500.050.10.150.2 αcrit 1 23Figure 4. (a) Demonstration of ADD at dierent values of SOC parameter ratG= 0:05. Numbers indicate: 1 - r= 0:05; 2 -r= 0:1; 3 -r= 0:5; Arrows show critical  value, corresponded to the reversal in the dependence (b) Demonstration of ADD at dierent values of the Josephson to magnetic energy ratio Gatr= 0:05. Numbers indicate: 1 -G= 0:01; 2 -G= 0:1; 3 -G= 1. B. B. Dynamics and IV-characteristics of the '0 junction at small system parameters As it was discussed in Refs.[6, 29, 30], in case of G;r;<< 1 andmz1, rst three equations of the sys- tem (6) can be simplied. Taking into account '=!Jt and neglecting quadratic terms of mxandmy, we get ( _mx=!F[my+Grsin(!Jt)mx] _my=!F[mxmy];(7) This system of equations can be written as the second order dierential equation with respect to my my+ 2!F_my+!2 Fmy=!2 FGrsin!Jt: (8)5 Corresponding solution for myhas the form my(t) =!+! rsin(!Jt) ++ rcos(!Jt);(9) where !=Gr2!F 2!J!F ; (10) and =Gr2!F 2!J : (11) with = (!J!F)2+ (!J)2(see Ref.[6] and corre- sponded Erratum[31]). When the Josephson frequency !Jis approaching the ferromagnetic one !F,mydemonstrates the damped fer- romagnetic resonance. Dierential resistance in the res- onance region is decreasing and it is manifested in the IV{characteristic as a resonance branch [7]. Taking into account rmy<<1, we rewrite expression for superconducting current as Is(t) = sin(!Jtrmy(t)) = sin(!Jt)rmycos(!Jt) (12) Using solution (9) we can obtain Is(t) = sin!Jt!+! 2sin 2!Jt + ++ 2cos 2!Jt+I0() (13) where I0= ++ 2: (14) This superconducting current explains the appearance of the resonance branch in the IV{characteristic. The generated current I0can be expressed through the am- plitude ofmyand SOI parameter r I0=r 2mmax y(!J); (15) withmmax y(!J) being the frequency response of my. At small model parameters <<Gr<< 1 of SFS'0 Josephson junction, the states with a negative dierential resistance appear in the IV-characteristics in the FMR region. Due to the nonlinearity, the resonance peak is asymmetric. Increasing of nonlinearity leads to the bista- bility (foldover eect). A natural questions appears if the states with a negative dierential resistance are the origin of the foldover and ADD. In order to clarify this question, we show in Fig.5 a part of IV{characteristics of '0junc- tion together with IV-characteristic of SIS junction in the ferromagnetic resonance region and numerically cal- culated superconducting current through this junction. The total IV{characteristics are demonstrated in the in- set to this gure. Figure 5. IV-characteristic of '0and SIS junctions and calcu- lated average superconducting current through the '0junc- tion. We see the correlation of the foldover eect in super- conducting current (blue) with the NDR part of IV-curve. The peak in the superconducting current and minimum of IV-curve are at the same voltage value. So, both ef- fects re ect the nonlinear features of the ferromagnetic resonance in '0junction. But in contrast to the foldover and ADD eects, which start manifesting themselves at relatively small deviations from linear case, the nonlin- earity in case of the NDR plays a more essential role. We note that in the resonance region for considered limit of model parameters the myamplitude are coupled to the value of superconducting current (see Eq.(15)). We stress the importance of the performed analysis demon- strating the analytical coupling of time independent su- perconducting current and magnetization, re ecting the Dung oscillator features of the '0junction. As it is well known, the states with negative dieren- tial resistance appear in IV-characteristics of Josephson structures in dierent physical situations. In particular, the nonlinear superconducting structures being driven far from equilibrium exhibit NDR states [32]. The NDR states plays an essential role in the applications related to the THz radiation emission [33]. A detailed expla- nation for dierent types of negative dierential resis- tance (NDR) in Josephson junction (i.e., N-shaped and S-shaped) is introduced in[34]. The authors emphasize that the nonlinear behavior of the Josephson junction plays a key role in the NDR feature. In our case, the NDR states appear as a result of system's nonlinearity at small values of '0junction parameters, such as SOC, ratio of Josephson to magnetic energy and Gilbert damp- ing. We demonstrate these eects here by by presented results of detailed investigation of the NDR state at dif- ferent system parameters and discuss possibility of their control near the ferromagnetic resonance. Figure 6 shows the eect of the spin-orbit coupling on the IV-characteristic at G= 0:05 and= 0:01. We see the NDR feature which is getting more pronounced with increase in r. A further increase in rleads to the jump down in voltage and then practically linear growth6 IV 0.49 0.495 0.5 0.505 0.510.460.470.480.490.09 0.15 IV 0 0.5 100.20.40.60.811.2r =0.15 Figure 6. Enlarged parts of the IV-curves in the resonance region at dierent values of SOC parameter rat= 0:01 andG= 0:05. The numbers indicate the increasing of rfrom 0.09 to 0.15 with an increment 0.01. Inset demonstrates total IV-characteristic ar r= 0:15 of IV-characteristic. An interesting question is concerning the eect of Gilbert damping. Results of IV-characteristics simula- tions in the resonance region at a certain range of the damping parameter atG= 0:05 andr= 0:13 are shown in Fig.7(a). In this case, the most pronounced characteristic appears at = 0:01. At chosen set of pa- rametersG= 0:05 andr= 0:13, the range of with pronounced NDR features is 0 :016<0:014. The maximal amplitude mmax yas a function of volt- age is shown in the Fig.7(b). Based on the results, pre- sented in Fig.7(a) and Fig.7(b), we came to the important conclusion that the foldover eect (bistability) and NDR state have strong correlations and have the same origin related to the nonlinearity at small system parameters. But anomalous damping dependence does not show a one-to-one correlation with either negative dierential re- sistance or foldover eect. The resonance peak positions ofmmax yin bias current Ipeak, and in voltage Vpeakas the functions of are demonstrated in Fig.7(c). According to our results, we can divide interval into two regions (see Fig.7(c)). Region I introduces the values of where the NDR feature is present, while in the Region II it dis- appears. In the Region II the foldover eect (bistability) disappears as well, but ADD is realized. C. C. Dung oscillator features of '0junction and critical damping The system of equations (6) is nonlinear and very com- plex, so in order to provide analytical study of dynamics of the'0junction we need to derive an approximated equation for some limited values of model parameters. In Ref. [14] was shown that the resonance curves demon- strate features of Dung oscillator, re ecting the nonlin- ear nature of Landau-Lifshitz-Gilbert-Josephson (LLGJ) IV 0.495 0.5 0.505 0.510.4650.470.4750.480.4850.49 0.01(a) 0.018 IV 0 0.5 100.20.40.60.811.2 α= 0.01 αIpeak, Vpeak 0.01 0.012 0.014 0.016 0.018 0.020.470.480.490.5Ipeak Vpeak(c) ΙΙΙFigure 7. (a) Enlarged parts of the IV-curves at dierent values of; (b) Voltage dependence of mmax yat dierent ; (c)-dependence of the resonance curve maximum in current (Ipeak) and voltage ( Vpeak). The numbers indicate the value offrom 0.01 to 0.018 in (a) and from 0.01 to 0.02 in (b) by an increment 0.001. Results are obtained at r= 0:13, and G= 0:05. system of equations. In this section, we present analyt- ical approach to describe the nonlinear dynamics of '0- junction and compare analytical results followed from ap- proximated Dung equation with numerical simulations of total system of equations (6). We show that in the limit of << G;r << 1, we arrive to the Dung os- cillator. We start by rst three equations of the (6) for7 magnetization components _mx !F=mymz+Grm zsin('rmy)mxm2 z _my !F=mxmzmym2 z (16) _mz !F=Grm xsin('rmy) +mz(m2 x+m2 y) Simplifying this system of equations by the same pro- cedure as it was done in Ref.[14] we can write equation formyas my+2_my+2(1+2)my2(1+24)m3 y=2 sin': (17) Finally, by neglecting the 2and4terms which are much smaller than 1, we come to well known Dung equation, my+ 2!F_my+!2 Fmy!2 Fm3 y=!2 FGrsin': (18) In the small parameter range, this Dung equation can describe the dynamics of my. We will have the full dynamics, once we consider the coupling with the Joseph- son equation, '+1 c[ _'r_my+ sin('rmy)] =1 cI: (19) The system of equations, (18) and (19) can replace the LLGJ equations in the limit of G;r<< 1 andG;r>> . Taking into account '=!Jtwe can right analytically obtained frequency response for equation (18) (mmax y)2= Gr
2  !21 +3 4(mmaxy)22+ 2!
2 (20) where!=!J=!F. From Eq. (20) we get (mmax y)6+8 3(!21)(mmax y)4 +4 32 (!21)2+ 2!
2 (mmax y)2 4 3Gr2 = 0: (21) This equation allows to determine analytically fre- quency dependence of the mmax yamplitude. To nd it we solve the equation (21) by the Newton method. Re- sults of analytical calculations (blue dots) corresponded to (21) and numerical one (red doted line) corresponded to the full system of equation (6) are demonstrated in Fig.8. Figure 8. Numerically (curve 1) and analytically (curve 2) calculated amplitude dependence of my. Figure 9. Numerically calculated superconducting current for SFS junction (plot 1) and analytical I0(plot 2) and super- conducting current for SIS junction (plot 3). We can see that they are close to each other which proves the correctness of the chosen approximation. Both curves demonstrate an asymmetric resonance peak, which is common for Dung oscillator. When a role of the cubic term is getting larger, we observe a bistability of the resonance curve, which is usually called a foldover eect. Note that the foldover eect can be also achieved by the damping decreasing; i.e., by the decreasing of dis- sipative term in (18), we can increase the in uence of the cubic term in this equation. The comparison of analytically and numerically cal- culated superconducting current as a function of the Josephson frequency is demonstrated in Fig. 9. We note that in our normalization V=!J. We can see the man- ifestation of the asymmetric resonance peak in the fre- quency dependence of superconducting current. So, the approximated system of equations 7 re ects one of the main feature of Dung oscillator. Figure (10) compares anomalous damping dependence of the resonance peak of mmax y(V) calculated numeri- cally according to the full LLGJ system of equations (6) with calculated numerically according to the generalized Dung model (equations (17, 19)). We see that in the damping parameter interval [0.001 { 0.2] the coincidence8 Figure 10. The -dependence of the resonance maximum of mmax y(V) in the damping parameter interval [0.001 { 0.12]. Green squares show results calculated numerically according to the full system of equations (6), blue circles show results calculated numerically according to the generalized Dung and Josephson equations (17,19). The dashed line connects the symbols to guide eyes. Solid line show analytical - dependence calculated according to the Eq. (22). All calcu- lation have been done at c= 25, G=0.05, r=0.05, !F= 0:5. of the dependences is enough good. Using equation (18) with '=!Jt, we can nd (see Supplementary materials ??) a relation between posi- tion of the resonance peak in mmax y(V) dependence and damping !peak=s 132 2+1 2r (12)212(Gr 4)2(22) where!peak=!J;peak !Fdetermines the position of the res- onance peak. Equation (22) allows to nd the formula for critical dampingcritwhich is an important parameter deter- mining the reversal point in damping dependence of the resonance peak of mmax y(V) . Taking into account equation (22) we can write equa- tion with respect of Gr=(4) (See supplementary mate- rials??). 9Gr 4crit4 + 32 crit(102 crit1)Gr 4crit2 (23) 24 crit(2 crit1)2= 0 Using approximation 10 2 crit<<1 and2 crit<<1 it gives (see Supplementary Materials) crit1 2sr 3 2Gr (24) Figure 11. Numerical calculations according to Eq. (6) (squares), analytical according to Eq. (23)(solid line) and approximated analytical according to Eq. (24) (dashed line). Table 1: A comparison between the numerical and an- alytical values of crit:at dierent values of Gandr. G r Grcrit:;numerics crit:;analytics 0.01 0.05 0.0005 0.0100 0.0123 0.05 0.05 0.0025 0.0300 0.0276 0.05 0.10 0.0050 0.0400 0.0391 0.05 0.30 0.0150 0.0700 0.0677 0.05 0.50 0.0250 0.0900 0.0874 0.10 0.05 0.0050 0.0391 0.0391 0.60 0.05 0.0300 0.0950 0.0958 0.70 0.05 0.0350 0.1000 0.1035 1.00 0.05 0.0500 0.1200 0.1237 Figure 11 presents comparison of numerical and ana- lytical results critversusGr. As we see, it shows a good agreement of numerical and analytical results of calculations at small product of Josephson to magnetic energy ratio and spin-orbit inter- action. IV. IV. CONCLUSIONS The understanding of the nonlinear features of magnetization dynamics in superconductor-ferromagnet- superconductor Josephson junction and their manifesta- tion in the IV-characteristics has implications for super- conductor spintronics, and modern information technol- ogy. In'0junctions the nonlinear features can aect the control of magnetization precession by superconducting current and external electromagnetic radiation [28]. Here, using numerical and analytic approaches, we have demonstrated that at small system parameters,9 namely, the damping, spin-orbit interaction and Joseph- son to magnetic energy ratio in '0junction, magnetic dy- namics is reduced to the dynamics of the scalar Dung oscillator, driven by the Josephson oscillations. We have claried the role of increasing superconducting current in the resonance region leading to the foldover eect in the ferromagnet magnetization. We have demonstrated the parameter dependence of the anomalous ferromag- netic resonant shifting with anomalous damping depen- dence due to nonlinearity of the full LLGJ equation and in its dierent approximations. We have derived the an- alytical expression for critical damping value. Also, we demonstrated appearance of negative dierential resis- tance in the IV-characteristics and the correlation with occurrence of the foldover eect in the magnetization of ferromagnet. We have stressed that the manifestation of negative dierential resistance is related to the nonlinear features of the system[34, 35]. It was demonstrated that in the small model parameters case the equation for magnetic subsystem takes form of Dung equation where nonlin- earity manifest itself as the cubic term. We have shown that the appearance of negative dierential resistance in the I-V curve is related to the appearance of foldover inthemmax y-Vcurve. We believe that the experimentally measured IV- characteristics of '0junction with manifestations dis- cussed in detail in the present paper, would allow close investigations of its nonlinear features important for su- perconductor electronics and spintronics. V. 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2022-10-01

We demonstrate the manifestations of the nonlinear features in magnetic dynamics and IV-characteristics of the $\varphi_0$ Josephson junction in the ferromagnetic resonance region. We show that at small values of system parameters, namely, damping, spin-orbit interaction, and Josephson to magnetic energy ratio, the magnetic dynamics is reduced to the dynamics of the scalar Duffing oscillator, driven by the Josephson oscillations. The role of increasing superconducting current in the resonance region is clarified. Shifting of the ferromagnetic resonant frequency and the reversal of its damping dependence due to nonlinearity are demonstrated by the full Landau-Lifshitz-Gilbert-Josephson system of equations, and in its different approximations. Finally, we demonstrate the negative differential resistance in the IV--characteristics, and its correlation with the foldover effect.

Nonlinear features of the superconductor--ferromagnet--superconductor $\varphi_0$ Josephson junction in ferromagnetic resonance region

2210.00366v1

arXiv:1002.4958v1 [cond-mat.mes-hall] 26 Feb 2010Correlation Effects in Stochastic Ferromagnetic Systems Thomas Bose and Steffen Trimper Institute of Physics, Martin-Luther-University, D-06099 Halle, Germany∗ (Dated: June 16, 2018) Abstract We analyze the Landau-Lifshitz-Gilbert equation when the p recession motion of the magnetic moments is additionally subjected to an uniaxial anisotrop y and is driven by a multiplicative cou- pled stochastic field with a finite correlation time τ. The mean value for the spin wave components offers that the spin-wave dispersion relation and its damping is strongly influenced by the deter- ministic Gilbert damping parameter α, the strength of the stochastic forces Dand its temporal rangeτ. The spin-spin-correlation function can be calculated in t he low correlation time limit by deriving an evolution equation for the joint probability fu nction. The stability analysis enables us to find the phase diagram within the α−Dplane for different values of τwhere damped spin wave solutions are stable. Even for zero deterministic Gilbert d amping the magnons offer a finite life- time. We detect a parameter range wherethe deterministic an d the stochastic damping mechanism are able to compensate each other leading to undamped spin-w aves. The onset is characterized by a critical value of the correlation time. An enhancement of τleads to an increase of the oscillations of the correlation function. PACS numbers: 75.10.Hk, 05.40.-a, 75.30.Ds,72.70.+m,76.60.Es ∗thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de 1I. INTRODUCTION Magnetism can be generally characterized and analyzed on different length and time scales. The description of fluctuations of the magnetization, the occurre nce of damped spin waves and the influence of additional stochastic forces are successfully performed on a mesoscopic scale where the spin variables are represented by a continuous spa tio-temporal variable [1]. In this case a well established approach isbased uponthe Landau-L ifshitz equation [2] which describes the precession motion of the magnetization in an effective magnetic field. This field consists of a superposition of an external field and internal fie lds, produced by the in- teracting magnetic moments. The latter one is strongly influenced b y the isotropic exchange interaction and the magnetocrystalline anisotropy, for a recent r eview see [3]. The studies using this frame are concentrated on different dynamical aspects as the switching behav- ior of magnetic nanoparticles which can be controlled by external tim e-dependent magnetic fields [4] and spin-polarized electric currents [5, 6]. Such a current- induced spin transfer allows the manipulation of magnetic nanodevices. Recently, it has bee n demonstrated that an electric current, flowing through a magnetic bilayer, can induce a coupling between the layers [7]. Likewise, such a current can also cause the motion of magn etic domain walls in a nanowire [8]. Another aspect is the dynamical response of ferrom agnetic nanoparticles as probed by ferromagnetic resonance, studied in [9]. In describing all this more complex behavior of magnetic systems, the Landau-Lifshitz equation has t o be extended by the in- clusion of dissipative processes. A damping term is introduced pheno menologically in such a manner, that the magnitude of the magnetization /vectorSis preserved at any time. Furthermore, the magnetization should align with the effective field in the long time limit. A realization is given by [2] ∂S ∂t=−γ[S×Beff]−ε[S×(S×Beff)]. (1) The quantities γandεare the gyromagnetic ratio and the damping parameter, respectiv ely. An alternative equation for the magnetization dynamics had been pr oposed by Gilbert [10]. The Gilbert equation yields an implicit form of the evolution of the magne tization. A com- bination of both equations, called Landau-Lifshitz-Gilbert equation (LLG) will be used as the basic relation for our studies, see Eq. (2). The origin of the dam ping term as a non- relativistic expansion of the Dirac equation has been discussed in [11 ] and a generalization of the LLG for conducting ferromagnetics is offered in [12]. The form of the damping seems 2to be quite general as it has been demonstrated in [13] using symmet ry arguments for fer- roelectric systems. As a new aspect let us focus on the influence of stochastic fields. Th e interplay between current and magnetic fluctuations and dissipation has been studied recently in [14]. Via the spin-transfer torque, spin-current noise causes a significant en hancement of the magnetiza- tion fluctuations. Such a spin polarized current may transfer mome ntum to a magnet which leads to a spin-torque phenomenon. The shot noise associated with the current gives rise to a stochastic force [15]. In our paper we discuss the interplay betwe en different dissipation mechanism, namely the inherent deterministic damping in Eq. (1) and t he stochastic mag- netic field originated for instance by defect configurations giving ris e to a different coupling strength between the magnetic moments. Assuming further, tha t the stochastic magnetic field is characterized by a finite correlation time, the system offers m emory effects which might lead to a decoherent spin precession. To that aim we analyze a f erromagnet in the classical limit, i.e., the magnetic order is referred to single magnetic at oms which occupy equivalent crystal positions, and the mean values of their spins exh ibit a parallel orientation. The last one is caused by the isotropic exchange interaction which will be here supplemented by a magneto-crystalline anisotropy that defines the direction of t he preferred orientation. Especially, we discuss the influence of an uniaxial anisotropy. The co upling between differ- ent dissipation mechanisms, mentioned above, leads to pronounced correlations, which are discussed below. Due to the multiplicative coupling of the stochastic fi eld and the finite correlation time the calculation of the spin-spin correlation function is more complicated. To that aim we have to derive an equivalent evolution equation for the joint probability distribution function. Within the small correlation time limit this approa ch can be fulfilled in an analytical manner. Our analysis is related to a recent paper [16] in which likewise the stochastic dynamics of the magnetization in ferromagnetic nanopa rticles has been studied. Further, we refer also to a recent paper [17] where the mean first passage time and the relaxation of magnetic moments has been analyzed. Different to tho se papers our approach is concentrated on the correlation effects in stochastic system wit h colored noise. Our paper is organized as follows: In Sec.II we discuss the LLG and ch aracterize the ad- ditional stochastic field. The equations for the single and the two pa rticle joint probability distribution are derived in Sec.III. Using these functions we obtain t he mean value of the spin wave variable and the spin-spin correlation function. The phase diagram, based on the 3stability analysis, is presented in Sec.IV. In Sec.V we finish with some co nclusions. II. MODEL In order to develop a stochastic model for the spin dynamics in ferr omagnetic systems let us first consider the deterministic part of the equation of motion. W e focus on a description based upon the level of Landau-Lifshitz phenomenology [2], for a r ecent review see [3]. To follow this line we consider a high spin systems in a ferromagnet sufficien tly below the Curie temperature. In that regime the dynamics of the magnet are dominated by transverse fluctuationsofthespatio-temporalvaryinglocalmagnetization. Theweakexcitations, called spin waves or magnons, are determined by a dispersion relation, the wavelength of which should be large compared to the lattice constant a, i.e., the relation q·a≪1 is presumed to be satisfied, where qis the wavenumber. In this limit the direction of the spin varies slowly while its magnitude |S|=msremains constant in time. A proper description for such a situation is achieved by applying the Landau-Lifshitz-Gilbert equatio n (LLG) [4, 10, 18]. The spin variable is represented by S=msˆ n, whereˆ n(r,t) is a continuous variable which characterizes the local orientation of the magnetic moment. The e volution equation for that local orientation reads ∂ˆ n ∂t=−γ 1+α2ˆ n×[Beff+α[ˆ n×Beff]]. (2) The quantities γandαare the gyromagnetic ratio and the dimensionless Gilbert damping parameter, respectively, where αis related to εintroduced in Eq. (1). Beffis the effective magnetic field that drives the motion of the spin density. Generally, it consists of an internal part originated by the interaction of the spins and an external field . This effective field is related to the Hamiltonian of the system by functional variation with respect to ˆ n Beff=−m−1 sδH δˆ n. (3) In absence of an external field the Hamiltonian can be expressed as [19, 20] H=/integraldisplay d3r{wex+wan},with wex=1 2msκ(∇ˆ n)2andwan=1 2msΓ sin2θ.(4) Thereby, the constants κand Γ denote the exchange energy density and the magneto- crystalline anisotropy energy density. To be more precise, κ∝Ja2,Jbeing the coupling 4strength that measures theinteraction between nearest neighb ors inthe isotropic Heisenberg model [21]. Once again ais thelattice constant. Notice that the formof theexchange ener gy in the Hamiltonian (4) arises from the Heisenberg model in the classica l limit. The quantity θrepresents the angle between ˆ nand the anisotropy axis ˆν= (0,0,1), where ˆνpoints in the direction of the easy axis in the ground state in the case of zer o applied external field. Thus, the constant Γ >0 characterizes anisotropy as a consequence of relativistic interactions (spin-orbital and dipole-dipole ones [20]). In deriving Eq . (4) we have used ˆ n2= 1. Although it is more conventional to introduce the angular coord inates (θ,Φ) [2, 4], we find it more appropriate to use Cartesian coordinates. To proce ed, we divide the vector ˆ ninto a static and a dynamic part designated by µandϕ, respectively. In the linearized spin wave approach let us make the ansatz ˆ n(r,t) =µ(r)+ϕ(r,t) =µˆν+ϕ, µ= const., (5) whereˆ n2= 1 is still valid. The effective field can now be obtained from Eqs. (3) an d (4). This yields Beff=κ∇2ϕ−Γϕ′;ϕ′= (ϕ1,ϕ2,0). (6) Eq. (2) together with Eqs. (3) and (4) represent the determinist ic model for a classical ferromagnet. In order to extent the model let us supplement the effective magnetic field in Eq. (6) by a stochastic component yielding an effective random field Beff=Beff+η(t). The stochastic process η(t) is assumed to be Gaussian distributed with zero mean and obeying a colored correlation function ˜χij(t,t′) =∝an}b∇acketle{tηi(t)ηj(t′)∝an}b∇acket∇i}ht=˜Dij ˜τijexp/bracketleftbigg −|t−t′| ˜τij/bracketrightbigg . (7) Here,˜Dijand ˜τijare the noise strength and the finite correlation time of the noise η. Due to the coupling of the effective field to the spin orientation ˆ nthe stochastic process is a multiplicative one. Microscopically, such a random process might be originated by a fluctuating coupling strength for instance. The situation associa ted with our model is illustrated in Fig. 1 and can be understood as follows: The stochastic vector fieldη(t) is able to change the orientation of the localized moment at different times. Therefore, fixed phase relations between adjacent spins might be destroyed. Moreover, theη(tk) are interrelated due to the finite correlation time τ. The anisotropy axis defines the preferred orientation of the mean value of magnetization. Due to the inclusion of η(t) the deterministic Eq. (2) is 5xyz anisotropy axis ˆν exchange ∝Jaη(t1)η(t2)η(t3)random field at different times ti FIG. 1. Part of a ferromagnetic domain influenced by stochast ic forces for the example of cubic symmetry with lattice constant a. The black spin in the center only interacts with its nearest neighbors (green), where Jis a measure for the exchange integral. transformed into the stochastic LLG. Using Eq. (5) it follows ∂ϕ ∂t=−γ 1+α2(µ+ϕ)×[Beff+α[(µ+ϕ)×Beff]]. (8) The random magnetic field is defined by Beff=κ∇2ϕ−Γϕ′+η(t), (9) whereϕ′is given in Eq. (6). With regard to the following procedure we suppose the random field to be solely generated dynamically, i.e., ˆ n×η(t) =ϕ×η(t). So far, the dynamics of our model (Eqs. (8) and (9)) are reflected by a nonlinear, stoc hastic partial differential equation (PDE). Using Fourier transformation, i.e., ψ(q,t) =F{ϕ(r,t)}and introducing the following dimensionless quantities β= (l0q)2+1, l2 0=κ Γ, ω=γΓ,¯t=ωt ,λ(t) =η(t) Γ,(10) the components ψi(q,t) fulfill the equation d dtψi(q,t) = Ωi(ψ(q,t))+Λ ij(ψ(q,t))λj(t). (11) 6The quantity l0is the characteristic magnetic length [22]. The vector Ωand the matrix Λ are given by Ω=ξµβ −(αµψ1+ψ2) ψ1−αµψ2 0 , ξ=1 1+α2, (12) and Λ =ξ αµψ3ψ3−(ψ2+αµψ1) −ψ3αµψ3ψ1−αµψ2 ψ2−ψ1 0 . (13) For convenience we have substituted ¯t→tagain. The statistical properties of λ(t) are expressed as ∝an}b∇acketle{tλ(t)∝an}b∇acket∇i}ht= 0 and χkl(t,t′) =∝an}b∇acketle{tλk(t)λl(t′)∝an}b∇acket∇i}ht=Dkl τklδklexp/bracketleftbigg −|t−t′| τkl/bracketrightbigg τkl→0− −− →2Dklδklδ(t−t′).(14) Incidentally, in the limit τ→0 the usual white noise properties are recovered. We empha- size that although we regard the long-wavelength limit ( a·q≪1), wave vectors for which l0·q≫1 (in Eq. (10)) can also occur [22]. But this case is not discussed in the present paper and will be the content of future work. Whereas, in what follo ws we restrict our considerations to the case q→0 so that, actually, l0·q≪1 is fulfilled. Hence, we can set β= 1 approximately in Eq. (10). Due to the anisotropy the spin wave dis persion relation offers a gap at q= 0. Owing to this fact ψis studied at zero wave vector. For this situation the assumption of a space-independent stochastic force ηi(t), compare Eq. (7), is reasonable. For non-zero wave vector the noise field should be a spatiotempora l fieldηi((r,t). Because our model is based on a short range interaction we expect that the corresponding noise correlation function is δ-correlated, i.e. instead of (14) we have χkl(r,t;r′,t′) =Dkl τklδklexp/bracketleftbigg −|t−t′| τkl/bracketrightbigg 2Mδ(r−r′), whereMis the strength of the spatial correlation. Using this relation we are able to study also the case of small qwhich satisfies l0·q≪1. In the present paper we concentrate on the case of zero wave vector q= 0. 7III. CORRELATION FUNCTIONS In the present section let us discuss the statistical behavior of th e basic Eqs. (11)-(14). They describe a non-stationary, non-Markovian process attributed t o the finite correlation time. Due to their common origin both characteristics can not be analyzed separately. In the limitτ→0, Eq. (11) defines a Markovian process which provides also station arity by an appropriate choice of initial conditions [23]. However, the present s tudy is focused on the effectofnonzerocorrelationtimes. Tothatpurposeweneedapro perprobabilitydistribution function which reflects the stochastic process defined by Eqs. (1 1)-(14). In deriving the relevant joint probability distribution function we follow the line given in [24], where the detailedcalculationshadbeencarriedout, seealsothereferences citedtherein. Inparticular, it has been underlined in those papers that in order to calculate corr elation functions of type ∝an}b∇acketle{tψi(t)ψj(t′)∝an}b∇acket∇i}hta single probability distribution function P(ψ,t) is not sufficient. Instead of that one needs a joint probability distribution of the form P(ψ,t;ψ′,t′). Before proceeding let us shortly summarize the main steps to get the joint probability dis tribution function. To simplify the calculation we assume τkl=τδklandDkl=Dδkl. Notice that our system has no ergodic properties what would directly allow us to relate the st ochastic interferences with temperature fluctuations by means of a fluctuation-dissipatio n theorem. Based on Eq. (11) the appropriate joint probability distribution is defined by [2 4, 25], for a more general discussion compare also [26]: P(ψ,t;ψ′,t′) =∝an}b∇acketle{tδ(ψ(t)−ψ)δ(ψ(t′)−ψ′)∝an}b∇acket∇i}ht. (15) Here the average is performed over all realizations of the stochas tic process. In defining the joint probability distribution function we follow the convention to indic ate the stochastic process by the function ψ(t) whereas the quantity without arguments ψstands for the special values of the stochastic variable. These values are even re lalized with the probaility P(ψ,t;ψ′,t′). The equation of motion for this probability distribution reads acco rding to 8[24] ∂ ∂tP(ψ,t;ψ′,t′) =−∂ ∂ψit/integraldisplay 0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t) δλk(t1)/bracketrightbigg ψ(t)=ψ·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg dt1 −∂ ∂ψ′it′/integraldisplay 0χjk(t,t1)/angbracketleftigg/bracketleftbiggδψi(t′) δλk(t1)/bracketrightbigg ψ(t′)=ψ′·δ(ψ(t)−ψ)δ(ψ(t′)−ψ′)/angbracketrightigg dt1,(16) where Novikov’s theorem [27] has been applied. Expressions for the response functions δψi(t)/δλk(t1) andδψi(t′)/δλk(t1) can be found by formal integration of Eq. (11) and iterating the formal solution. After a tedious but straightforwar d calculation including the computation of the response functions to lowest order in ( t−t1) and (t′−t1) and the evaluation of several correlation integrals referring to χklfrom Eq. (14), Eq. (16) can be rewritten in the limit of small correlation time τas ∂ ∂tPs(ψ,t;ψ′,t′) =/braceleftbig L0(ψ,τ) +exp[−(t−t′)/τ]D∂ ∂ψiΛik(ψ)∂ ∂ψ′ nΛnk(ψ′)/bracerightbigg Ps(ψ,t;ψ′,t′).(17) Thereby, transient terms and terms of the form ∝τexp[−(t−t′)/τ] (these terms would lead to terms of order τ2in Eq. (22)) have been neglected. The result is valid in the stationary case characterized by t→ ∞andt′→ ∞but finites=t−t′. In Eq. (17) L0is the operator appearing in the equation for the single probability density. Following [2 4, 28] the operator reads L0(ψ,τ) =−∂ ∂ψiΩi(ψ)+∂ ∂ψiΛik(ψ)∂ ∂ψn/braceleftigg D/bracketleftbig Λnk(ψ)−τMnk(ψ)/bracketrightbig +D2τ/bracketleftbigg Knkm(ψ)∂ ∂ψlΛlm(ψ)+1 2Λnm(ψ)∂ ∂ψlKlkm(ψ)/bracketrightbigg/bracerightigg ,(18) with Mnk= Ωr∂Λnk ∂ψr−Λrk∂Ωn ∂ψr Knlk= Λrk∂Λnl ∂ψr−∂Λnk ∂ψrΛrl.(19) The equation of motion for the expectation value ∝an}b∇acketle{tψi∝an}b∇acket∇i}htscan be evaluated from the single probability distribution in the stationary state ∂ ∂tPs(ψ,t) =L0Ps(ψ,t). (20) 9One finds d dt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=∝an}b∇acketle{tΩi∝an}b∇acket∇i}hts+D/angbracketleftbigg∂Λik ∂ψn/parenleftbig Λnk−τMnk/parenrightbig/angbracketrightbigg s−D2τ/braceleftigg/angbracketleftbigg∂ ∂ψr/parenleftbigg∂Λik ∂ψnKnkm/parenrightbigg Λrm/angbracketrightbigg s +1 2/angbracketleftbigg∂ ∂ψr/parenleftbigg∂Λik ∂ψnΛnm/parenrightbigg Krkm/angbracketrightbigg s/bracerightigg .(21) The knowledge of the evolution equation of the joint probability distr ibutionP(ψ,t;ψ′,t′) due to Eqs. (17) and (18) allows us to get the corresponding equat ion for the correlation functions. Following again [24], it results d dt∝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 ∂ψn/parenleftbig Λnk−τMnk/parenrightbig/bracketrightbigg tψj(t′)/angbracketrightbigg s −D2τ/braceleftigg/angbracketleftbigg/bracketleftbigg∂ ∂ψr/parenleftbigg∂Λik ∂ψnKnkm/parenrightbigg Λrm/bracketrightbigg tψj(t′)/angbracketrightbigg s +1 2/angbracketleftbigg/bracketleftbigg∂ ∂ψr/parenleftbigg∂Λik ∂ψnΛnm/parenrightbigg Krkm/bracketrightbigg tψj(t′)/angbracketrightbigg s/bracerightigg +Dexp/bracketleftbigg −t−t′ τ/bracketrightbigg ∝an}b∇acketle{tΛik(ψ(t))Λjk(ψ(t′))∝an}b∇acket∇i}hts,(22) where the symbol [ ...]tdenotes the quantity [ ...] at timet. As mentioned above the result is valid for t, t′→ ∞whiles=t−t′>0 remains finite. The quantities MnkandKklmare defined in Eq. (19). The components Ω iand Λ ijare given in Eqs. (12) and (13). Performing the summation over double-indices according to Eqs. (21) and (22) we obtain the evolution equations for the mean value and the correlation function d dt∝an}b∇acketle{tψi(t)∝an}b∇acket∇i}hts=Gik∝an}b∇acketle{tψk(t)∝an}b∇acket∇i}hts, (23) and d dsCij(s) =d ds∝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 +Dexp/bracketleftig −s τ/bracketrightig ∝an}b∇acketle{tΛik(ψ(t′+s))Λjk(ψ(t′))∝an}b∇acket∇i}hts.(24) Notice, that in the steady state one gets Cij(t,t′) =Cij(s) withs=t−t′. The matrix components of Gikare given by Gik= −A1A20 −A2−A10 0 0 −A3 , (25) 10where A1=−D2τ(6µ2α2−1)ξ4+2µ2αDτξ3−D(µ2α2−2)ξ2+µ2αξ A2=1 2µαD2τ/parenleftbig 11−3µ2α2/parenrightbig ξ4+µDτ/parenleftbig µ2α2−1/parenrightbig ξ3+3µDαξ2−µξ A3= +D2τ/parenleftbig 3µ2α2+1/parenrightbig ξ4−4µ2αDτξ3+2Dξ2,(26) andξis defined in Eq. (12). At this point let us stress that in the case t′= 0 the term ∝exp[−(t−t′)/τ] on the rhs. in Eqs. (22) and (24), respectively, would vanish in the steady state, i.e. ∝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. Theoccurrenceofsuchatermisastrongindicationforthenon-st ationarityofourmodel. An explicit calculation shows, that in general this inequality holds for non -stationary processes [23]. IV. RESULTS The solution of Eq. (23) can be found by standard Greens function methods and Laplace transformation. As the result we find ∝an}b∇acketle{tψ(t)∝an}b∇acket∇i}hts= e−A1tcos(A2t)e−A1tsin(A2t) 0 −e−A1tsin(A2t)e−A1tcos(A2t) 0 0 0 e−A3t ·∝an}b∇acketle{tψ0∝an}b∇acket∇i}hts, (27) where∝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 in Eqs. (26) play the roles of the magnon lifetime and the frequency o f the spin wave at zero wave vector, respectively. As can be seen in Eq. (26) all of th ese three parameters are affected by the correlation time τand the strength Dof the random force. Moreover, the Gilbert damping parameter αinfluences the system as well. The solution of Eq. (24) for the correlation function in case of t′= 0 is formal identical to that of Eq. (27). The more general situation t′∝ne}ationslash= 0 allows no simple analytic solution and hence the behavior of the correlation function C(s) is studied numerically. In order to analyze the mean values and the correlation function let us first examine the parameter range w here physical accessible solutions 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 the solutions for ψ1(t) andψ2(t) on the one hand and ψ3(t) on the other hand are decoupled 11in Eq. (27). Therefore, spin wave solutions only exists for non-zer o averages ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htand ∝an}b∇acketle{tψ2(t)∝an}b∇acket∇i}ht. The existence of such non-trivial solutions are determined in depe ndence on the noise parameters Dandτand the deterministic damping parameter α. Notice, that the dimensionless quantity D=˜D/Γ, i.e.,Dis the ratio between the strength of the correlation function (Eq. (7)) and the anisotropy field in the original units. The stability of spin wave solutions is guaranteed for positive parameters A1andA3. According to Eqs. (26) the phase diagrams are depicted in Fig. 2 within the α−Dplane for different values of the correlation time τ. The separatrix between stable and unstable regions is determined by the condition A1= 0. The second condition A3= 0 is irrelevant due to the imposed initial conditions. As the result of the stability analysis the phase space dia gram is subdivided into four regions where region IV does not exist in case of τ= 0, see Fig. 2(a). For generality, we take into account both positive and negative values of Dindicating correlations and anti-correlations of the stochastic field. Damped spin waves are ob served in the areas I and IV, whereas the sectors II and III reveal non-accessible solutio ns. In those regions the spin wave amplitude, proportional to exp[ −A1t], tends to infinity which should not be realized, compare Figs. 2(b)-2(d). Actually, a reasonable behavior is obser ved in regions I and IV. As visible from Fig. 2 damped spin waves will always emerge for D>0 even in the limit of zero damping parameter αand vanishing correlation time τ. This behavior is shown in Fig. 3, where theevolution of ∝an}b∇acketle{tψ1(t)∝an}b∇acket∇i}htis depicted for different valuesof α. As canbeseen in Fig.2(a) thesolutionfor D<0isunlimitedandconsequently, itshouldbeexcludedfurther. Contr ary to this situation, additional solutions will be developed in region IV in ca se ofτ >0 and simultaneously α= 0, see Figs. 2(b)-2(d). Thereby the size of area IV grows with inc reasing τ. Likewise, the extent of region I decreases for an enhanced τ. However, in the limit of D= 0 and consequently for τ= 0, too, only damped spin waves are observed. Immediately on the separations line undamped periodic solutions will evolve, compa re the sub-figures in Fig. 2. This remarkable effect can be traced back to the interplay be tween the deterministic damping and the stochastic forces. Both damping mechanism are co mpensated mutually which reminds of a kind of resonance phenomenon. The difference to conventional resonance behavior consists of the compensation of the inherent determinist ic Gilbert damping and the stochastic one originated from the random field. This statement is e mphasized by the fact that undamped periodic solutions do not develop in the absence of st ochastic interferences, i.e.,D= 0. The situation might be interpreted physically as follows: the requ ired energy 12(a)τ= 0 (b)τ= 0.1 (c)τ= 1 (d)τ= 10 FIG. 2.α−Dplane for fixed magnetization µ= 0.9 and different values of τ. that enables the system to sustain the deterministic damping mecha nisms is delivered by the stochastic influences due to the interaction with the environme nt. To be more precise, in general, the Gilbert damping enforces the coherent alignment of th e spin density along the precession axis. Contrary, the random field supports the dephas ing of the orientation of the classical spins. Surprisingly, the model predicts the existence of a critical value τ=τc≥0 13FIG. 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 (dash-dotted line), 0 .05 (solid line), 0 .5 (dotted line) and 1 (dashed line). depending on αandDwhich determines the onset of undamped periodic solutions. Notice, that negative values of τcare excluded. The critical value is τc=−[µ2(α3−Dα2+α)+2D](1+α2)2 2Dµ2(α3−3Dα2+α)+D2. (28) Hence, this result could imply the possibility of the cancellation of both damping processes. Examples according to the damped and the periodic case are displaye d in Fig. 4. An increas- ingτfavors the damping process as it is visible in Fig. 4(a). Based on estima tions obtained for ferromagnetic materials [29] and references therein, the Gilbe rt damping parameter can range between 0 .04<α<0.22 in thin magnetic films, whereas the bulk value for Co takes αb≈0.005. The phase space diagram in Fig. 2 offers periodic solutions only fo r values of αlarger than those known from experiments. Therefore such perio dic solutions seem to be hard to see experimentally. We proceed further by analyzing the be havior of the correlation function by numerical computation of the solution of Eq. (24) with E qs. (25) and (26). As initial values we choose Cik(t=t′,t′) =Cik(s= 0) =C0for every combination i,k={1,2,3}. The results are depicted in Figs. 5 and 6. Inspecting Figs. 5(a)-5(c ) one recognizes that an enhancement of the correlation time τleads to an increase of the oscillations within the correlation functions C1k,k={1,2,3}. Moreover, Fig. 5(d) reveals that the oscillatory 14(a) (b) FIG. 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 from 10 (solid line), 1 (dotted line) and 0 (dash-dotted line ). (b):D= 2,α= 1 andτ=τc≈1.79 (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. behavior of C31seems to be suppressed. Obviously, the decay of the correlation f unction is enhanced if τgrowths up. The pure periodic case for τ=τc, corresponding to Fig. 4(b), is depicted in Fig. 6. Exemplary, C12andC31are illustrated. The behavior of the latter is similar to the damped case, displayed in Fig. 5(d), unless slight oscillatio ns occur. However, if one compares the form of C12in Fig. 5(b) and Fig. 6 the differences are obvious. The am- plitude of the correlation function for the undamped case grows to the fourfold magnitude in comparison with C0, whereas the damped correlation function approaches zero. Fur ther, a periodic behavior is shown in Fig. 6, and therefore the correlation w ill oscillate about zero but never vanish for all s=t−t′>0. V. CONCLUSIONS In this paper we have analyzed the dynamics of a classical spin model with uniaxial anisotropy. Aside from the deterministic damping due to the Landau -Lifshitz-Gilbert equation the system is subjected to an additional dissipation proce ss by the inclusion of a stochastic field with colored noise. Both dissipation processes are a ble to compete leading to 15(a) (b) (c) (d) FIG. 5. Correlation functions Cik(s) forµ= 0.9,D= 0.1 andα= 0.005.τtakes 0 (dotted line), 1 (solid line) and 10 (dash-dotted line). a more complex behavior. To study this one we derive an equation for the joint probability distribution which allows us to find the corresponding spin-spin-corr elation function. This program can be fulfilled analytically and numerically in the spin wave appr oach and the small correlation time limit. Based on the mean value for the spin wave c omponent and 16FIG. 6. Correlation functions Cik(s) forτ=τc≈1.79 (Eq. (28)), µ= 0.9,D= 2 andα= 1. The dotted line represents C12and the solid line is C31. the correlation function we discuss the stability of the system in ter ms of the stochastic parameters, namely the strength of the correlated noise Dand the finite correlation time τ, as well as the deterministic Gilbert damping parameter α. The phase diagram in the α−Dplane offers that the system develops stable and unstable spin wave solutions due to the interplay between the stochastic and the deterministic damping mechanism. So stable solutions evolve for arbitrary positive Dand moderate values of the Gilbert damping α. Further, we find that also the finite correlation time of the stochas tic field influences the evolution of the spin waves. In particular, the model reveals for fix edDandαa critical valueτcwhich characterizes the occurrence of undamped spin waves. The different situa- tions are depicted in Fig. 2. Moreover, the correlation time τaffects the damped spin wave which can be observed in regions I and IV in the phase diagram. If the parameters Dand αchanges within these regions, an increasing τleads to an enhancement of the spin wave damping, cf. Fig. 4(a). The influence of τon the correlation functions is similar as shown in Figs. 5(a)-5(c). The study could be extended by the inclusion of fi nite wave vectors and using an approach beyond the spin wave approximation. 17ACKNOWLEDGMENTS One of us (T.B.) is grateful to the Research Network ’Nanostructu red Materials’, which is supported by the Saxony-Anhalt State, Germany. 18[1] L. D. Landau, E. Lifshitz, and L. Pitaevskii, Electrodynamics of continuous media (Pergamon Press, Oxford, 1989). [2] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935). [3] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halpe rin, Rev. Mod. Phys. 77, 1375 (2005). [4] A. Sukhov and J. Berakdar, J. Phys. - Cond. Mat. 20, 125226 (2008). [5] J. C. Slonczewski, J. Magn. and Mag. Mat. 159, L1 (1996). [6] L. Berger, Phys. Rev. B 54, 9353 (1996). [7] S. Urazhdin, Phys. Rev. B 78, 060405 (2008). [8] B. Kr¨ uger, D. Pfannkuche, M. Bolte, G. Meier, andU. Merk t, Phys. Rev. B 75, 054421 (2007). [9] K. D. Usadel, Phys. Rev. B 73, 212405 (2006). [10] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [11] M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102, 137601 (2009). [12] S. F. Zhang and S. S. L. Zhang, Phys. Rev. Lett. 102, 086601 (2009). [13] S. Trimper, T. Michael, and J. M. Wesselinowa, Phys. Rev . B76, 094108 (2007). [14] J. Foros, A. Brataas, G. E. W. Bauer, and Y. Tserkovnyak, Phys. Rev. B 79, 214407 (2009). [15] A. L. Chudnovskiy, J. Swiebodzinski, and A. Kamenev, Ph ys. Rev. Lett. 101, 066601 (2008). [16] D. M. Basko and M. G. Vavilov, Phys. Rev. B 79, 064418 (2009). [17] S. I. Denisov, K. Sakmann, P. Talkner, and P. H¨ anggi, Ph ys. Rev. B 75, 184432 (2007). [18] M. Daniel and M. Lakshmanan, Physica A 120, 125 (1983). [19] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53, 2497 (1984). [20] V. G. Bar’Yakhtar, M. V. Chetkin, B. A. Ivanov, andS. N. G adetskii, Dynamics of Topological Magnetic Solitons: Experiment and Theory (Springer Tracts i n Modern Physics) (Springer, 1994). [21] M. Lakshmanan, T. W. Ruijgrok, and C. J. Thompson, Physi ca A84, 577 (1976). [22] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys. Rep .194, 117 (1990). [23] A. Hernandez-Machado and M. San Miguel, J. Math. Phys. 25, 1066 (1984). [24] A. Hernandez-Machado, J. M. Sancho, M. San Miguel, and L . Pesquera, Zeitschr. f. Phys. B 52, 335 (1983). 19[25] N. G. van Kampen, Braz. J. Phys. 28, 90 (1998). [26] N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amster- dam, 1981). [27] E. A. Novikov, Sov. Phys. JETP 20, 1290 (1965). [28] H. Dekker, Phys. Lett. A 90, 26 (1982). [29] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev . Lett.88, 117601 (2002). 20

2010-02-26

We analyze the Landau-Lifshitz-Gilbert equation when the precession motion of the magnetic moments is additionally subjected to an uniaxial anisotropy and is driven by a multiplicative coupled stochastic field with a finite correlation time $\tau$. The mean value for the spin wave components offers that the spin-wave dispersion relation and its damping is strongly influenced by the deterministic Gilbert damping parameter $\alpha$, the strength of the stochastic forces $D$ and its temporal range $\tau$. The spin-spin-correlation function can be calculated in the low correlation time limit by deriving an evolution equation for the joint probability function. The stability analysis enables us to find the phase diagram within the $\alpha-D$ plane for different values of $\tau$ where damped spin wave solutions are stable. Even for zero deterministic Gilbert damping the magnons offer a finite lifetime. We detect a parameter range where the deterministic and the stochastic damping mechanism are able to compensate each other leading to undamped spin-waves. The onset is characterized by a critical value of the correlation time. An enhancement of $\tau$ leads to an increase of the oscillations of the correlation function.

Correlation Effects in the Stochastic Landau-Lifshitz-Gilbert Equation

1002.4958v1

arXiv:2310.18878v1 [math.AP] 29 Oct 2023ASYMPTOTIC PROFILES FOR THE CAUCHY PROBLEM OF DAMPED BEAM EQUATION WITH TWO VARIABLE COEFFICIENTS AND DERIVATIVE NONLINEARITY MOHAMED ALI HAMZA1, YUTA WAKASUGI2∗AND SHUJI YOSHIKAWA3 1Basic Sciences Department, Deanship of Preparatory Year an d Supporting Studies, P. O. Box 1982, Imam Abdulrahman Bin Faisal Univers ity, Dammam, KSA. 2Laboratory of Mathematics, Graduate School of Advanced Sci ence and Engineering, Hiroshima University, Higashi-Hiroshima, 7 39-8527, Japan 3Division of Mathematical Sciences, Faculty of Science and T echnology, Oita University, Oita, 870-1192, Japan Abstract. In this article we investigate the asymptotic profile of solu tions for the Cauchy problem of the nonlinear damped beam equation wit h two variable coefficients: ∂2 tu+b(t)∂tu−a(t)∂2 xu+∂4 xu=∂x(N(∂xu)). In the authors’ previous article [ 17], the asymptotic profile of solutions for linearized problem ( N≡0) was classified depending on the assumptions for the coefficients a(t) andb(t) and proved the asymptotic behavior in effective damping cases. We heregive the conditions ofthe coefficients andthe nonlinear term in order that the solution behaves as the solution for th e heat equation: b(t)∂tu−a(t)∂2 xu= 0 asymptotically as t→ ∞. 1.Introduction We study the Cauchy problem of nonlinear damped beam equation /braceleftBigg ∂2 tu+b(t)∂tu−a(t)∂2 xu+∂4 xu=∂x(N(∂xu)), t∈(0,∞),x∈R, u(0,x) =u0(x), ∂tu(0,x) =u1(x), x ∈R,(1.1) whereu=u(t,x)isareal-valuedunknown, a(t) andb(t)aregivenpositivefunctions oft,N(∂xu) denotes the nonlinear function, and u0andu1are given initial data. E-mail address :mahamza@iau.edu.sa, wakasugi@hiroshima-u.ac.jp, yoshikawa@oita-u.ac.jp . Date: October 31, 2023. Key words and phrases. Nonlinear damped beam equations; asymptotic behavior; glo bal ex- istence; variable coefficients. 2010 Mathematics Subject Classification. 35G25; 35B40; 35A 01 ∗Corresponding author 12 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA Before giving more precise assumptions for a(t),b(t) andNand our result, we first mention the physical background and the mathematical mo tivations of the problem. The equation ( 1) corresponds to the so-called Falk model under isothermal assumption with the damping term. The Falk model is one o f the models for a thermoelastic deformation with austenite-martensite phase transitions on shape memory alloys: ∂2 tu+∂4 xu=∂x/braceleftbig (θ−θc)∂xu−(∂xu)3+(∂xu)5/bracerightbig , ∂tθ−∂2 xθ=θ∂xu∂t∂xu, t ∈(0,∞),x∈R, u(0,x) =u0(x), ∂tu(0,x) =u1(x), θ(0,x) =θ0(x), x ∈R, whereuandθare the displacement and the absolute temperature, respectively , andθcis a positive constant representing the critical temperature for t he phase transition. If we assume the temperature arecontrollableand unif ormly distributed with respect to the space, that is, θis given function uniform in xsuch asθ−θc= a(t) and setN(ε) =ε5−ε3, then the problem ( 1) is surely derived. Our interest directs to the behavior of solution around the initial temperature θ0is closed to the critical temperature θc. Indeed, the Lyapunov stability for the solution of the Falk model is shown in [ 11], which claims that the temperature tends to the function uniformly distributed in xin the bounded domain case. For more precise information of the Falk model of shape memory alloys, we refer the r eader to Chapter 5 in [ 2]. We are also motivated by the extensible beam equation proposed by Woinovsky-Krieger [ 16]: ∂2 tu−/parenleftbigg/integraldisplay R|∂xu|2dx/parenrightbigg ∂2 xu+∂4 xu= 0. In [1], the model with the damping term was proposed and the stability res ult was shown. The problem ( 1) corresponds to the nonlinear generalization for the equa- tion. As the observation similar to the Kirchhoff equation, the lineariz ed problem substituting the given function a(t) into the nonlocal term was also studied by e.g. [4], [10], [17] and [6]. Next, let us explain the mathematical background of our problem. I t is well- known that the solution of the Cauchy problem for the damped wave equation ∂2 tu+∂tu−∂2 xu= 0 behaves as the solution for the heat equation ∂tu−∂2 xu= 0 asymptotically as t→ ∞(see e.g. [ 7]). Roughly speaking, this implies that ∂2 tu decays faster than ∂tuast→ ∞. From the same observation, the solution for the beam equation ∂2 tu+∂tu−∂2 xu+∂4 xu= 0 behaves as the solution for the heat equation ∂tu−∂2 xu= 0 asymptotically as t→ ∞, because∂4 xudecays faster than∂2 xu. The above observation induces the investigation of the solution fo r the equation with time variable coefficient: ∂2 tu+b(t)∂tu−∂2 xu= 0 withb(t)∼(1+t)β. The precise analysis implies that the solution behaves as the solution f or the heat equationb(t)∂tu−∂2 xu= 0 whenβ <−1, and on the other hand, that the solution behaves as the solution for the wave equation ∂2 tu−∂2 xu= 0 when −1< β <1 (see e.g. [ 8], [9], [14] and [15]). Correspondingly, the authors in [ 17] studied the asymptotic behavior of the solution for the linearized problem of ( 1) ∂2 tu+b(t)∂tu−a(t)∂2 xu+∂4 xu= 0.NONLINEAR DAMPED BEAM EQUATION 3 As in Figure 1, we divide the two-dimensional regions Ω jfor (α,β) (j= 1,2,3,4,5) by Ω1:=/braceleftbig (α,β)∈R2| −1<β <min{α+1,2α+1}/bracerightbig , Ω2:=/braceleftbig (α,β)∈R2|max{−1,2α+1}<β <1/bracerightbig , Ω3:=/braceleftbig (α,β)∈R2|β <−1<α/bracerightbig , Ω4:=/braceleftbig (α,β)∈R2|β <−1,α<−1/bracerightbig , Ω5:=/braceleftbig (α,β)∈R2|max{1,α+1}<β/bracerightbig . αβ 01 −1−1 2 β=−1β=α+1 β= 2α+1Ω1Ω2 Ω3Ω4Ω5 Figure 1. By a scaling argument, the result gives the conjecture for the clas sification of the asymptotic behavior of the solution: (1) In (α,β)∈Ω1,u(t) behaves as the solution for b(t)ut−a(t)uxx= 0. (2) In (α,β)∈Ω2,u(t) behaves as the solution for b(t)ut+uxxxx= 0. (3) In (α,β)∈Ω3,u(t) behaves as the solution for utt−a(t)uxx= 0. (4) In (α,β)∈Ω4,u(t) behaves as the solution for utt+uxxxx= 0. (5) In (α,β)∈Ω5,u(t) behaves as the solution for over damping case. As a partial answer, the authors proved the effective damping cas es (1) and (2) in [17]. Here we shall give the result for the nonlinear problem ( 1) in the case (1). From now on, we shall give our main result. To state it precisely, we pu t the following assumptions. Assumption (A) The coefficients a(t) andb(t) are smooth positive functions satisfying C−1(1+t)α≤a(t)≤C(1+t)α, C−1(1+t)β≤b(t)≤C(1+t)β and |a′(t)| ≤C(1+t)α−1,|b′(t)| ≤C(1+t)β−1 with some constant C≥1 and the parameters α,β∈R. Moreover, we assume (α,β)∈Ω1:=/braceleftbig (α,β)∈R2| −1<β <min{α+1,2α+1}/bracerightbig . Assumption (N) The function N(∂xu) is a linear combination of ( ∂xu)2andp-th order terms with p≥3. More precisely, the function Nhas the form N(z) =µz2+˜N(z),4 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA with someµ∈R, where ˜N∈C2(R) satisfies ˜N(j)(0) = 0 and |˜N(j)(z)−˜N(j)(w)| ≤C(|z|+|w|)p−1−j|z−w|(z,w∈R) holds with some p≥3 forj= 0,1,2. A typical example of our nonlinearity is ∂x(N(∂xu)) =∂x(∂xu)2+∂x/parenleftbig |∂xu|p−1∂xu/parenrightbig withp≥3. Remark 1.1. The assumption (A) implies −β+1 α−β+1<2<p. This means that the nonlinearity is supercritical (see the a rgument in Section 4.1). We further prepare the following notations. Let G=G(t,x) be the Gaussian, that is, G(t,x) =1√ 4πtexp/parenleftbigg −x2 4t/parenrightbigg . We define r(t) =a(t) b(t)andR(t) =/integraldisplayt 0r(τ)dτ. Remark that C−1(1+t)α−β+1≤R(t)≤C(1+t)α−β+1 holds with some constant C≥1, thanks to the assumption (A). Theorem 1.2. Under the assumptions (A) and (N), there exists a constant ε0>0 such that if (u0,u1)∈/parenleftbig H2,1(R)∩H3,0(R)/parenrightbig ×/parenleftbig H0,1(R)∩H1,0(R)/parenrightbig and /bardblu0/bardblH2,1∩H3,0+/bardblu1/bardblH0,1∩H1,0≤ε0, then there exists a unique solution u∈C([0,∞);H2,1(R)∩H3,0(R))∩C1([0,∞);H0,1(R)∩H1,0(R)).(1.2) Moreover, the solution uhas the asymptotic behavior /bardblu(t,·)−m∗G(R(t),·)/bardblL2≤C(R(t)+1)−1 4−λ 2(/bardblu0/bardblH2,1∩H3,0+/bardblu1/bardblH0,1∩H1,0) with some constants C >0,m∗∈R, andλ>0. Remark 1.3. From the proof, λis taken to be arbitrary so that 0<λ<min/braceleftbigg1 2,2(β+1) α−β+1,2α−β+1 α−β+1/bracerightbigg . The proof is based on the method by Gallay and Raugel [ 5] using the self-similar transformation and the standard energy method. This paper is organized as follows. In Section 2, we rewrite the proble m through the self-similar transformation. Thereafter, we show several en ergy estimates in Section 3, and give a priori estimates through the estimates for th e nonlinear terms in Section 4. In the appendix, we also give a lemma for energy identities which is frequently used in the proof and the proof of local-in-time existenc e of solution for the readers’ convenience.NONLINEAR DAMPED BEAM EQUATION 5 At the end of this section we prepare notation and several definitio ns used throughout this paper. We denote by Ca positive constant, which may change from line to line. The symbol a(t)∼b(t) means that C−1b(t)≤a(t)≤Cb(t) holds for some constant C≥1.Lp=Lp(R) stands for the usual Lebesgue space, and Hk,m=Hk,m(R) fork∈Z≥0andm∈Ris the weighted Sobolev space defined by Hk,m(R) =/braceleftBigg f∈L2(R);/bardblf/bardblHk,m=k/summationdisplay ℓ=0/bardbl(1+|x|)m∂ℓ xf/bardblL2<∞/bracerightBigg . 2.Scaling variables Thelocalexistenceofthe solutioninthe class( 1.2) isstandard(seeAppendix B). Thus, it suffices to show the a priori estimate. To prove it, following t he argument of Gallay and Raugel [ 5], we introduce the scaling variables s= log(R(t)+1), y=x/radicalbig R(t)+1, and definev=v(s,y) andw=w(s,y) by u(t,x) =1/radicalbig R(t)+1v/parenleftBigg log(R(t)+1),x/radicalbig R(t)+1/parenrightBigg , ut(t,x) =R′(t) (R(t)+1)3/2w/parenleftBigg log(R(t)+1),x/radicalbig R(t)+1/parenrightBigg . Then, by a straightforward computation, the Cauchy problem ( 1) is rewritten as vs−y 2vy−1 2v=w, r2e−s a/parenleftbigg ws−y 2wy−3 2w/parenrightbigg +/parenleftbigg 1+r′ a/parenrightbigg w=vyy−e−s avyyyy+es a∂y/parenleftbig N/parenleftbig e−svy/parenrightbig/parenrightbig , v(0,y) =v0(y), w(0,y) =w0(y), (2.1) wherev0(y) =u0(y) andw0(y) =1 r(0)u1(y). Here, we also note that the functions a,b,r,r′appearing the above precisely mean such as a(t(s)) =a(R−1(es−1)). Remark 2.1. WhenN(z) =|z|p−1z, the nonlinearity has a bound /vextendsingle/vextendsingle/vextendsingle/vextendsinglees a(t(s))∂y/parenleftbig N(e−svy)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce(−β+1 α−β+1−p)s|vy|p−1|vyy|. Since−β+1 α−β+1<2, the assumption (N) implies that the nonlinearity can be tre ated as remainder. To investigate the asymptotic behavior of the solution of ( 2), we define m(s) :=/integraldisplay Rv(s,y)dy. By the first equation of ( 2) and the integration by parts, we have ms(s) =d dsm(s) =/integraldisplay Rvsdy=/integraldisplay R/parenleftbiggy 2vy+1 2v+w/parenrightbigg dy=/integraldisplay Rwdy.6 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA We also define φ(y) :=G(1,y) =1√ 4πexp/parenleftbigg −y2 4/parenrightbigg , ψ(y) :=φyy(y). Using them, we decompose ( v,w) as v(s,y) =m(s)φ(y)+f(s,y), w(s,y) =ms(s)φ(y)+m(s)ψ(y)+g(s,y),(2.2) and we expect that the functions fandgdefined above can be treated as remain- ders. By a direct calculation, we obtain the following equation for m(s): Lemma 2.2. We have r2e−s a(mss−ms) =−/parenleftbigg 1+r′ a/parenrightbigg ms. From the above lemma and the straightforward computation, we ca n see that (f,g) satisfies the following equations. fs−y 2fy−1 2f=g, r2e−s a/parenleftbigg gs−y 2gy−3 2g/parenrightbigg +/parenleftbigg 1+r′ a/parenrightbigg g=fyy−e−s afyyyy+es a∂y/parenleftbig N/parenleftbig e−svy/parenrightbig/parenrightbig +h, (2.3) where h=−r2e−s a/parenleftbigg 2msψ−y 2mψy−3 2mψ/parenrightbigg −r′ amψ−e−s amψyy.(2.4) They satisfy /integraldisplay Rf(s,y)dy=/integraldisplay Rg(s,y)dy=/integraldisplay Rh(s,y)dy= 0. (2.5) Therefore, our goal is to give energy estimates of the solutions ( f,g) to the equation (2) under the condition ( 2). 3.Energy estimates In this section, we give energy estimates of ( f,g) defined by ( 2). We first prepare the following general lemma for energy identities. Lemma 3.1. Letl,m∈R,n∈N∪ {0}, and letc1(s),c2(s),c4(s)be smooth functions defined on [0,∞). We consider a system for two functions f=f(s,y) andg=g(s,y)given by fs−y 2fy−lf=g, c1(s)/parenleftBig gs−y 2gy−mg/parenrightBig +c2(s)g+g=fyy−c4(s)fyyyy+h(s,y)∈(0,∞)×R, (3.1)NONLINEAR DAMPED BEAM EQUATION 7 whereh=h(s,y)is a given smooth function belonging to C([0,∞);H0,n(R)). We define the energies E1(s) =1 2/integraldisplay Ry2n/parenleftbig f2 y+c4(s)f2 yy+c1(s)g2/parenrightbig dy, E2(s) =/integraldisplay Ry2n/parenleftbigg1 2f2+c1(s)fg/parenrightbigg dy. Then, we have d dsE1(s) =−/integraldisplay Ry2ng2dy+/parenleftbigg −2n−1 4+l/parenrightbigg/integraldisplay Ry2nf2 ydy+/parenleftbigg −2n−3 4+l/parenrightbigg c4(s)/integraldisplay Ry2nf2 yydy +/parenleftbigg −2n+1 4+m/parenrightbigg c1(s)/integraldisplay Ry2ng2dy−c2(s)/integraldisplay Ry2ng2dy −2n/integraldisplay Ry2n−1fygdy−2n(2n−1)c4(s)/integraldisplay Ry2n−2fyygdy−4nc4(s)/integraldisplay Ry2n−1fyygydy +c′ 4(s) 2/integraldisplay Ry2nf2 yydy+c′ 1(s) 2/integraldisplay Ry2ng2dy+/integraldisplay Ry2nghdy and d dsE2(s) =−/integraldisplay Ry2nf2 ydy−c4(s)/integraldisplay Ry2nf2 yydy+/parenleftbigg −2n+1 4+l/parenrightbigg/integraldisplay Ry2nf2dy +c1(s)/integraldisplay Ry2ng2dy+/parenleftbigg −2n+1 2+l+m/parenrightbigg c1(s)/integraldisplay Ry2nfgdy−c2(s)/integraldisplay Ry2nfgdy −2n/integraldisplay Ry2n−1ffydy−4nc4(s)/integraldisplay Ry2n−1fyfyydy−2n(2n−1)c4(s)/integraldisplay Ry2n−2ffyydy +c′ 1(s)/integraldisplay Ry2nfgdy+/integraldisplay Ry2nfhdy. The casen= 0 is given by [ 17, Lemma 3.1]. We will prove a slightly more general version of this lemma in Appendix A. To bring out the decay property of the solutions ( f,g) to (2) from the condition (2), we define the auxiliary functions F(s,y) :=/integraldisplayy −∞f(s,z)dz, G(s,y) :=/integraldisplayy −∞g(s,z)dz, H(s,y) :=/integraldisplayy −∞h(s,z)dz. Then, bythefollowingHardyinequality, theconditions( 2)andf(s),g(s)∈H0,1(R) ensureF(s),G(s)∈L2(R). Lemma 3.2 (Hardy-type inequality [ 13, Lemma 3.9]) .Letf=f(y)∈H0,1(R) and satisfy/integraltext Rf(y)dy= 0. LetF(y) =/integraltexty −∞f(z)dz. Then, we have /integraldisplay RF(y)2dy≤4/integraldisplay Ry2f(y)2dy.8 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA From (2),FandGsatisfy the following system. Fs−y 2Fy=G, r2e−s a/parenleftBig Gs−y 2Gy−G/parenrightBig +/parenleftbigg 1+r′ a/parenrightbigg G=Fyy−e−s aFyyyy+es aN/parenleftbig e−svy/parenrightbig +H. We define the energies of ( F,G) by E01(s) :=1 2/integraldisplay RFy(s,y)2dy+e−s 2a/integraldisplay RFyy(s,y)2dy+r2e−s 2a/integraldisplay RG(s,y)2dy, E02(s) :=1 2/integraldisplay RF(s,y)2dy+r2e−s a/integraldisplay RF(s,y)G(s,y)dy. Lemma 3.3. We have d dsE01(s)+/integraldisplay RG2dy=1 2E01(s)−a′ 2ra2/integraldisplay RF2 yydy−ra′ 2a2/integraldisplay RG2dy +/integraldisplay RGes aN/parenleftbig e−svy/parenrightbig dy+/integraldisplay RGHdy, d dsE02(s)+1 2E02(s)+2E01(s) = 2r2e−s a/integraldisplay RG2dy+/parenleftbiggr′ a−ra′ a2/parenrightbigg/integraldisplay RFGdy +/integraldisplay RFes aN/parenleftbig e−svy/parenrightbig dy+/integraldisplay RFHdy. Proof.We apply Lemma 3.1asf=F,g=Gandh=es aN(e−svy)+Hwithl= 0, m= 1,n= 0,c1(s) =r2e−s/a,c2(s) =r′/a, andc4(s) =e−s/a. Noting that d dsr(t(s)) =d dsr(R−1(es−1)) =r′(t(s)) r(t(s))es, we first have c′ 1(s) =1 a2/parenleftbigg 2rdr dsae−s−r2ae−s−r2da dse−s/parenrightbigg =2r′ a−r2e−s a−ra′ a2,(3.2) c′ 4(s) =1 a2/parenleftbigg −ae−s−da dse−s/parenrightbigg =−e−s a−a′ ra2. (3.3) Thus, we obtain d dsE01(s) =−/integraldisplay RG2dy+1 4/integraldisplay RF2 ydy+3e−s 4a/integraldisplay RF2 yydy +3r2e−s 4a/integraldisplay RG2dy−r′ a/integraldisplay RG2dy−1 2/parenleftbigge−s a+a′ ra2/parenrightbigg/integraldisplay RF2 yydy +1 2/parenleftbigg2r′ a−r2e−s a−ra′ a2/parenrightbigg/integraldisplay RG2dy +/integraldisplay RGes aN/parenleftbig e−svy/parenrightbig dy+/integraldisplay RGHdy =−/integraldisplay RG2dy+1 2E01(s)−a′ 2ra2/integraldisplay RF2 yydy−ra′ 2a2/integraldisplay RG2dy +/integraldisplay RGes aN/parenleftbig e−svy/parenrightbig dy+/integraldisplay RGHdy.NONLINEAR DAMPED BEAM EQUATION 9 Next, we compute the derivative of E02(s). By Lemma 3.1, we obtain d dsE02(s) =−/integraldisplay RF2 ydy−e−s a/integraldisplay RF2 yydy−1 4/integraldisplay RF2dy +r2e−s a/integraldisplay RG2dy+r2e−s 2a/integraldisplay RFGdy−r′ a/integraldisplay RFGdy +/integraldisplay RFes aN/parenleftbig e−svy/parenrightbig dy+/integraldisplay RFHdy +/parenleftbigg2r′ a−r2e−s a−ra′ a2/parenrightbigg/integraldisplay RFGdy =−1 2E02(s)−2E01(s) +2r2e−s a/integraldisplay RG2dy+/parenleftbiggr′ a−ra′ a2/parenrightbigg/integraldisplay RFGdy +/integraldisplay RFes aN/parenleftbig e−svy/parenrightbig dy+/integraldisplay RFHdy. This completes the proof. /square Next, forn= 0,1, we define the energies of ( f,g) by E(n) 11(s) :=1 2/integraldisplay Ry2nfy(s,y)2dy+e−s 2a/integraldisplay Ry2nfyy(s,y)2dy+r2e−s 2a/integraldisplay Ry2ng(s,y)2dy, E(n) 12(s) :=1 2/integraldisplay Ry2nf(s,y)2dy+r2e−s a/integraldisplay Ry2nf(s,y)g(s,y)dy. Lemma 3.4. Forn= 0,1, we have d dsE(n) 11(s)+/integraldisplay Ry2ng2dy=3−2n 2E(n) 11(s)−2n/integraldisplay Ry2n−1fygdy −2n(2n−1)e−s a/integraldisplay Ry2n−2fyygdy−4ne−s a/integraldisplay Ry2n−1fyygydy −a′ 2ra2/integraldisplay Ry2nf2 yydy−ra′ 2a2/integraldisplay Ry2ng2dy +/integraldisplay Ry2ng/parenleftbigges a∂y(N(e−svy))+h/parenrightbigg dy and d dsE(n) 12(s)+1 2E(n) 12(s)+2E(n) 11(s) = (1−n)E(n) 12(s)+2r2e−s a/integraldisplay Ry2ng2dy−2n/integraldisplay Ry2n−1ffydy −4ne−s a/integraldisplay Ry2n−1fyfyydy−2n(2n−1)e−s a/integraldisplay Ry2n−2ffyydy +/parenleftbiggr′ a−ra′ a2/parenrightbigg/integraldisplay Ry2nfgdy+/integraldisplay Ry2nf/parenleftbigges a∂y(N(e−svy))+h/parenrightbigg dy.10 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA Proof.Forn= 0,1,weapplyLemma 3.1asf=f,g=gandh=es a∂yN(e−svy)+h withl=1 2,m=3 2,c1(s) =r2e−s/a,c2(s) =r′/a, andc4(s) =e−s/a. Using ( 3) and (3), we have d dsE(n) 11(s) =−/integraldisplay Ry2ng2dy+3−2n 4/integraldisplay Ry2nf2 ydy+5−2n 4e−s a/integraldisplay Ry2nf2 yydy +5−2n 4r2e−s a/integraldisplay Ry2ng2dy−r′ a/integraldisplay Ry2ng2dy−2n/integraldisplay Ry2n−1fygdy −2n(2n−1)e−s a/integraldisplay Ry2n−2fyygdy−4ne−s a/integraldisplay Ry2n−1fyygydy −1 2/parenleftbigge−s a+a′ ra2/parenrightbigg/integraldisplay Ry2nf2 yydy+1 2/parenleftbigg2r′ a−r2e−s a−ra′ a2/parenrightbigg/integraldisplay Ry2ng2dy +/integraldisplay Ry2ng/parenleftbigges a∂y(N(e−svy))+h/parenrightbigg dy =−/integraldisplay Ry2ng2dy+3−2n 2E(n) 11(s) −2n/integraldisplay Ry2n−1fygdy−2n(2n−1)e−s a/integraldisplay Ry2n−2fyygdy−4ne−s a/integraldisplay Ry2n−1fyygydy −a′ 2ra2/integraldisplay Ry2nf2 yydy−ra′ 2a2/integraldisplay Ry2ng2dy +/integraldisplay Ry2ng/parenleftbigges a∂y(N(e−svy))+h/parenrightbigg dy. Similarly, we calculate d dsE(n) 12(s) =−/integraldisplay Ry2nf2 ydy−e−s a/integraldisplay Ry2nf2 yydy+1−2n 4/integraldisplay Ry2nf2dy +r2e−s a/integraldisplay Ry2ng2dy+3−2n 2r2e−s a/integraldisplay Ry2nfgdy−r′ a/integraldisplay Ry2nfgdy −2n/integraldisplay Ry2n−1ffydy−4ne−s a/integraldisplay Ry2n−1fyfyydy−2n(2n−1)e−s a/integraldisplay Ry2n−2ffyydy +/parenleftbigg2r′ a−r2e−s a−ra′ a2/parenrightbigg/integraldisplay Ry2nfgdy+/integraldisplay Ry2nf/parenleftbigges a∂y(N(e−svy))+h/parenrightbigg dy =1−2n 2E(n) 12(s)−2E(n) 11(s)+2r2e−s a/integraldisplay Ry2ng2dy −2n/integraldisplay Ry2n−1ffydy−4ne−s a/integraldisplay Ry2n−1fyfyydy−2n(2n−1)e−s a/integraldisplay Ry2n−2ffyydy +/parenleftbiggr′ a−ra′ a2/parenrightbigg/integraldisplay Ry2nfgdy+/integraldisplay Ry2nf/parenleftbigges a∂y(N(e−svy))+h/parenrightbigg dy. This completes the proof. /squareNONLINEAR DAMPED BEAM EQUATION 11 Finally, to control the bad term −4e−s a/integraldisplay Ryfyygydyind dsE(1) 11(s), we consider the energies E21(s) =1 2/integraldisplay R/parenleftbigg f2 yy+e−s af2 yyy+r2e−s ag2 y/parenrightbigg dy, E22(s) =/integraldisplay R/parenleftbigg1 2f2 y+r2e−s afygy/parenrightbigg dy. Since (fy,gy) satisfies the equations (fy)s−y 2(fy)y−fy=gy, r2e−s a/parenleftBig (gy)s−y 2(gy)y−2gy/parenrightBig +gy+r′ agy= (fy)yy−e−s a(fy)yyyy+es a∂2 yN(e−svy))+hy, we have the following lemma. Lemma 3.5. We have d dsE21(s)+/integraldisplay Rg2 ydy=5 2E21(s)−a′ 2ra2/integraldisplay Rf2 yyydy−ra′ 2a2/integraldisplay Rg2 ydy +/integraldisplay Rgy/parenleftbigges a∂2 y(N(e−svy))+hy/parenrightbigg dy and d dsE22(s)+2E21(s) =3 2E22(s)+2r2e−s a/integraldisplay Rg2 ydy+/parenleftbiggr′ a−ra′ a2/parenrightbigg/integraldisplay Rfygydy +/integraldisplay Rfy/parenleftbigges a∂2 y(N(e−svy))+hy/parenrightbigg dy. Proof.Applying Lemma 3.1asf=fy,g=gyandh=es a∂yyN(e−svy)+hywith l= 1,m= 2,n= 0,c1(s) =r2e−s a,c2(s) =r′ a, andc4(s) =e−s a, and also using ( 3) and (3), we have d dsE21(s) =−/integraldisplay Rg2 ydy+5 4/integraldisplay Rf2 yydy+7 4e−s a/integraldisplay Rf2 yyydy +7 4r2e−s a/integraldisplay Rg2 ydy−r′ a/integraldisplay Rg2 ydy −1 2/parenleftbigge−s a+a′ ra2/parenrightbigg/integraldisplay Rf2 yyydy+1 2/parenleftbigg2r′ a−r2e−s a−ra′ a2/parenrightbigg/integraldisplay Rg2 ydy +/integraldisplay Rgy/parenleftbigges a∂2 y(N(e−svy))+hy/parenrightbigg dy =−/integraldisplay Rg2 ydy+5 2E21(s) −a′ 2ra2/integraldisplay Rf2 yyydy−ra′ 2a2/integraldisplay Rg2 ydy +/integraldisplay Rgy/parenleftbigges a∂2 y(N(e−svy))+hy/parenrightbigg dy.12 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA Similarly, we have d dsE22(s) =−/integraldisplay Rf2 yydy−e−s a/integraldisplay Rf2 yyydy+3 4/integraldisplay Rf2 ydy +r2e−s a/integraldisplay Rg2 ydy+5 2r2e−s a/integraldisplay Rfygydy−r′ a/integraldisplay Rfygydy +/parenleftbigg2r′ a−r2e−s a−ra′ a2/parenrightbigg/integraldisplay Rfygydy+/integraldisplay Rfy/parenleftbigges a∂2 y(N(e−svy))+hy/parenrightbigg dy =3 2E22(s)−2E21(s) +2r2e−s a/integraldisplay Rg2 ydy+/parenleftbiggr′ a−ra′ a2/parenrightbigg/integraldisplay Rfygydy +/integraldisplay Rfy/parenleftbigges a∂2 y(N(e−svy))+hy/parenrightbigg dy. This completes the proof. /square Finally, we define Em1(s) :=1 2r2e−s ams(s)2, Em2(s) :=1 2m(s)2+r2e−s am(s)ms(s). Then, we have the following energy identities. Lemma 3.6. We have d dsEm1(s)+1 2Em1(s)+ms(s)2=/parenleftbigg3r2e−s 4a−ra′ 2a2/parenrightbigg ms(s)2 and d dsEm2(s) = 2Em1(s)+/parenleftbiggr′ a−ra′ a2/parenrightbigg m(s)ms(s). Proof.By (3) and Lemma 2.2, we have d dsEm1(s) =1 2d ds/parenleftbiggr2e−s a/parenrightbigg m2 s+r2e−s amsmss =1 2/parenleftbigg2r′ a−r2e−s a−ra′ a2/parenrightbigg m2 s +r2e−s am2 s−/parenleftbigg 1+r′ a/parenrightbigg m2 s =1 2r2e−s am2 s−1 2ra′ a2m2 s−m2 s =−1 2Em1(s)−m2 s+/parenleftbigg3r2e−s 4a−ra′ 2a2/parenrightbigg m2 s.NONLINEAR DAMPED BEAM EQUATION 13 Similarly, we have d dsEm2(s) =mms+d ds/parenleftbiggr2e−s a/parenrightbigg mms+r2e−s am2 s+r2e−s ammss =mms+/parenleftbigg2r′ a−r2e−s a−ra′ a2/parenrightbigg mms +r2e−s am2 s+r2e−s amms−/parenleftbigg 1+r′ a/parenrightbigg mms =r2e−s am2 s+/parenleftbiggr′ a−ra′ a2/parenrightbigg mms = 2Em1(s)+/parenleftbiggr′ a−ra′ a2/parenrightbigg mms. /square 3.1.Energy estimates. Now, we combine the energy identities in the previous subsection to obtain the energyestimates. First, we prepare the following estimates for remainders. Lemma 3.7. Set δ:= min/braceleftbiggβ+1 α−β+1,2α−β+1 α−β+1/bracerightbigg , which is positive if (α,β)∈Ω1. Then, we have r2e−s a∼e−β+1 α−β+1s≤Ce−δs,e−s a∼e−2α−β+1 α−β+1s≤Ce−δs, and /vextendsingle/vextendsingle/vextendsingle/vextendsinglea′ ra2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce−s a,/vextendsingle/vextendsingle/vextendsingle/vextendsinglera′ a2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cr2e−s a,/vextendsingle/vextendsingle/vextendsingle/vextendsingler′ a/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cr2e−s a. Define E0(s) :=E01(s)+c0E02(s), wherec0>0 is a sufficiently large constant determined later. Lemma 3.8. There exists a constant c0>0satisfying the following: For any η>0, there exists s0>0such that for any s≥s0, we have E0(s)≥C/parenleftbigg/integraldisplay RF2 ydy+e−s 2a/integraldisplay RF2 yydy+r2e−s 2a/integraldisplay RG2dy+/integraldisplay RF2dy/parenrightbigg and d dsE0(s)+1 2E0(s)+1 4/integraldisplay RG2dy ≤ηE0(s)+C(η)/parenleftbigg /bardblH(s)/bardbl2 L2+e2s a2/bardblN(e−svy)/bardbl2 L2/parenrightbigg .14 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA Proof.Letη >0 be arbitrary. Lemmas 3.3and3.7, and the Schwarz inequality imply d dsE01(s)+/integraldisplay RG2dy≤1 2E01(s)+C1e−s a/integraldisplay RF2 yydy+C1r2e−s a/integraldisplay RG2dy +1 2/integraldisplay RG2dy+C/parenleftbigg /bardblH(s)/bardbl2 L2+e2s a2/bardblN(e−svy)/bardbl2 L2/parenrightbigg ≤/parenleftbigg1 2+2C1/parenrightbigg E01(s)+1 2/integraldisplay RG2dy +C/parenleftbigg /bardblH(s)/bardbl2 L2+e2s a2/bardblN(e−svy)/bardbl2 L2/parenrightbigg with someC1>0 and d dsE02(s)+1 2E02(s)+2E01(s)≤C2(η1)e−δs/integraldisplay RG2dy+η1/integraldisplay RF2dy +C(η1)/parenleftbigg /bardblH(s)/bardbl2 L2+e2s a2/bardblN(e−svy)/bardbl2 L2/parenrightbigg , with someC2(η1)>0, whereη1is an arbitrary small positive number determined later. We take c0sufficiently large so that 2 c0−1 2−2C1≥1 2. Then, letting s0 sufficiently large so that c0C2(η1)e−δs≤1 4holds for any s≥s0, we conclude d dsE0(s)+1 2E0(s)+1 4/integraldisplay RG2dy ≤2c0η1/integraldisplay RF2dy+C/parenleftbigg /bardblH(s)/bardbl2 L2+e2s a2/bardblN(e−svy)/bardbl2 L2/parenrightbigg . (3.4) On the other hand, we remark that r2e−s a/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay RF(s,y)G(s,y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1 4/integraldisplay RF(s,y)2dy+C/parenleftbiggr2e−s a/parenrightbigg2/integraldisplay RG(s,y)2dy ≤1 4/integraldisplay RF(s,y)2dy+Ce−δsr2e−s 2a/integraldisplay RG(s,y)2dy. From this, retaking s0larger if needed, we have for s≥s0, E0(s)≥C/parenleftbigg/integraldisplay RF2 ydy+e−s 2a/integraldisplay RF2 yydy+r2e−s 2a/integraldisplay RG2dy+/integraldisplay RF2dy/parenrightbigg , which shows the first assertion. In particular, it gives/integraltext RF2 ydy≤CE0(s) Applying this to the right-hand side of ( 3.1) and taking η1so thatη= 2c0Cη1, we have the desired estimate. /square Next, forn= 0,1, we define E(n) 1(s) :=E(n) 11(s)+c(n) 1E(n) 12(s), wherec(0) 1andc(1) 1are sufficiently large constants determined later. The following two lemmas are the estimates for E(0) 1(s) andE(1) 1(s), respectively.NONLINEAR DAMPED BEAM EQUATION 15 Lemma 3.9. There exist positive constants c(0) 1ands(0) 1such that for any s≥s(0) 1, we have E(0) 1(s)≥C/parenleftbigg/integraldisplay Rf2 ydy+e−s 2a/integraldisplay Rf2 yydy+r2e−s 2a/integraldisplay Rg2dy+/integraldisplay Rf2dy/parenrightbigg and d dsE(0) 1(s)+1 2E(0) 1(s)+1 4/integraldisplay Rg2dy ≤CE0(s)+C/parenleftbigg /bardblh(s)/bardbl2 L2+e2s a2/vextenddouble/vextenddouble∂yN/parenleftbig e−svy/parenrightbig/vextenddouble/vextenddouble2 L2/parenrightbigg . Proof.By Lemmas 3.4and3.7, and the Schwarz inequality, we have d dsE(0) 11(s)+1 2E(0) 11(s)+/integraldisplay Rg2dy ≤2E(0) 11(s)+Ce−s a/integraldisplay Rf2 yydy+Cr2e−s a/integraldisplay Rg2dy +1 2/integraldisplay Rg2dy+C/parenleftbigg /bardblh(s)/bardbl2 L2+e2s a2/bardbl∂y/parenleftbig N/parenleftbig e−svy/parenrightbig/parenrightbig /bardbl2 L2/parenrightbigg , which implies d dsE(0) 11(s)+1 2E(0) 11(s)+1 2/integraldisplay Rg2dy ≤(2+C1)E(0) 11(s)+C/parenleftbigg /bardblh(s)/bardbl2 L2+e2s a2/bardbl∂y/parenleftbig N/parenleftbig e−svy/parenrightbig/parenrightbig /bardbl2 L2/parenrightbigg with some constant C1>0. In a similar way, we also obtain d dsE(0) 12(s)+1 2E(0) 12(s)+2E(0) 11(s) ≤C/integraldisplay Rf2dy+C2e−δs/integraldisplay Rg2dy +C/parenleftbigg /bardblh(s)/bardbl2 L2+e2s a2/bardbl∂y/parenleftbig N/parenleftbig e−svy/parenrightbig/parenrightbig /bardbl2 L2/parenrightbigg with some constant C2>0. Therefore, taking c(0) 1ands(0) 1sufficiently large so that 2c(0) 1−(2+C1)≥1 2andc(0) 1C2e−δs≤1 4holds for any s≥s(0) 1, we conclude d dsE(0) 1(s)+1 2E(0) 1(s)+1 4/integraldisplay Rg2dy ≤C/integraldisplay Rf2dy+C/parenleftbigg /bardblh(s)/bardbl2 L2+e2s a2/vextenddouble/vextenddouble∂yN/parenleftbig e−svy/parenrightbig/vextenddouble/vextenddouble2 L2/parenrightbigg . Finally,by/integraltext Rf2dy=/integraltext RF2 ydy≤CE0, theproofofthesecondassertioniscomplete. The first assertion is proved in the same way as the previous lemma an d we omit the detail. /square Lemma 3.10. There exists a constant c(1) 1>0satisfying the following: for any η′>0, there exists a constant s(1) 1>0such that for any s≥s(1) 1, we have E(1) 1(s)≥C/parenleftbigg/integraldisplay Ry2f2 ydy+e−s 2a/integraldisplay Ry2f2 yydy+r2e−s 2a/integraldisplay Ry2g2dy+/integraldisplay Ry2f2dy/parenrightbigg16 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA and d dsE(1) 1(s)+1 2E(1) 1(s)+1 4/integraldisplay Ry2g2dy ≤η′E(1) 1(s)+CE(0) 1(s)−4e−s a/integraldisplay Ryfyygydy+Ce−δs/integraldisplay Rg2dy +C(η′)/parenleftbigg /bardblyh(s)/bardbl2 L2+e2s a2/bardbly∂y(N(e−svy))/bardbl2 L2/parenrightbigg . Proof.Letη′>0 be arbitrary. By Lemmas 3.4and3.7, and the Schwarzinequality, we have d dsE(1) 11(s)+1 2E(1) 11(s)+/integraldisplay Ry2g2dy ≤E(1) 11(s)+Ce−s a/integraldisplay Ry2f2 yydy+Ce−s a/integraldisplay Rf2 yydy+Cr2e−s a/integraldisplay Ry2g2dy +1 2/integraldisplay Ry2g2dy+C/integraldisplay Rf2 ydy+Ce−δs/integraldisplay Rg2dy −4e−s a/integraldisplay Ryfyygydy+C/parenleftbigg /bardblyh(s)/bardbl2 L2+e2s a2/bardbly∂y(N(e−svy))/bardbl2 L2/parenrightbigg , which implies d dsE(1) 11(s)+1 2E(1) 11(s)+1 2/integraldisplay Ry2g2dy ≤(1+C′ 1)E(1) 11(s)+CE(0) 1(s)+Ce−δs/integraldisplay Rg2dy −4e−s a/integraldisplay Ryfyygydy+C/parenleftbigg /bardblyh(s)/bardbl2 L2+e2s a2/bardbly∂y(N(e−svy))/bardbl2 L2/parenrightbigg with some constant C′ 1>0. Next, for E(1) 12(s), Lemmas 3.4and3.7and the Schwarz inequality imply d dsE(1) 12(s)+1 2E(1) 12(s)+2E(1) 11(s) =2r2e−s a/integraldisplay Ry2g2dy−2/integraldisplay Ryffydy−4e−s a/integraldisplay Ryfyfyydy−2e−s a/integraldisplay Rffyydy +/parenleftbiggr′ a−ra′ a2/parenrightbigg/integraldisplay Ry2fgdy+/integraldisplay Ry2f/parenleftbigges a∂y(N(e−svy))+h/parenrightbigg dy ≤η′ 1/integraldisplay Ry2f2dy+E(1) 11(s)+C′ 2(η′ 1)e−δs/integraldisplay Ry2g2dy +C/integraldisplay Rf2 ydy+C/parenleftbigge−s a/parenrightbigg2/integraldisplay Rf2 yydy+C/integraldisplay Rf2dy +C/parenleftbigg /bardblyh(s)/bardbl2 L2+e2s a2/bardbly∂y(N(e−svy))/bardbl2 L2/parenrightbigg ≤η′ 1/integraldisplay Ry2f2dy+E(1) 11(s)+C′ 2(η′ 1)e−δs/integraldisplay Ry2g2dy+CE(0) 1(s) +C/parenleftbigg /bardblyh(s)/bardbl2 L2+e2s a2/bardbly∂y(N(e−svy))/bardbl2 L2/parenrightbiggNONLINEAR DAMPED BEAM EQUATION 17 for arbitrary small η′ 1>0 determined later and some constant C′ 2(η′ 1)>0. There- fore, taking c(1) 1ands(1) 1so thatc(1) 1−(1+C′ 1)≥1 2andc(1) 1C′ 2(η′ 1)e−δs≤1 4holds for anys≥s(1) 1, we conclude d dsE(1) 1+1 2E(1) 1+1 4/integraldisplay Ry2g2dy ≤η′ 1c(1) 1/integraldisplay Ry2f2dy−4e−s a/integraldisplay Ryfyygydy+C/integraldisplay Rf2 ydy+Ce−δs/integraldisplay Rg2dy +CE(0) 1(s)+C/parenleftbigg /bardblyh(s)/bardbl2 L2+e2s a2/bardbly∂y(N(e−svy))/bardbl2 L2/parenrightbigg . Takingη′ 1so that the first term of the right-hand side is bounded by η′E(1) 1(s) and using/integraltext Rf2 ydy≤CE(0) 1(s), we complete the proof of the second assertion. The first assertion is proved in the same way as before and we omit the detail. /square Next, we define E2(s) :=E21(s)+c2E22(s), wherec2is a sufficiently large constant determined later. Lemma 3.11. There exist positive constants c2ands2such that for any s≥s2, we have E2(s)≥C/parenleftbigg/integraldisplay Rf2 yydy+e−s 2a/integraldisplay Rf2 yyydy+r2e−s 2a/integraldisplay Rg2 ydy+/integraldisplay Rf2 ydy/parenrightbigg and d dsE2(s)+1 2E2(s)+1 4/integraldisplay Rg2 ydy ≤CE(0) 1(s)+C/parenleftbigg /bardbl∂yh(s)/bardbl2 L2+e2s a2/bardbl∂2 y(N(e−svy))/bardbl2 L2/parenrightbigg . Proof.By Lemmas 3.5and3.7and the Schwarz inequality, we have d dsE21(s)+/integraldisplay Rg2 ydy≤5 2E21(s)+C1e−s a/integraldisplay Rf2 yyydy+C1r2e−s a/integraldisplay Rg2 ydy +1 2/integraldisplay Rg2 ydy+C/parenleftbigg /bardblhy(s)/bardbl2 L2+e2s a2/bardbl∂2 yN(e−svy)/bardbl2 L2/parenrightbigg ≤/parenleftbigg5 2+2C1/parenrightbigg E21(s)+1 2/integraldisplay Rg2 ydy +C/parenleftbigg /bardblhy(s)/bardbl2 L2+e2s a2/bardbl∂2 yN(e−svy)/bardbl2 L2/parenrightbigg .18 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA with someC1>0 and d dsE22(s)+1 2E22(s)+2E21(s) ≤2E22(s)+Cr2e−s a/integraldisplay Rg2 ydy+Cr2e−s a/integraldisplay Rf2 ydy +C/integraldisplay Rf2 ydy+C/parenleftbigg /bardblhy(s)/bardbl2 L2+e2s a2/bardbl∂2 yN(e−svy)/bardbl2 L2/parenrightbigg ≤C2e−δs/integraldisplay Rg2 ydy+C/integraldisplay Rf2 ydy +C/parenleftbigg /bardblhy(s)/bardbl2 L2+e2s a2/bardbl∂2 yN(e−svy)/bardbl2 L2/parenrightbigg with someC2>0. We take c2sufficiently large so that 2 c2−5 2−2C1≥1 2. Then, lettings2sufficiently large so that c2C2e−δs≤1 4holds for any s≥s2, we conclude d dsE2(s)+1 2E2(s)+1 4/integraldisplay Rg2 ydy ≤C/integraldisplay Rf2 ydy+C/parenleftbigg /bardblhy(s)/bardbl2 L2+e2s a2/bardbl∂2 yN(e−svy)/bardbl2 L2/parenrightbigg for anys≥s2. This and/integraltext Rf2 ydy≤CE(0) 1(s) complete the proof of the second assertion. The first assertion is proved in the same way as before a nd we omit the detail. /square Finally, let us combine the estimates in Lemmas 3.8–3.11. Fix λ∈/parenleftbigg 0,min/braceleftbigg1 2,2(β+1) α−β+1,2α−β+1 α−β+1/bracerightbigg/parenrightbigg and lets′ ∗= max{s0,s(0) 1,s(1) 1,s2}. We first note that the Schwarz inequality and Lemma3.7imply −4e−s a/integraldisplay Ryfyygydy≤η′′E(1) 1(s)+C(η′′)e−δs/integraldisplay Rg2 ydy for anyη′′>0. We take η,η′ 1in Lemmas 3.8and3.10andη′′above so that 1 2−η≤λandη′+η′′≤1 2−λ. Then, we take ˜ c0≫˜c(0) 1≫˜c(1) 1≫1 and define E(s) = ˜c0E0(s)+˜c(0) 1E(0) 1(s)+˜c(1) 1E(1) 1(s)+E2(s)+Em1(s), G(s) =/integraldisplay R/parenleftBig ˜c0G2+˜c(0) 1g2+˜c(1) 1y2g2+g2 y/parenrightBig dy, /tildewideE(s) =E(s)+Em2(s).NONLINEAR DAMPED BEAM EQUATION 19 Then, adding the estimates in Lemmas 3.6and3.8–3.11, we conclude that d dsE(s)+λE(s)+1 4G(s)+ms(s)2 ≤Ce−δs/integraldisplay Rg2dy+Ce−δs/integraldisplay Rg2 ydy+/parenleftbigg3r2e−s 4a−ra′ 2a2/parenrightbigg ms(s)2 +C/parenleftbig /bardblH(s)/bardbl2 L2+/bardblh(s)/bardbl2 H0,1+/bardblhy(s)/bardbl2 L2/parenrightbig +C/parenleftbigges a/parenrightbigg2/parenleftBig /bardblN(e−svy)/bardbl2 L2+/vextenddouble/vextenddouble∂yN/parenleftbig e−svy/parenrightbig/vextenddouble/vextenddouble2 H0,1+/bardbl∂2 y(N(e−svy))/bardbl2 L2/parenrightBig holds fors≥s′ ∗. Moreover, Lemma 3.7leads to /parenleftbigg3r2e−s 4a−ra′ 2a2/parenrightbigg ≤Ce−δs. Therefore, we finally reach the following energy estimate. Proposition 3.12. There exist constants s∗>0andC >0such that for any s≥s∗, we have d dsE(s)+λE(s)+1 8/parenleftbig G(s)+m2 s/parenrightbig ≤C/parenleftbig /bardblH(s)/bardbl2 L2+/bardblh(s)/bardbl2 H0,1+/bardblhy(s)/bardbl2 L2/parenrightbig +C/parenleftbigges a/parenrightbigg2/parenleftBig /bardblN(e−svy)/bardbl2 L2+/vextenddouble/vextenddouble∂yN/parenleftbig e−svy/parenrightbig/vextenddouble/vextenddouble2 H0,1+/bardbl∂2 y(N(e−svy))/bardbl2 L2/parenrightBig . 4.Estimates of remainder terms and the proof of a priori estima te In this section, we give estimates of the right-hand side of Proposit ion3.12, and complete the a priori estimate, which ensures the existence of the global solution. 4.1.Estimates of remainder terms. First, by the Hardy-type inequality in Lemma3.2, we have /bardblH(s)/bardbl2 L2≤4/bardblyh(s)/bardbl2 L2,/bardblN(e−svy)/bardbl2 L2≤4/bardbly∂yN(e−svy)/bardbl2 L2. Hence, it suffices to estimate /bardblh(s)/bardbl2 H0,1,/bardblhy(s)/bardbl2 L2,/parenleftbigges a/parenrightbigg2/vextenddouble/vextenddouble∂yN/parenleftbig e−svy/parenrightbig/vextenddouble/vextenddouble2 H0,1,/parenleftbigges a/parenrightbigg2 /bardbl∂2 y(N(e−svy))/bardbl2 L2. First, from the definition of h(see (2)) and Lemma 3.7, we easily obtain /bardblh(s)/bardbl2 H0,1+/bardblhy(s)/bardbl2 L2≤Ce−2δs/parenleftbig m(s)2+ms(s)2/parenrightbig ≤e−2δs/tildewideE(s). Next, we estimate the nonlinear term. By Assumption (N), we see th at ∂yN(e−svy) = 2µe−2svyvyy+˜N′(e−svy)e−svyy, ∂2 yN(e−svy) = 2µe−2s(v2 yy+vyvyyy)+˜N′′(e−svy)e−2sv2 yy+˜N′(e−svy)e−svyyy. Therefore, by |˜N′(z)| ≤C|z|p−1, the Sobolev embedding theorem, and /bardblf/bardbl2 H2,1≤C/parenleftbigge−s a/parenrightbigg−1/parenleftbig E(0) 1(s)+E(1) 1(s)/parenrightbig ≤Ce2α−β+1 α−β+1s/tildewideE(s),20 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA we have /parenleftbigges a/parenrightbigg2/vextenddouble/vextenddouble∂yN/parenleftbig e−svy/parenrightbig/vextenddouble/vextenddouble2 H0,1 ≤C/parenleftbigges a/parenrightbigg2 e−4s/bardblvyvyy/bardbl2 H0,1+C/parenleftbigges a/parenrightbigg2 e−2ps/bardbl|vy|p−1vyy/bardbl2 H0,1 ≤Ce−2(1+α α−β+1)s/bardblvy/bardbl2 L∞/bardblvyy/bardbl2 H0,1+Ce−2(p−2)se−2(1+α α−β+1)s/bardblvy/bardbl2(p−1) L∞/bardblvyy/bardbl2 H0,1 ≤Ce−2(2α−β+1) α−β+1s/bardblvy/bardbl2 H1/bardblvyy/bardbl2 H0,1+Ce−2(p−2)se−2(2α−β+1) α−β+1s/bardblvy/bardbl2(p−1) H1/bardblvyy/bardbl2 H0,1 ≤C/parenleftBig e−2(2α−β+1) α−β+1s(/bardblf/bardbl2 H2,0+m(s)2)+e−2(p−2)se−2(2α−β+1) α−β+1s(/bardblf/bardbl2 H2,0+m(s)2)p−1/parenrightBig ×(/bardblf/bardbl2 H2,1+m(s)2) ≤Ce−2α−β+1 α−β+1s/tildewideE(s)2+e−[2(p−2)+2α−β+1 α−β+1]s/tildewideE(s)p. Similarly, by |˜N′′(z)| ≤C|z|p−2, the Sobolev embedding theorem, and /bardblf/bardbl2 H3,0≤C/parenleftbigge−s a/parenrightbigg−1/parenleftbig E(0) 1(s)+E21(s)/parenrightbig ≤Ce2α−β+1 α−β+1s/tildewideE(s), we obtain /parenleftbigges a/parenrightbigg2/vextenddouble/vextenddouble∂2 yN/parenleftbig e−svy/parenrightbig/vextenddouble/vextenddouble2 L2 ≤C/parenleftbigges a/parenrightbigg2 e−4s/parenleftbig /bardblv2 yy/bardbl2 L2+/bardblvyvyyy/bardbl2 L2/parenrightbig +C/parenleftbigges a/parenrightbigg2 e−2ps/parenleftbig /bardbl|vy|p−2v2 yy/bardbl2 L2+/bardbl|vy|p−1vyyy/bardbl2 L2/parenrightbig ≤Ce−2(1+α α−β+1)s(/bardblvyy/bardbl2 L∞/bardblvyy/bardbl2 L2+/bardblvy/bardbl2 L∞/bardblvyyy/bardbl2 L2) +Ce−2(p−2)se−2(1+α α−β+1)s(/bardblvy/bardbl2(p−2) L∞/bardblvyy/bardbl2 L∞/bardblvyy/bardbl2 L2+/bardblvy/bardbl2(p−1) L∞/bardblvyyy/bardbl2 L2) ≤Ce−2(2α−β+1) α−β+1)s(/bardblvyy/bardbl2 H1,0/bardblvyy/bardbl2 L2+/bardblvy/bardbl2 H1,0/bardblvyyy/bardbl2 L2) +Ce−2(p−2)se−2(2α−β+1) α−β+1)s(/bardblvy/bardbl2(p−2) H1,0/bardblvyy/bardbl2 L∞/bardblvyy/bardbl2 L2+/bardblvy/bardbl2(p−1) H1,0/bardblvyyy/bardbl2 L2) ≤C/parenleftBig e−2(2α−β+1) α−β+1)s(/bardblf/bardbl2 H2,0+m(s)2)+e−2(p−2)se−2(2α−β+1) α−β+1)s(/bardblf/bardbl2 H2,0+m(s)2)p−1/parenrightBig ×(/bardblf/bardbl2 H3,0+m(s)2) ≤Ce−2α−β+1 α−β+1s/tildewideE(s)2+e−[2(p−2)+2α−β+1 α−β+1]s/tildewideE(s)p. 4.2.Proof of a priori estimate. Combining the energy estimates obtained in Proposition 3.12with the estimates of remainder terms given in the previous sub- section, we deduceNONLINEAR DAMPED BEAM EQUATION 21 d dsE(s)+λE(s)+1 8/parenleftbig G(s)+ms(s)2/parenrightbig ≤Ce−2δs/tildewideE(s)+Ce−2α−β+1 α−β+1s/tildewideE(s)2+Ce−[2(p−2)+2α−β+1 α−β+1]s/tildewideE(s)p(4.1) From Lemmas 3.6and3.7, we see that d dsEm2(s) = 2Em1(s)+/parenleftbiggr′ a−ra′ a2/parenrightbigg m(s)ms(s) ≤Ce−δsms(s)2+1 16ms(s)2+Ce−2δsm(s)2. Therefore, there exists constants sm≥s∗andc>0 such that for any s≥sm, we have d ds/tildewideE(s)+λ/tildewideE(s)+c(G(s)+ms(s)2) ≤C3e−2δs/tildewideE(s)+C3/parenleftBig e−2α−β+1 α−β+1s/tildewideE(s)2+e−[2(p−2)+2α−β+1 α−β+1]s/tildewideE(s)p/parenrightBig (4.2) with someC3>0. Define Λ(s) = exp/parenleftbigg −C3/integraldisplays sme−2δσdσ/parenrightbigg . Note that Λ(s) = exp/parenleftbiggC3 2δ/parenleftbig e−2δs−e−2δsm/parenrightbig/parenrightbigg ∼1 and Λ( sm) = 1. Multiplying ( 4.2) by Λ(s), we deduce d ds/bracketleftBig Λ(s)/tildewideE(s)/bracketrightBig +λΛ(s)E(s)+cΛ(s)/parenleftbig G(s)+ms(s)2/parenrightbig ≤C3Λ(s)/parenleftBig e−2α−β+1 α−β+1s/tildewideE(s)2+e−[2(p−2)+2α−β+1 α−β+1]s/tildewideE(s)p/parenrightBig . Integrating the above over [ sm,s], we have Λ(s)/tildewideE(s)≤/tildewideE(sm)+C3/integraldisplays smΛ(σ)/parenleftBig e−2α−β+1 α−β+1s/tildewideE(σ)2+e−[2(p−2)+2α−β+1 α−β+1]s/tildewideE(σ)p/parenrightBig dσ Finally, we put /tildewideEmax(s) = max σ∈[sm,s]/tildewideE(σ) fors≥sm. Then, the above estimate implies /tildewideEmax(s)≤C0/tildewideE(sm)+C′ 0/parenleftBig /tildewideEmax(s)2+/tildewideEmax(s)p/parenrightBig with some constants C0,C′ 0>0, where we have used δ >0 andp >−β+1 α−β+1(see Remark 1.1). Thus, we conclude the a priori estimate /tildewideEmax(s)≤2C0/tildewideE(sm) (4.3) for alls≥sm, provided that /tildewideE(sm) is sufficiently small. From the local existence result (Proposition B.2), we see that, for sufficiently small initial dat a, the local solution uniquely exists over [0 ,sm], and it satisfies /tildewideE(sm)≤C(/bardblu0/bardblH2,1∩H3,0+ /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 can be controlled by the norm of initial data. This and Proposition B.2 ( iii) (blow- upalternative)indicatetheexistenceoftheglobalsolutioniftheinit ialdata(u0,u1) is sufficiently small. It remains to prove the asymptotic estimate. To this end, we go bac k to the estimate ( 4.2). By virtue of the a priori estimate ( 4.2), we have d dsE(s)+λE(s)+1 8/parenleftbig G(s)+ms(s)2/parenrightbig ≤Ce−min{2δ,2α−β+1 α−β+1,2(p−2)+2α−β+1 α−β+1}s/tildewideE(sm) =Ce−min{2(β+1) α−β+1,2α−β+1 α−β+1}s/tildewideE(sm), where we have also used /tildewideE(sm), which can be assumed without loss of generality. Now, recall λ∈/parenleftbigg 0,min/braceleftbigg1 2,2(β+1) α−β+1,2α−β+1 α−β+1/bracerightbigg/parenrightbigg , and multiply the above estimate by eλs. Then, we obtain d ds/bracketleftbig eλsE(s)/bracketrightbig +1 8eλs/parenleftbig G(s)+ms(s)2/parenrightbig ≤Ceλ−min{2(β+1) α−β+1,2α−β+1 α−β+1}s/tildewideE(sm). Integrating this over [ sm,s] implies eλsE(s)+1 8/integraldisplays smeλσ/parenleftbig G(σ)+ms(σ)2/parenrightbig dσ≤C/tildewideE(sm). Therefore, we have E(s)≤Ce−λs/tildewideE(sm) (4.4) for alls≥sm. Moreover, we deduce /integraldisplays smeλσms(σ)2dσ≤C/tildewideE(sm). This shows, for any s≥s′≥sm, |m(s)−m(s′)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplays s′ms(σ)dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤/parenleftbigg/integraldisplays s′e−λσdσ/parenrightbigg1/2/parenleftbigg/integraldisplays s′eλσms(σ)2dσ/parenrightbigg1/2 ≤/parenleftbigg1 λ(e−λs′−e−λs)/parenrightbigg1/2 C/tildewideEm(sm)1/2 →0 (s′,s→ ∞). This means that the limit m∗= lims→∞m(s) exists and satisfies |m∗−m(s)|2≤C/tildewideE(sm)e−λs for alls≥sm. Consequently, by the above estimate and ( 4.2), we have /bardblv(s)−m∗ϕ/bardbl2 L2=/bardblm(s)ϕ+f(s)−m∗ϕ/bardbl2 L2 ≤C/parenleftbig |m∗−m(s)|2/bardblϕ/bardbl2 L2+/bardblf(s)/bardbl2 L2/parenrightbig ≤Ce−λs/tildewideE(sm) ≤Ce−λs/parenleftbig /bardblu0/bardbl2 H2,1∩H3,0+/bardblu1/bardblH0,1∩H1,0/parenrightbig2NONLINEAR DAMPED BEAM EQUATION 23 fors≥sm, which implies /bardblu(t)−m∗G(R(t))/bardbl2 L2≤C(R(t)+1)−1 2−λ/parenleftbig /bardblu0/bardbl2 H2,1∩H3,0+/bardblu1/bardblH0,1∩H1,0/parenrightbig2 fort≥tm:=R−1(es−1). This completes the proof of the asymptotic estimate. Appendix A.A general lemma for the energy identity In this appendix, we give a proof of Lemma 3.1. Actually, we give a slightly more general version of it and prove the following lemma. If we take k=1 2and c3(s)≡1, then we have Lemma 3.1. Lemma A.1. Letk,l,m∈R,n∈N∪ {0}, and letcj=cj(s) (j= 1,2,3,4) be smooth functions defined on [0,∞). We consider a system for two functions f=f(s,y)andg=g(s,y)given by /braceleftBigg fs−kyfy−lf=g, c1(s)(gs−kygy−mg)+c2(s)g+g=c3(s)fyy−c4(s)fyyyy+h(s,y)∈(0,∞)×R, (A.1) whereh=h(s,y)is a given smooth function belonging to C([0,∞);H0,n(R)). We define the energies E1(s) =1 2/integraldisplay Ry2n/parenleftbig c3(s)f2 y+c4(s)f2 yy+c1(s)g2/parenrightbig dy, E2(s) =/integraldisplay Ry2n/parenleftbigg1 2f2+c1(s)fg/parenrightbigg dy. Then, we have d dsE1(s) =−/integraldisplay Ry2ng2dy+/parenleftbigg −2n−1 2k+l/parenrightbigg c3(s)/integraldisplay Ry2nf2 ydy+/parenleftbigg −2n−3 2k+l/parenrightbigg c4(s)/integraldisplay Ry2nf2 yydy +/parenleftbigg −2n+1 2k+m/parenrightbigg c1(s)/integraldisplay Ry2ng2dy−c2(s)/integraldisplay Ry2ng2dy −2nc3(s)/integraldisplay Ry2n−1fygdy−2n(2n−1)c4(s)/integraldisplay Ry2n−2fyygdy−4nc4(s)/integraldisplay Ry2n−1fyygydy +c′ 3(s) 2/integraldisplay Ry2nf2 ydy+c′ 4(s) 2/integraldisplay Ry2nf2 yydy+c′ 1(s) 2/integraldisplay Ry2ng2dy+/integraldisplay Ry2nghdy24 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA and d dsE2(s) =−c3(s)/integraldisplay Ry2nf2 ydy−c4(s)/integraldisplay Ry2nf2 yydy+/parenleftbigg −2n+1 2k+l/parenrightbigg/integraldisplay Ry2nf2dy +c1(s)/integraldisplay Ry2ng2dy+(−(2n+1)k+l+m)c1(s)/integraldisplay Ry2nfgdy−c2(s)/integraldisplay Ry2nfgdy −2nc3(s)/integraldisplay Ry2n−1ffydy−4nc4(s)/integraldisplay Ry2n−1fyfyydy−2n(2n−1)c4(s)/integraldisplay Ry2n−2ffyydy +c′ 1(s)/integraldisplay Ry2nfgdy+/integraldisplay Ry2nfhdy. Proof of Lemma A.1.We calculate d dsE1(s) =d ds/bracketleftbigg1 2/integraldisplay Ry2n/parenleftbig c3(s)f2 y+c4(s)f2 yy+c1(s)g2/parenrightbig dy/bracketrightbigg =c3(s)/integraldisplay Ry2nfyfysdy+c′ 3(s) 2/integraldisplay Ry2nf2 ydy +c4(s)/integraldisplay Ry2nfyyfyysdy+c′ 4(s) 2/integraldisplay Ry2nf2 yydy +c1(s)/integraldisplay Ry2nggsdy+c′ 1(s) 2/integraldisplay Ry2ng2dy. Using the equation ( A.1), we rewrite the above identity as d dsE1(s) =c3(s)/integraldisplay Ry2nfy(kyfy+lf+g)ydy+c′ 3(s) 2/integraldisplay Ry2nf2 ydy +c4(s)/integraldisplay Ry2nfyy(kyfy+lf+g)yydy+c′ 4(s) 2/integraldisplay Ry2nf2 yydy +c1(s)/integraldisplay Ry2ng(kygy+mg)dy−c2(s)/integraldisplay Ry2ng2dy−/integraldisplay Ry2ng2dy +c3(s)/integraldisplay Ry2ngfyydy−c4(s)/integraldisplay Ry2ngfyyyydy+/integraldisplay Ry2nghdy +c′ 1(s) 2/integraldisplay Ry2ng2dy.NONLINEAR DAMPED BEAM EQUATION 25 By noting the relations y2nfy(yfy)y=/parenleftbiggy2n+1 2f2 y/parenrightbigg y−2n−1 2y2nf2 y, y2nfyy(yfy)yy=/parenleftbiggy2n+1 2f2 yy/parenrightbigg y−2n−3 2y2nf2 yy, y2ng(ygy) =/parenleftbiggy2n+1 2g2/parenrightbigg y−2n+1 2y2ng2, y2ngfyy=/parenleftbig y2ngfy/parenrightbig y−y2nfygy−2ny2n−1fyg, y2ngfyyyy=/parenleftbig y2ngfyyy/parenrightbig y−/parenleftbig (y2ng)yfyy/parenrightbig y +/parenleftbig 2n(2n−1)y2n−2g+4ny2n−1gy+y2ngyy/parenrightbig fyy, we have d dsE1(s) =/parenleftbigg −2n−1 2k+l/parenrightbigg c3(s)/integraldisplay Ry2nf2 ydy+c3(s)/integraldisplay Ry2nfygydy+c′ 3(s) 2/integraldisplay Ry2nf2 ydy +/parenleftbigg −2n−3 2k+l/parenrightbigg c4(s)/integraldisplay Ry2nf2 yydy+c4(s)/integraldisplay Ry2nfyygyydy+c′ 4(s) 2/integraldisplay Ry2nf2 yydy +/parenleftbigg −2n+1 2k+m/parenrightbigg c1(s)/integraldisplay Ry2ng2dy−c2(s)/integraldisplay Ry2ng2dy−/integraldisplay Ry2ng2dy −c3(s)/integraldisplay Ry2nfygydy−2nc3(s)/integraldisplay Ry2n−1fygdy −2n(2n−1)c4(s)/integraldisplay Ry2n−2fyygdy−4nc4(s)/integraldisplay Ry2n−1fyygydy−c4(s)/integraldisplay Ry2nfyygyydy +/integraldisplay Ry2nghdy+c′ 1(s) 2/integraldisplay Ry2ng2dy. Thus, we conclude d dsE1(s) =−/integraldisplay Ry2ng2dy+/parenleftbigg −2n−1 2k+l/parenrightbigg c3(s)/integraldisplay Ry2nf2 ydy+/parenleftbigg −2n−3 2k+l/parenrightbigg c4(s)/integraldisplay Ry2nf2 yydy +/parenleftbigg −2n+1 2k+m/parenrightbigg c1(s)/integraldisplay Ry2ng2dy−c2(s)/integraldisplay Ry2ng2dy −2nc3(s)/integraldisplay Ry2n−1fygdy−2n(2n−1)c4(s)/integraldisplay Ry2n−2fyygdy−4nc4(s)/integraldisplay Ry2n−1fyygydy +c′ 3(s) 2/integraldisplay Ry2nf2 ydy+c′ 4(s) 2/integraldisplay Ry2nf2 yydy+c′ 1(s) 2/integraldisplay Ry2ng2dy+/integraldisplay Ry2nghdy.26 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA This gives the desired identity for E1(s). Next, we compute d dsE2(s) =d ds/bracketleftbigg/integraldisplay Ry2n/parenleftbigg1 2f2+c1(s)fg/parenrightbigg dy/bracketrightbigg =/integraldisplay Ry2nffsdy+c1(s)/integraldisplay Ry2nfsgdy+c1(s)/integraldisplay Ry2nfgsdy+c′ 1(s)/integraldisplay Ry2nfgdy. Using the equation ( A.1), we rewrite the above identity as d dsE2(s) =/integraldisplay Ry2nf(kyfy+lf+g)dy+c1(s)/integraldisplay Ry2n(kyfy+lf+g)gdy +c1(s)/integraldisplay Ry2nf(kygy+mg)dy−c2(s)/integraldisplay Ry2nfgdy−/integraldisplay Ry2nfgdy +c3(s)/integraldisplay Ry2nffyydy−c4(s)/integraldisplay Ry2nffyyyydy +/integraldisplay Ry2nfhdy+c′ 1(s)/integraldisplay Ry2nfgdy. By noting the relations y2nf(yfy) =/parenleftbiggy2n+1 2f2/parenrightbigg y−2n+1 2y2nf2, y2nf(ygy) =/parenleftbig y2n+1fg/parenrightbig y−y2n+1fyg−(2n+1)y2nfg, y2nffyy=/parenleftbig y2nffy/parenrightbig y−y2nf2 y−2ny2n−1ffy, y2nffyyyy=/parenleftbig y2nffyyy/parenrightbig y−/parenleftbig (y2nf)yfyy/parenrightbig y +(2n(2n−1)y2n−2f+4ny2n−1fy+y2nfyy)fyy, we have d dsE2(s) =/parenleftbigg −2n+1 2k+l/parenrightbigg/integraldisplay Ry2nf2dy+/integraldisplay Ry2nfgdy +kc1(s)/integraldisplay Ry2n+1fygdy+lc1(s)/integraldisplay Ry2nfgdy+c1(s)/integraldisplay Ry2ng2dy −kc1(s)/integraldisplay Ry2n+1fygdy+(−(2n+1)k+m)c1(s)/integraldisplay Ry2nfgdy −c2(s)/integraldisplay Ry2nfgdy−/integraldisplay Ry2nfgdy −c3(s)/integraldisplay Ry2nf2 ydy−2nc3(s)/integraldisplay Ry2n−1ffydy −2n(2n−1)c4(s)/integraldisplay Ry2n−2ffyydy−4nc4(s)/integraldisplay Ry2n−1fyfyydy−c4(s)/integraldisplay Ry2nf2 yydy +/integraldisplay Ry2nfhdy+c′ 1(s)/integraldisplay Ry2nfgdy.NONLINEAR DAMPED BEAM EQUATION 27 Thus, we conclude d dsE2(s) =−c3(s)/integraldisplay Ry2nf2 ydy−c4(s)/integraldisplay Ry2nf2 yydy+/parenleftbigg −2n+1 2k+l/parenrightbigg/integraldisplay Ry2nf2dy +c1(s)/integraldisplay Ry2ng2dy+(−(2n+1)k+l+m)c1(s)/integraldisplay Ry2nfgdy−c2(s)/integraldisplay Ry2nfgdy −2nc3(s)/integraldisplay Ry2n−1ffydy−4nc4(s)/integraldisplay Ry2n−1fyfyydy−2n(2n−1)c4(s)/integraldisplay Ry2n−2ffyydy +c′ 1(s)/integraldisplay Ry2nfgdy+/integraldisplay Ry2nfhdy. This completes the proof. /square Appendix B.Local existence We discuss the local existence and basic properties of solutions to ( 1). Let X=H3,0(R)×H1,0(R) and U:=/parenleftbigg u ∂tu/parenrightbigg , U0:=/parenleftbigg u0 u1/parenrightbigg . LetD(A) =H5,0(R)×H3,0(R) and define A=/parenleftbigg 0 1 −∂4 x0/parenrightbigg ,T(t) = exp(tA). We also define K(σ;U(s)) =/parenleftbigg 0 −b(σ)∂tu(s)+a(σ)∂2 xu(s)+∂xN(∂xu(s))/parenrightbigg , namely,σandsdenote the variables for the coefficients a(t),b(t) and the unknown u, respectively. Now, we introduce the definition of the strong solution and the mild so lution. Definition B.1. LetI= [0,T]with someT >0orI= [0,∞). We say that a functionu(orU=t(u,∂tu)) is a strong solution to (1)onIif U∈C(I;D(A))∩C1(I;X), d dtU(t) =AU(t)+K(t;U(t))onI, U(0) =U0. Also, we say that a function u(orU=t(u,∂tu)) is a mild solution to (1)onIif U∈C(I;X), U(t) =T(t)U0+/integraldisplayt 0T(t−s)K(s;U(s))dsinC(I;X). Proposition B.2. (i) (Local existence) For anyU0∈X, there exists T >0such that there exists a mild solution to (1)on[0,T]. (ii) (Uniqueness) LetT >0. IfUandVare mild solutions in C([0,T];X)with the same initial condition U(0) =V(0) =U0, thenU=V.28 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA (iii) (Blow-up alternative) LetTmax=Tmax(U0)be Tmax= sup{T∈(0,∞];∃U∈C([0,T];X) :a mild solution to (1)}. IfTmax<∞, thenlimt→Tmax−0/bardblU(t)/bardblX=∞. (iv) (Continuous dependence on the initial data) LetU0∈Xand{U(j) 0}∞ j=1a sequence in Xsatisfying limj→∞/bardblU(j) 0−U0/bardblX= 0. LetUandU(j)be the corre- sponding mild solutions to the initial data U0andU(j) 0, respectively. Then, for any fixedT∈(0,Tmax(U0)), we haveTmax(U(j) 0)>Tfor sufficiently large jand lim j→∞sup t∈[0,T]/bardblU(j)(t)−U(t)/bardblX= 0. (v) (Regularity) LetT >0. IfU0∈D(A), then the mild solution in (i)on[0,T] becomes a strong solution on [0,T]. (vi) (Small data almost global existence) For anyT >0, there exists ε0>0such that if/bardblU0/bardblX< ε0, then the corresponding mild solution Ucan be extended to [0,T]. (vii) (Boundedness of weighted norm) LetT >0andY:=H2,1(R)×H0,1(R). If U0∈X∩Y, then the corresponding mild solution Uon[0,T]belongs toC([0,T];X∩ Y). Proof.LetT0>0 be fixed. Then, a(t),b(t) are positive, and they and their first derivativesarebounded by someconstant CT0>0on [0,T0]. LetT∈(0,T0]. Then, for anyU=t(u,v)∈Xandt∈[0,T], we have /bardblK(t;U)/bardblX=/vextenddouble/vextenddouble−b(t)∂tu+a(t)∂2 xu+∂xN(∂xu)/vextenddouble/vextenddouble H1 ≤CT0/parenleftbig /bardbl∂tu/bardblH1+/bardbl∂2 xu/bardblH1/parenrightbig +C/parenleftBig /bardbl∂xu/bardblW1,∞+/bardbl∂xu/bardblp−1 W1,∞/parenrightBig /bardbl∂2 xu/bardblH1<∞, that is,K(t;·) :X→X. Moreover, for M >0 andU=t(u,v),W=t(w,z)∈ BM={U∈X;/bardblU/bardblX≤M}, we calculate /bardblK(t;U)−K(t;W)/bardblX≤CT0(/bardblv−z/bardblH1+/bardblu−w/bardblH3) +C(/bardblu/bardblW2,∞+/bardblw/bardblW2,∞)/bardblu−w/bardblH3 +C(|u/bardblW2,∞+/bardblw/bardblW2,∞)p−1/bardblu−w/bardblH3 ≤CT0,M/bardblU−W/bardblX. Therefore,K(t;·) islocally Lipschitzcontinuousin X. Therefore, fromthe proofs of [3, Lemmas 4.3.2, Proposition 4.3.3], there exist T >0 and a unique mild solution uonI= [0,T]. Also, [ 3, Theorem 4.3.4] shows the property (iii). Moreover, by [3, Proposition 4.3.7], the continuous dependence on the initial data. T his proves (i)–(iv). Next, we prove (iv) along with the argument of [ 3, Lemma 4.3.9]. Take U0∈ D(A) andT∈(0,Tmax). Leth >0,t∈[0,T−h], andM:= sups∈[0,T]/bardblU(s)/bardblX.NONLINEAR DAMPED BEAM EQUATION 29 Consider U(t+h)−U(t) =T(h)T(t)U0−T(t)U0 +/integraldisplayt 0T(s){K(t+h−s;U(t+h−s))−K(t−s;U(t−s)))}ds +/integraldisplayh 0T(t+s)K(h−s;U(h−s))ds =:J1+J2+J3. ForJ1,J2, we estimate /bardblJ1/bardblX≤ /bardblT(h)U0−U0/bardblX=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayh 0T(s)AU0ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble X≤h/bardblAU0/bardblX, /bardblJ3/bardblX≤hsup s∈[0,T]/bardblK(s;U(s))/bardblX. ForJ2, using the Lipschitz continuity of a,b: [0,T]→RandK(s:·) :X→X, we can show /bardblJ2/bardblX≤/integraldisplayt 0/bardblK(t+h−s;U(t+h−s))−K(t+h−s;U(t−s))/bardblXds +/integraldisplayt 0/bardblK(t+h−s;U(t−s))−K(t−s;U(t−s))/bardblXds ≤CT0,Mh+CT0,M/integraldisplayt 0/bardblU(s+h)−U(s)/bardblXds. Then, the Gronwall inequality implies /bardblU(t+h)−U(t)/bardblX≤CT0,Mh, that is,U: [0,T]→Xis Lipschitz continuous. This further leads to /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 ≤CT0,M|t−s|, i.e.,K(·;U(·)) : [0,T]→Xis Lipschitz continuous, and hence, K(·;U(·))∈ W1,1((0,T);X). This enables us to apply [ 3, Lemma 4.16] and ubecomes a strong solution. This proves (v). Next, we prove (vi). Let T >0 be arbitrary fixed, I:= [0,T], and CT,a,b:=/integraldisplayT 0(|a(s)|+|b(s)|)ds. Letε >0 be sufficiently small so that 2(1 + CT,a,b)ε <1 and let Bε={U∈ C([0,T];X); supt∈[0,T]/bardblU(t)/bardblX≤2(1 +CT,a,b)ε}. Define a map Φ : C(I:X)→ C(I;X) by Φ[U](t) :=T(t)U0+/integraldisplayt 0T(t−s)K(s;U(s))ds.30 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA Then, forU0satisfying /bardblU0/bardblX≤εandU=t(u,v)∈ Bε, we see that /bardblΦ[U](t)/bardblX≤ /bardblT(t)U0/bardblX+/integraldisplayt 0/bardblT(t−s)K(s;U(s))/bardblXds ≤ /bardblU0/bardblX+/integraldisplayt 0/parenleftbig |b(s)|/bardblv(s)/bardblH1+|a(s)|/bardbl∂2 xu(s)/bardblH1/parenrightbig ds +/integraldisplayt 0/bardbl∂xN(∂xu(s))/bardblH1ds ≤(1+CT,a,b)ε+TCN(2(1+CT,a,b)ε)2, whereCN>0 is a constant depending only on the nonlinearity N. Similarly, we have, forU,V∈ Bε, /bardblΦ[U](t)−Φ[V](t)/bardblX≤/integraldisplayt 0/bardblK(s;U(s))−K(s;V(s))/bardblXds ≤T˜CN(2(1+CT,a,b)ε) sup s∈[0,T]/bardblU(s)−V(s)/bardblX, ˜CN>0 is a constant depending only on the nonlinearity N. Therefore, taking ε further small so that TCN(2(1+CT,a,b)ε)≤1, T˜CN(2(1+CT,a,b)ε)≤1 2, we see that Φ is a contraction mapping on Bε. This and the uniqueness of mild solution imply that the mild solution obtained in (i) can be extended to [0 ,T]. Finally, we prove (vii). Let T >0,I= [0,T],U0∈Y, and letUbe the corre- spondingmildsolutionon[0 ,T]totheinitialdata U0. WeputM:= supt∈I/bardblU(t)/bardblX. In order to justify the following energy method, we take a sequenc e{U(j) 0}∞ j=1from [C∞ 0(R)]2such that lim j→∞U(j) 0=U0inX∩Y. Then, the corresponding strong solutionU(j)∈C(I;D(A))∩C1(I;X) to the data U(j) 0satisfies lim j→∞U(j)=U inC(I;X) by the continuous dependence on the initial data. In particular, t aking sufficiently large j, we may suppose that supj∈N,t∈I/bardblU(j)(t)/bardblX≤2M. Let χ∈C∞ 0(R),0≤χ≤1, χ(x) =/braceleftBigg 1 (|x| ≤1), 0 (|x| ≥2), χn(x) :=χ/parenleftBigx n/parenrightBig (n∈N). By suppχn⊂[−2n,2n], we easily see that |∂x(x2χn(x)2)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle2xχn(x)2+2x2 nχ′/parenleftBigx n/parenrightBig χn(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|x|χn(x), |∂2 x(x2χn(x)2)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle2χn(x)2+4x nχ′/parenleftBigx n/parenrightBig χn(x)+2x2 n2/parenleftbigg/parenleftBig χ′/parenleftBigx n/parenrightBig/parenrightBig2 +χ′′/parenleftBigx n/parenrightBig χn(x)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤CNONLINEAR DAMPED BEAM EQUATION 31 with some constant C >0. Denote U=t(u,∂tu),U(j)=t(u(j),∂tu(j)), and consider En(t;u) :=/integraldisplay Rx2χn(x)2/parenleftbig |∂tu(t,x)|2+a(t)|∂xu(x)|2+|∂2 xu(t,x)|2+|u(t,x)|2/parenrightbig dx, E(t;u) :=/integraldisplay Rx2/parenleftbig |∂tu(t,x)|2+a(t)|∂xu(x)|2+|∂2 xu(t,x)|2+|u(t,x)|2/parenrightbig dx. Note thatEn(t;u(j)) is finite thanks to χn. Differentiating it, we have d dtEn(t;u(j)) = 2/integraldisplay Rx2χn(x)2/parenleftBig ∂tu(j)∂2 tu(j)+a(t)∂xu(j)∂t∂xu(j)+∂2 xu(j)∂t∂2 xu(j)/parenrightBig dx +2/integraldisplay Rx2χn(x)2u(j)∂tu(j)dx+/integraldisplay Rx2χn(x)2a′(t)|∂xu(j)|2dx. By the integration by parts and using the equation ( 1), the right-hand side can be written as 2/integraldisplay Rx2χn(x)2∂tu(j)/parenleftBig −b(t)∂tu(j)+∂xN(∂xu(j))/parenrightBig dx −2/integraldisplay R∂x(x2χn(x)2)a(t)∂xu(j)∂tu(j)dx +4/integraldisplay R∂x(x2χn(x)2)a(t)∂3 xu(j)∂tu(j)dx+2/integraldisplay R∂2 x(x2χn(x)2)a(t)∂2 xu(j)∂tu(j)dx +2/integraldisplay Rx2χn(x)2u(j)∂tu(j)dx+/integraldisplay Rx2χn(x)2a′(t)|∂xu(j)|2dx. The above quantity can be further estimated by C(2M)2+CT,a,b,MEn(t;u(j)) with some constants C,CT,a,b,M>0. Hence, the Gronwall inequality implies En(t;u(j))≤˜CT,a,b,M, where the constant ˜CT,a,b,Mis independent of nandj. Lettingj→ ∞first and using the continuous dependence on the initial data, we have /integraldisplay Rx2χn(x)2/parenleftbig |∂tu(t,x)|2+a(t)|∂xu(x)|2+|∂2 xu(t,x)|2+|u(t,x)|2/parenrightbig dx≤˜CT,a,b,M. Then, letting n→ ∞, we conclude /integraldisplay Rx2/parenleftbig |∂tu(t,x)|2+a(t)|∂xu(x)|2+|∂2 xu(t,x)|2+|u(t,x)|2/parenrightbig dx≤˜CT,a,b,M, which shows U(t)∈Yfor anyt∈[0,T]. The continuity of /bardblU(t)/bardblYintfollows from the estimate |En(t;u(j))−En(s;u(j))| ≤/integraldisplayt s/vextendsingle/vextendsingle/vextendsingle/vextendsingled dσEn(σ;u(j))/vextendsingle/vextendsingle/vextendsingle/vextendsingledσ≤CT,a,b,M(t−s) fors<tand taking the limits j→ ∞andn→ ∞. /square32 M. A. HAMZA, Y. WAKASUGI AND S. YOSHIKAWA Acknowledgements This work was supported by JSPS KAKENHI Grant Numbers JP18H01 132, JP20K14346. References [1]J. M. Ball, Stability theory for an extensible beam, J. Differential Equations, 14(1973), 399–418. [2]M. Brokate, J. Sprekels , Hysteresis and Phase Transitions, Springer, New York, 199 6. [3]Th. Cazenave, A. Haraux ,An introduction to semilinear evolution equations , Oxford Uni- versity Press, 1998. [4]M. D’Abbicco, M.-R. Ebert ,Asymptotic profiles and critical exponents for a semilinear damped plate equation with time-dependent coefficients , Asymptotic Analysis 123(2021), 1–40. [5]Th. Gallay, G. Raugel, Scaling variables and asymptotic expansions in damped wave equa- tions, J. Differential Equations, 150(1998), 42–97. [6]C. Lizama, M. Murillo-Arcila ,On the dynamics of the damped extensible beam 1D- equation , J. Math. Anal. Appl., 522(2023), Paper No. 126954, 10 pp. [7]P. Marcati, K. Nishihara ,TheLp-Lqestimates of solutions to one-dimensional damped wave equations and their application to the compressible flo w through porous media , J. Dif- ferential Equations, 191(2003), 445–469. [8]K. Mochizuki ,Scattering theory for wave equations with dissipative term s, Publ. Res. Inst. Math. Sci., 12(1976), 383–390. [9]K. Nishihara, Asymptotic profile of solutions for 1-D wave equation with ti me-dependent damping and absorbing semilinear term, Asymptot. Anal., 71(2011), 185–205. [10]R. Racke, S. Yoshikawa ,Singular limits in the Cauchy problem for the damped extensi ble beam equation , J. Differential Equations, 259(2015), 1297–1322. [11]T. Suzuki, S. Yoshikawa ,Stability of the steady state for the Falk model system of sha pe memory alloys , Math. Methods Appl. Sci., 30(2007), 2233–2245. [12]H. Takeda, S. Yoshikawa ,On the initial value problem of the semilinear beam equation with weak damping II: asymptotic profiles, J. Differential Equations, 253(2012), 3061–3080. [13]Y. Wakasugi ,Scaling variables and asymptotic profiles for the semilinea r damped wave equation with variable coefficients , J. Math. Anal. Appl., 447(2017), 452–487. [14]J. Wirth ,Wave equations with time-dependent dissipation I. Non-effe ctive dissipation , J. Differential Equations, 222(2006), 487–514. [15]J. Wirth ,Wave equations with time-dependent dissipation II. Effecti ve dissipation , J. Dif- ferential Equations, 232(2007), 74–103. [16]S. Woinovsky-Krieger ,The effect of axial force on the vibration of hinged bars, J. Appl. Mech.,17(1950), 35–36. [17]S. Yoshikawa, Y. Wakasugi ,Classification of asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients: Effectiv e damping case , J. Differential Equations 272(2021), 938–957.

2023-10-29

In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients: \[ \partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u + \partial_x^4 u = \partial_x \left( N(\partial_x u) \right). \] In the authors' previous article [17], the asymptotic profile of solutions for linearized problem ($N \equiv 0$) was classified depending on the assumptions for the coefficients $a(t)$ and $b(t)$ and proved the asymptotic behavior in effective damping cases. We here give the conditions of the coefficients and the nonlinear term in order that the solution behaves as the solution for the heat equation: $b(t) \partial_t u - a(t) \partial_x^2 u=0$ asymptotically as $t \to \infty$.

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

2310.18878v1

Negative Gilbert damping in cavity optomagnonics Yunshan Caoand Peng Yany School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China Exceptional point (EP) associated with the parity-time ( PT) symmetry breaking is receiving considerable recent attention by the broad physics community. By introducing balanced gain and loss, it has been realized in photonic, acoustic, and electronic structures. However, the observation of magnonic EP remains elusive. The major challenge is to experimentally generate the negative Gilbert damping, which was thought to be highly unlikely but is demanded by the PT symmetry. In this work, we study the magneto-optical interaction of circularly-polarized lasers with a submicron magnet placed in an optical cavity. We show that the o -resonant coupling between the driving laser and cavity photon in the far-blue detuning can induce the magnetic gain (or negative damping) exactly of the Gilbert type. A hyperbolic-tangent function ansatz is found to well describe the time-resolved spin switching as the intrinsic magnetization dissipation is overcome. When the optically pumped magnet interacts with a purely lossy one, we observe a phase transition from the imbalanced to passive PTsymmetries by varying the detuning coe cient. Our findings provide a feasible way to manipulate the sign of the magnetic damping parameter and to realize the EP in cavity optomagnonics. Introduction. —One of the most fundamental principles in quantum mechanics is that a physical observable should be described by a Hermitian operator to guarantee real eigenval- ues [1]. However, Bender and Boettcher [2] reported a class of non-Hermitian Hamiltonians that allow entirely real spec- trum as long as the combined parity ( P) and time (T)-reversal symmetries are respected. By tuning system parameters, both the eigenvalues and eigenstates of the PT-symmetric Hamil- tonian simultaneously coalesce [3, 4], giving rise to a non- Hermitian degeneracy called exceptional point (EP). The na- ture around the EP that is accompanied by a phase transi- tion can trigger many intriguing phenomena, such as unidi- rectional invisibility [5, 6], loss-induced laser suppression and revival [7] and optical transparency [8], laser mode selection [9], and EP enhanced sensing [10–13]. Over the past decades, the experimental observation of EPs has been realized in a broad field of photonics [14–17], acoustics [18, 19], and elec- tronics [20–22]. Very recently, the concept of PTsymmetry is attracting significant attention in spintronics and magnonics [23–31]. The simplest way to obtain a PT-symmetric system consists in coupling two identical subsystems, one with gain and the other with equal amount of loss. The composite sys- tem isPT symmetric because space reflection interchanges the subsystems, and time reversal interchanges gain and loss. Indeed, aPT-symmetric magnetic structure composed of two identical ferromagnets with balanced gain and loss was first proposed by Lee et al. [23] and subsequently investigated by Yang et al. [27]. One recent breakthrough was made by Liu et al. [32] who reported EP in passive PT-symmetric devices in the form of a trilayer structure with two magnetic layers of dierent (positive) Gilbert damping. However, the exper- imental observation of genuine PT symmetry for magnons (the quanta of spin waves)—as elementary excitations in or- dered magnets—is still elusive. The di culty lies in that the Gilbert damping can hardly be tuned to be negative [33, 34]. The past ten years have witnessed the development and application of spin cavitronics, allowing cavity photons res- onantly coupled to magnons with the same microwave fre-quency [35–46]. One recent trend beyond microwaves is the realization of the parametric coupling between optical lasers and magnons, that would generate interesting new opportuni- ties. Tantalizing physics indeed has been demonstrated, such as nonreciprocal Brillouin light scattering [47], microwave- to-optical converting [48, 49], optical cooling of magnons [50], etc. In these studies, considerable interests have been drawn to the scalar properties of magnons, e.g., magnon num- ber (population), temperature, and chemical potential, which is successful to describe the small-angle spin precession. In contrast, their vectorial behavior, i.e., the full time-evolution of the magnetic moment driven by optical lasers, remains largely unexplored, with few exceptions [51]. It has been shown that a ferromagnetic-to-antiferromangetic phase tran- sition may emerge in the vicinity of the magnonic EP [27]. In such case, the magnetic moment would significantly deviate from its equilibrium direction, and a vectorial field descrip- tion becomes more relevant than a scalar one. S z x y Circularly polarized laser beamOptical cavity(a) (b) (c)ωlas > ωcav ωlas‘ωm Red-detuningBlue-detuning ωlas < ωcav ωlas‘ ωm FIG. 1: (a) Schematic illustration of a macrospin Sinteracting with three orthogonally propagating circularly-polarized lasers (red beams) in an optical cavity. O -resonant coupling between the driv- ing laser (!las) and the cavity photon ( !cav) mediated by magnons (!m!cav) in the blue (b) and red (c) detuning regimes. In this Letter, we propose to realize the negative GilbertarXiv:2006.16510v1 [cond-mat.mtrl-sci] 30 Jun 20202 damping by considering the optomagnonic interaction be- tween three orthogonally propagating circularly-polarized lasers and a submicron magnet placed in an optical cavity [see Fig. 1(a)]. By solving the coupled equations of motion and integrating the photon’s degree of freedom, we derive the an- alytical formula of the optical torque acting on the macrospin. In the far-blue detuning, we find that the optical torque exactly takes the Gilbert form opt S˙SSwithopt>0 (see below). The total Gilbert damping becomes negative when the intrin- sic dissipation is overcome. In such case, a hyperbolic-tangent function ansatz is found to well describe the time-resolved spin switching. We further study the optically pumped spin interacting with a purely lossy one, and observe a phase transi- tion from the imbalanced to passive PTsymmetries by vary- ing the detuning parameter. Model. —The proposed setup is schematically plotted in Fig. 1(a). Three circularly-polarized laser beams propagat- ing respectively along x;y;zdirections drive the parametric coupling with a macrospin S=(ˆSx;ˆSy;ˆSz) inside the optical cavity. The Hamiltonian reads H=~!0ˆSz~X j=x;y;z
jgjˆSj ˆcy jˆcj+Hdr; (1) where!0= B0is the Larmor frequency around the exter- nal magnetic field B0pointing to the negative z-direction with being the gyromagnetic ratio,
j=!las;j!cavis the de- tuning between the laser frequency !las;jand the cavity reso- nant frequency !cav, and ˆ cy j(ˆcj) is the creation (annihilation) operator of the optical cavity photons, with j=x;y;z. The coupling strength gjbetween the spin and optical photon orig- inates from the Faraday-induced modification of the electro- magnetic energy in ferromagnets [52]. The last term describes the interaction between the driving laser and the cavity pho- tonHdr=i~P j(Ajˆcy jh:c:), where Aj=(2jPj=~!las;j)1=2is the field amplitude, with jthe laser loss rate and Pjbeing the driving power. The Heisenberg-Langevin equations of motion for coupled photons and spins are expressed as ( ohˆoi), ˙cj=(i
jj)cjigjSjcj+Aj; (2a) ˙Sx=!0Sy+gynySzgznzSy; (2b) ˙Sy=!0SxgxnxSz+gznzSx; (2c) ˙Sz=gynySx+gxnxSy; (2d) where nj=hˆcy jˆcjiis the average photon number in the cav- ity. Because the spin dynamics usually is much slower than optical photons, one can expand the cavity photon operator as cj(t)cj0(t)+cj1(t)+


, in orders of ˙Sj. Equation (2a) then can be recast in series 0=(i
jj)cj0igjSjcj0+Aj; (3a) ˙cj0=(i
jj)(cj0+cj1)igjSj(cj0+cj1)+Aj;(3b) by keeping up to the first-order terms. We can therefore derivethe formula of photon number in the cavity nj(t) jcj0j2+2Re[ c j0cj1] =A2 j (
jgjSj)2+2 j4jA2 jgj(
jgjSj) h (
jgjSj)2+2 ji3˙Sj:(4) Substituting (4) into Eqs. (2b)-(2d), we obtain ˙S= SBe+ S(˙SS)optS; (5) where the e ective magnetic field Be=B0ez+Boptin- cludes both the external magnetic field and the optically in- duced magnetic field Bopt=X j 1gjA2 j (
jgjSj)2+2 jej; (6) which is the zeroth-order of ˙Sj. The second term in the right hand side of (5) is the intrinsic Gilbert damping torque, with S=jSjthe total spin number and  > 0 being the intrinsic Gilbert damping constant. The last term in (5) represents the optical torque with the anisotropic e ective field opt=X j4jA2 jg2 j(
jgjSj) h (
jgjSj)2+2 ji3˙Sjej; (7) which is linear with the first-order time-derivative of Sj. Be- low, we show that the anisotropic nature of (7) can be smeared out under proper conditions. Negative Gilbert damping. —To obtain the optical torque of exactly the Gilbert form, we make two assumptions: (i) the three laser beams are identical, i.e., Aj=A;gj=g;j=; and
j=
; (ii) the optomagnonic coupling works in the far detuning regime, i.e., jj1 with=
=(gS), which allows us to drop the gjSjterms in Eq. (7). The optically induced eective fields then take the simple form Bopt= 1gA2
2+2X jej; (8) and opt=opt S˙S;withopt=4A2g2S
(
2+2)3(9) being the laser-induced magnetic gain or loss that depends the sign of the detuning
. Based on the above results, we finally obtain the optically modulated spin dynamics ˙S= SBe+e S(˙SS); (10) withe=opt+. One can observe that a negative ef- fective Gilbert constant ( e<0) emerges in the far-blue de- tuning regime, i.e., 1 <  <  c. In case of the red detun- ing ( < 0), we have opt<0, which indicates the enhance- ment of the magnetic attenuation. In the deep-blue detuning3 ηηηPTηC ηC=7.11η P (W)P (μWĎ r (m)αopt /α Bopt (μT)αeff =0 Bopt =333 μT(a) (b)(c) (d) ηPTηCBopt =440 μTαeff =-α × FIG. 2: Optically induced magnetic gain (a) and magnetic field (b) vs. the optical detuning parameter . (c)PT(orange) and C(green) as a function of the driving laser power. (d) Radius dependence of the laser power at the compensation point C=7:11. regime (> c), driving lasers can still generate the magnetic gain (opt>0) but cannot compensate the intrinsic dissipa- tion, i.e., 0 <  opt< . Herecis the critical detuning pa- rameter at which the e ective Gilbert damping vanishes. The physics can be understood from the diagram plotted in Figs. 1(b) and 1(c): In the blue detuning regime ( !las> ! cav), mi- crowave magnons are emitted in the non-resonant interaction between the driving laser and the cavity photon, representing a magnetic gain. On the contrary, they are absorbed in the red detuning (!las< ! cav), manifesting a magnon absorption or cooling. Below we discuss practical materials and parameters to realize this proposal. Materials realizations. —For a ferromagnetic insulator like yttrium ion garnet (YIG), the intrinsic Gilbert constant typ- ically ranges 103105[53–55]. We take =104 in the following calculations. The magneto-optical coupling strength is determined by the Faraday rotation coe cientF of the materials gS'cF=pr, with cthe speed of light and rthe relative permittivity (for YIG, we choose r=15 [56] andF=188=cm [57]). We thus have gS=21 GHz. The optical cavity is set at the resonant frequency !cav=2=100 THz with the loss rate =2=1 GHz. For a YIG sphere of radius r=10 nm and spin density s1028m3, we esti- mate the total spin number S=sr3104and the coupling strength g=20:1 MHz. Materials parameters are summa- rized in Table I. Because g, all interesting physics occurs in the weak coupling regime. A negative eis demanded for realizing thePTsymmetry in magnetic system. Considering the driving laser with a fixed power P=1W, the e ective TABLE I: Parameters for optical cavity and YIG. !cav=2 = 2 ! 0=2 gS=2 r 100 THz 1 GHz 10 GHz 1 GHz 10 nm 104Gilbert-type magnetic gain is e=atPT'6:16, and the critical gain-loss point e=0 occurs atC'7:11, indeed satisfying the large-detuning condition jj1 in deriving (9). Figure 2(a) shows the monotonically decreasing dependence of the optically induced magnetic gain opton the detuning pa- rameter. The-dependence of the optical field is plotted in Fig. 2(b), showing that it monotonically decreases with the in- creasing of the detuning, too. Enhancing the laser power will push the two critical points CandPTinto the deep detuning region, as demonstrated in Fig. 2(c). For a magnetic sphere of larger volume (1 m)3(1 mm)3that contains a total spin number S=10101019with the reduced magneto-optical coupling strength g=2=1011010Hz, the required laser power then should be 6 15 orders of magnitude higher than the nm-scale sphere case, as shown in Fig. 2(d). Time-resolved spin flipping. —To justify the approximation adopted in deriving the Gilbert-type magnetic gain, we di- rectly simulate the time evolution of the unit spin components (sjSj=S) based on both Eq. (5) and Eq. (10). Numer- ical results are, respectively, plotted in Figs. 3(a) and 3(b) for the same detuning parameter =1:8 (corresponding to an eective magnetic gain e=0:0453) and!0=2=10 GHz. Both figures show that the very presence of the negative Gilbert damping can flip the spin in a precessional manner, with similar switching curves. The fast Fourier transforma- tion (FFT) analysis of the spatiotemporal oscillation of sxalso confirms this point (see the insets). Although the analytical form of sz(t) by solving (5) generally is unknown [58, 59], we find an ansatz that can well describe the time-resolved spin switching sj sj szη=1.8 (αeff =-0.0453 ) (a) (c) (d) (b)szsysx ττ τ ηEq. (5) Eq. (10)τ0=98.9 (a) fitting (b) fitting Theoryτ0=107.8’’ τ0=100.4’τp=22.1τp=22.4’’τp=14.9’ tanh(- )τ-τ0τp τ0’ τ0’’ Theory τp’ τp’’ Theory46810121401020 9.779.71 Frequency (GHz)Frequency (GHz)FFT of s x FFT of s x46810121401020 FIG. 3: Time evolution of unit spin components ( sx;sy;sz) at de- tuning=1:8 based on Eq. (5) (a) and Eq. (10) (b). Insets show the FFT spectrum of sx. (c) Theoretical fittings of szusing the hyperbolic-tangent ansatz (11) (dashed curves). The solid green curve is the analytical formula without any fitting. (d) Numerical re- sults of the -dependence of the two characteristic times 0andp, comparing with formula (12) (solid curves).4 sz()'tanh 0 p! ; (11) which is reminiscent of the Walker solution for modeling the profile of 180magnetic domain wall [60] by replacing the time coordinate with the space coordinate x. Here0is the switching time, prepresents the life-time of uniform magnons, and =!0t. From perturbation theory, we derive the analytical form of these two parameters p=1+2 e e;and0=ptanh1vt 14B2 opt B2 e:(12) Figure 3(c) shows the time evolution of sz. Symbols repre- sent the numerical results, dashed curves label the theoretical fittings of ansatz (11), and the solid curve is the analytical for- mula without fitting. The fitted switching time 0 0=100:4 (00 0=107:8) and magnon life-time 0 p=14:9 (00 p=22:4) from from Eq. (5) [Eq. (10)] compare well with the analyt- ical formula (12) which gives 0=98:9 andp=22:1. We further show that the analytical ansatz agrees excellently with numerical results in a broad range of detuning parameters, as plotted in Fig. 3(d). Phase transition in spin dimers. —We have shown that un- der proper conditions, one can realize the Gilbert-type mag- netic gain which is essential for observing PT-symmetry in purely magnetic structures. Next, we consider the optically pumped spin Sinteracting with a lossy one S0, as shown in Fig. 4(a). The coupled spin dynamics is described by the Landau-Lifshitz-Gilbert equation ˙s= sBe+!exss0+e˙ss; (13a) ˙s0= s0B0 e+!exs0s+˙s0s0; (13b) where s(0)S(0)=Sis the unit spin vector. Since the optically induced magnetic field is the same order of magnitude with the geomagnetic field (much smaller than B0), it can be safely ignored. Spin s0is exchange coupled to the optically pumped spins, and su ers an intrinsic Gilbert damping. If e=, the two-spin system satisfies the PT-symmetry: Eqs. (13) are invariant in the combined operation of the parity P(s$s0 andBe$B0 e) and the time reversal T(t! t,s!s, s0!s0,Be!Be, and B0 e!B0 e). Assuming a harmonic time-dependence for the small-angle spin precession sx;y(t)=sx;yei!twithjsx;yj  1, one can solve the eigenspectrum of Eqs. (13). By tuning the spin- spin coupling strength !ex, we observe a transition from exact PT phase to the broken PT phase, separated by the EP at !c ex=2=1 MHz for=PT=6:16, as shown in Figs. 4(b) and 4(c). Interestingly, the unequal gain and loss, i.e., e<0 ande,, leads to an imbalanced parity-time ( IPT )- symmetry. In this region ( > IPT=5:66), the eigenfrequen- cies have di erent real parts but share the identical imaginary one, as plotted in Fig. 4(d). A passive parity-time ( PPT )- symmetry is further identified when e>0. In such case Re[(ω-ω0)/2π] (MHz)(b)(a) (d) (c) ωex /2π (MHz)Im[(ω-ω0)/2π] (MHz) ηη=ηPT ηPTηPPT ηIPTωex 2πc =1 MHz(e) ωexs s’ ωex 2π=1.5 MHzFIG. 4: (a) Spin dimmer consisting of an optically pumped spin sand a purely lossy one s0. Evolution of eigenfrequencies vs. the exchange coupling (b,c) at the detuning PT=6:16, and vs. the detuning pa- rameter (d,e) at the exchange coupling !ex=2=1:5 MHz. ( >  PPT=7:11), the imaginary part of both branches is smaller than their intrinsic damping [see Fig. 4(e)]. Discussion. —In the above derivation, we focus on the case that the intrinsic Gilbert damping is isotropic. Our approach can also be generalized to treat the case when the intrinsic damping is anisotrpic [61, 62]. The three propagating lasers then should be accordingly adjusted to match the tensor form of the intrinsic magnetic damping, by modulating the driving power or the frequency of each beam, for instance. The red- detuning region is appealing to cool magnons to the subtle quantum domain. Inspired by PT-symmetric optics [19], we envision a giant enhancement of the magnonic gain and an ultralow-threshold magnon lasing in a two-cavity system with balanced optical gain and loss, which is an open question for future study. While the magnonic passive PTsymmetry has been observed by Liu et al. [32], the exact and imbalanced PTphases are still waiting for the experimental discovery. Conclusion. —To summarize, we have proposed an opto- magnonic method to generate the negative Gilbert damp- ing in ferromagnets, by studying the parametric dynamics of a macrospin coupled with three orthogonally propagating circularly-polarized lasers in an optical cavity. We analyti- cally derived the formula of the optical torque on the spin and identified the condition for the magnetic gain exactly in the Gilbert form. We found a hyperbolic-tangent function reminiscent of the Walker ansatz to well describe the time- resolved spin switching when the intrinsic damping is over- come. We finally investigated the spectrum of exchange cou- pled spin dimers. By varying the detuning parameter, we ob- served phase transitions from imbalanced to exact, and to pas- sivePTsymmetries. Our findings suggest an experimentally5 feasible way to achieve negative Gilbert damping that is es- sential for studying the PTphysics and for observing the EP in magnonic systems. We acknowledge the helpful discussion with Z. Z. Sun. This work was supported by the National Natural Sci- ence Foundation of China (Grants No. 11704060 and No. 11604041) and the National Key Research Development Pro- gram under Contract No. 2016YFA0300801. Corresponding author: yunshan.cao@uestc.edu.cn yCorresponding author: yan@uestc.edu.cn [1] P. A. M. Dirac, The Principles of Quantum Mechanics , (4th ed., Oxford University Press, 1958). [2] C. M. 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2020-06-30

Exceptional point (EP) associated with the parity-time (PT) symmetry breaking is receiving considerable recent attention by the broad physics community. By introducing balanced gain and loss, it has been realized in photonic, acoustic, and electronic structures. However, the observation of magnonic EP remains elusive. The major challenge is to experimentally generate the negative Gilbert damping, which was thought to be highly unlikely but is demanded by the PT symmetry. In this work, we study the magneto-optical interaction of circularly-polarized lasers with a submicron magnet placed in an optical cavity. We show that the off-resonant coupling between the driving laser and cavity photon in the far-blue detuning can induce the magnetic gain (or negative damping) exactly of the Gilbert type. A hyperbolic-tangent function ansatz is found to well describe the time-resolved spin switching as the intrinsic magnetization dissipation is overcome. When the optically pumped magnet interacts with a purely lossy one, we observe a phase transition from the imbalanced to passive PT symmetries by varying the detuning coeffcient. Our findings provide a feasible way to manipulate the sign of the magnetic damping parameter and to realize the EP in cavity optomagnonics.

Negative Gilbert damping in cavity optomagnonics

2006.16510v1

arXiv:1401.6467v2 [cond-mat.mtrl-sci] 24 Jan 2016Wavenumber-dependent Gilbert damping in metallic ferroma gnets Y. Li and W. E. Bailey Dept. of Applied Physics & Applied Mathematics, Columbia University, New York NY 10027, USA (Dated: November 21, 2021) A wavenumber-dependentdissipative term to magnetization dynamics, mirroring the conservative term associated with exchange, has been proposed recently f or ferromagnetic metals. We present measurements ofwavenumber-( k-)dependentGilbert dampinginthree metallic ferromagnet s, NiFe, Co, and CoFeB, using perpendicular spin wave resonance up to 26 GHz. In the thinnest films accessible, where classical eddy-current damping is negli gible, size effects of Gilbert damping for the lowest and first excited modes support the existence of a k2term. The new term is clearly separable from interfacial damping typically attributed to spin pump ing. Higher-order modes in thicker films do not show evidence of enhanced damping, attributed to a com plicating role of conductivity and inhomogeneous broadening. Our extracted magnitude of the k2term, ∆α∗ kE= ∆α∗ 0+A∗ kk2where A∗ k=0.08-0.1 nm2in the three materials, is an order of magnitude lower than th at identified in prior experiments on patterned elements. The dynamical behavior of magnetization for ferro- magnets (FMs) can be described by the Landau-Lifshitz- Gilbert (LLG) equation[1]: ˙ m=−µ0|γ|m×Heff+αm×˙ m (1) whereµ0is the vacuum permeability, m=M/Msis the reduced magnetization unit vector, Heffis the effective magnetic field, γis the gyromagnetic ratio, and αis the Gilbert damping parameter. The LLG equation can be equivalently formulated, for small-angle motion, in terms of a single complex effective field along the equilibrium direction, as ˜Heff=Heff-iαω/|γ|; damping torque is in- cluded in the imaginary part of ˜Heff. For all novel spin-transport related terms to the LLG identified so far[2–7], each real (conservative) effective field term is mirrored by an imaginary (dissipative) counterpart. In spin-transfer torque, there exist both conventional[2, 3] and field-like[8] terms in the dynamics. In spin-orbit torques (spin Hall[4] and Rashba[6] effect) dampinglike and fieldlike components have been theoret- ically predicted[9] and most terms have been experimen- tally identified[5, 6]. For pumped spin current[7], theory predictsrealandimaginaryspinmixingconductances[10] g↑↓ randg↑↓ iwhich introduce imaginary and real effective fields, respectively. It is well known that the exchange interaction, respon- sible for ferromagnetism, contributes a real effective field (fieldlike torque) quadratic in wavenumber kfor spin waves[11]. It isthennaturaltoaskwhetheracorrespond- ing imaginary effective field might exist, contributing a dampinglike torque to spin waves. Theoretically such an interaction has been predicted due to the intralayer spin- current transport in a spin wave[12–15], reflected as an additional term in Eq. (1): ˙ m=···−(|γ|σ⊥/Ms)m×∇2˙ m (2) whereσ⊥is the transverse spin conductivity. This term represents a continuum analog of the well-established in-terlayer spin pumping effect[7, 16, 17]. For spin wave resonance (SWR) with well-defined wavenumber k, Eq. (2) generates an additional Gilbert damping ∆ α(k) = (|γ|σ⊥/Ms)k2. In this context, Gilbert damping refers to an intrinsic relaxation mechanism in which the field- swept resonance linewidth is proportional to frequency. Remarkably, the possible existence of such a term has not been addressed in prior SWR measurements. Previ- ous studies of ferromagneticresonance (FMR) linewidths of spin waves[18–21] were typically operated at fixed fre- quency, not allowing separation of intrinsic (Gilbert) and extrinsic linewidths. Experiments have been carried out on thick FM films, susceptible to a large eddy current damping contribution[22]. Any wavenumber-dependent linewidth broadening in these systems has been at- tributed to eddy currents or inhomogeneous broadening, not intrinsic torques which appear in the LLG equation. In this Manuscript, wepresent a study of wavenumber- dependent Gilbert damping in the commonly applied ferromagnetic films Ni 79Fe21(Py), Co, and CoFeB. A broad range of film thicknesses (25-200 nm) has been studied in order to exclude eddy-current effects. We observe a thickness-dependent difference in the Gilbert damping for uniform and first excited spin wave modes which is explained well by the intralayer spin pump- ing model[14]. Corrections for interfacial damping, or conventional spin pumping, have been applied and are found to be small. The measurements show that the wavenumber-dependent damping, as identified in contin- uousfilms, isinreasonableagreementwith thetransverse spin relaxation lengths measured in Ref. [23], but an or- der of magnitude smaller than identified in experiments on sub-micron patterned Py elements[24]. Two different types of thin-film heterostructures were investigated in this study. Films were deposited by UHV sputtering with conditions given in Ref. [23, 25]. Multilayers with the structure Si/SiO 2(substrate)/Ta(5 nm)/Cu(5 nm)/ FM(tFM)/Cu(5 nm)/Ta(5 nm), where2 FM= Py, Co and CoFeB and tFM= 25-200 nm, were designed to separate the effects of eddy-current damp- ing and the intralayer damping mechanism proposed in Eq. (2). The minimum thickness investigated here is our detection threshold for the first SWR mode, 25 nm. A second type of heterostructure focused on much thin- ner Py films, with the structure Si/SiO 2(substrate)/Ta(5 nm)/Cu(5 nm)/Py( tPy)/Cu(5 nm)/ X(5 nm),tPy= 3-30 nm. Here the cap layer X= Ta or SiO 2was changed, for two series of this type, in order to isolate the effect of interfacial damping (spin pumping) from Cu/Ta inter- faces. To study the Gilbert damping behavior of finite- wavenumber spin waves in the samples, we have excited perpendicular standing spin wave resonance (PSSWR)[26] using a coplanar waveguide from 3 to 26 GHz. The spin-wave mode dispersion is given by the Kittel equation ω(k)/|γ|=µ0(Hres−Ms+Hex(k)); the effective field from exchange, µ0Hex(k) = (2Aex/Ms)k2 withAexas the exchange stiffness, gives a precise mea- surement of the wavenumber excited ((Fig. 1 inset)). PSSWR modes are indexed by the number of nodes p, withk=pπ/tFMin the limit of unpinned surface spins. The full-width half-maximum linewidth, ∆ H1/2, is fit- ted using µ0∆H1/2(ω) =µ0∆H0+ 2αω/|γ|to extract the Gilbert damping α. Forp= 1 modes we fix µ0∆H0 as the values extracted from the corresponding p= 0 modes for ( tFM≤40 nm), because frequency ranges are reduced due to large exchange fields. In unconstrained fits for films of this thickness, the inhomogeneous broad- eningµ0∆H0of thep= 1 modes does not exhibit a discernible trend with 1 /t2 FM(ork2)[19–21], justifying this approximation[27]. To fit our data, we have solved Maxwell’s equations and the LLG equation (Eq. 1), including novel torques such as those given in Eq. (2), according to the method of Rado[28]. The model (designated ’EM+LLG’) is de- scribed in the Supplemental Information. Values calcu- lated using the EM+LLG model are shown with curves in Fig. 1 and dashed lines in Fig. 4. Comparison with such a model has been necessary since in our first type of sample series, tFM= 25-200 nm, eddy-current damping is negligible for thinner films (25 nm), the Akk2contri- bution is negligible for thicker films (200 nm), but the two effects coexist for the intermediate region. In Fig. 1(a-c)we comparethe measuredGilbert damp- ing for the uniform ( p= 0,αu) and first excited ( p= 1, αs)spinwavemodes. Thedominantthickness-dependent contribution to Gilbert damping of the uniform modes of Py, Co, and CoFeB is clearly due to eddy currents which arequadraticinthickness. Notethateddy-currentdamp- ing is negligible for the thinnest films investigated (25 nm), but quite significant for the thickest films (200 nm). This term sums with the bulk Gilbert damping α0[29]. The simulation of αu, shown by black curves in Fig. 1, matches closely with the analytical expression for bulkand eddy-current damping only[30] of αu=αu0+αE0, whereαE0=µ2 0γMst2 FM/12ρcdenotes the eddy-current damping for uniform modes. Fittings of αuyield resis- tivitiesρc= 16.7, 26.4 and 36.4 µΩ·cm for Py, Co and CoFeB, respectively. Unlike the uniform-mode damping, the 1st SWR (×10 -3 ) uu s u s(a) (b) (c) Co CoFeB Py μ0HB (T). . .p=0 p=1 Py 75nm μ 0Hex x50 s (×10 -3 ) kus-= FIG. 1. Thickness dependence of αuandαsfor (a) Py, (b) Co and (c) CoFeB thin films. Curves are calculated from a combined solution of Maxwell’s equations and the LLG (EM+LLG). For αuthe values of µ0Ms,α(Table I), effective spin mixing conductance (Supplemental Information Sectio n C) g-factor (2.12 for Py and CoFeB and 2.15 for Co) and ρc (from analytical fitting) are used. For αsthe values of ∆ α∗ kE and ∆α∗ k0(Table I) are also included in the simulation. Inset: 10 GHz FMR spectra of p= 0 and p= 1 modes in Py 75 nm film. mode damping αsis found to exhibit a minimum as a function of thickness. For decreasing thicknesses below 75 nm,αsis increased. This behavior indicates an addi- tional source of Gilbert damping for the 1st SWR modes. In CoFeB the increased αsis less visible in Fig. 1(c) due to fluctuations in damping for samples of different thick- ness, but is evident in the difference, αs−αu, plotted in Fig. 2. In order to isolate this new damping mechanism, we plot in Fig. 2 the increased damping for the 1st SWR mode, ∆ αk=αs−αu, side-by-side with exchange field µ0Hexas a function of ( π/tFM)2taken as the wavenum- berk2. When π/tFMis large, a linear k2dependence of ∆αkin all three ferromagnets mirrors the linear de- pendence of µ0Hexonk2. This parallel behavior reflects the wavenumber-dependent imaginary and real effective fields acting on magnetization, respectively. To quantify the quadratic wavenumber term in ∆ αk, we also show the eddy-current-corrected values ∆ αkE=∆αk−∆αEin Fig. 2(a). Here ∆ αE=αE1−αE0denotes the differ- ence in eddy current damping between p= 1 and p= 0 modes according to the theory of Ref. [30], for weak sur- face pinning, where αE1≈0.23αE0(See Supplemental Information for more details). We then fit this eddy-3(×10 -3 ) Py Co CoFeB Py 150 nm (a) (b) (π/t FM )2 (×10 16 m -2 )Py Co CoFeB kk FIG. 2. Imaginary (damping, a) and real (exchange, b) ef- fective fields as a function of k2for Py, Co and CoFeB. (a) Additional SWR damping ∆ αk(circle) and eddy-current cor- rected value ∆ αkE(cross) as a function of ( π/tFM)2. Solid lines are guides to eye and dashed lines are fits to Eq. (3). (b) Exchange field µ0Hexas a function of ( π/tFM)2((pπ/tFM)2, p=0-6, for Py 150 nm). Lines are fits to µ0Hex= (2A/Ms)k2. current-corrected value to a linearization of Eq. (2), as: ∆αkE= ∆αk0+Akk2(3) withAk=|γ|σ⊥/Msand ∆αk0a constant offset. The values of Akestimated this way are 0 .128±0.022 nm2, 0.100±0.011 nm2and 0.100±0.018 nm2for Py, Co and CoFeB. Recently, Kapelrud et al.[31] have predicted that interface-localized (e.g. spin-pumping) damping terms will also be increased in SWR, with interfacial terms forp≥1 modes a factor of two greater than those for thep= 0 mode. Using the second series of thinner Py films, we have applied corrections for the interfacial term to our data, and find that these effects introduce only a minor ( ∼20%) correction to the estimate of Ak. Thep= 0 mode damping associated with the Cu/Ta interface has been measured from the increase in damping upon replacement of SiO 2with Ta at the top surface (Fig. 3, inset). Here Cu/SiO 2is taken as a reference with zero interfacial damping; insulating layers have been shown to have no spin pumping contribution[32]. We find the damping enhancement to be inversely proportional to tFM, indicating an interfa- cial damping term quantified as spin pumping into Ta[7] with ∆αsp=γ¯h(g↑↓/S)/4πMstFM. Using the values in Table I yields the effective spin mixing conductance asg↑↓ Py/Cu/Ta/S=2.5 nm−2, roughly a factor of three smaller than that contributed by Cu/Pt interfaces[17]. Using the fitted g↑↓ FM/Cu/Ta/S, we calculate andcorrect for the additional spin pumping contribution to damping of the p= 1 mode, 2∆ αsp(from top and bot- tom interfaces). The corrected values for the 1st SWR damping enhancement, ∆ α∗ kE= ∆αkE−2∆αsp, are plotted for Py(25-200nm)in Fig. 3. These correctionsdo not change the result significantly. We fit the k2depen- dence of ∆ α∗ kEto Eq. (3) to extract the corrected values A∗ kand ∆α∗ k0. The fitted value, A∗ k= 0.105±0.021 nm2 for Py, is slightly smaller than the uncorrected value Ak. Other extracted interfacial-corrected values A∗ kare listed in Table I. Note that the correction of wavenumber by finite surface anisotropy will only introduce a small correction of AkandA∗ kwithin errorbars. We also show the EM+LLG numerical simulation results for the uniform modes and the first SWR modes in Fig. 1 (solid curves). Those curves coincide with the analytical expressions of eddy-current damping plus k2damping (not shown) and fit the experimental data points nicely. The negative offsets ∆ α∗ k0between uniform modes (π/t FM )2 (×10 16 m -2 )(×10 -3 )*(= ) **u0 u0 tFM (nm) FIG. 3. Interfacial damping correction for Py. Main panel: ∆αkEand ∆α∗ kEas a function of ( π/tFM)2. Dashed lines are fits to k2-dependent equation as Eq. (3); ∆ α∗ k0are ex- tracted from ∆ α∗ kEfits.Inset:size effect of uniform-modes Gilbert damping in Py/Cu/Ta and Py/Cu/SiO 2samples (cir- cles). The dashed curve is the theoretical reproduction of Py/Cu/SiO 2usingαu0+ ∆αsp(tFM). The shadow is the same reproduction using αu0+ ∆αsp(tFM) +A∗ kk2where the error of shadow is from A∗ k. Here kis determined by Aexk2= 2Ks/tFM. and spin wave modes for Py and CoFeB are attributed to resistivitylike intrinsic damping[33]: because ˙ mis averaged through the whole film for uniform modes and maximized at the interfaces for unpinned boundary condition, the SWR mode experiences a lower resistivity near low-resistivity Cu and thus a reduced value of damping. For Co a transition state between resistivity- like and conductivitylike mechanisms[34] corresponds to negligible ∆ α∗ k0as observed in this work. In addition to the thickness-dependent comparison of p= 0 and p= 1 modes, we have also measured Gilbert4 damping for a series of higher-order modes in a thick Py (150 nm) film. Eddy-current damping ( αE∼0.003) is the dominant mode-dependent contribution in this film. The wavenumber kfor the mode p= 6 is roughly equal to that for the first SWR, p= 1, in the 25 nm film. Resonance positions are plotted with the dashed lines in Fig. 2(b), as a function of k, and are in good agreement with those found from the p= 1 data. In Fig. 4 we plot the mode-related Gilbert damping αpup to p= 6, which gradually decreases as pincreases. We have again conducted full numerical simulations using the EM+LLG method with ( A∗ k= 0.105 nm2) or without (A∗ k= 0) the intralayer spin pumping term, shown in red and black crosses, respectively. Neither scenario fits the data closely; an increase at p= 3 is closer to the model including the k2mechanism, but experimental α atp= 6 falls well below either calculation. We believe there are two possibilities why the α∝p2 damping term is not evident in this configuration. First, the effective exchange field increases with p, resulting in a weaker (perpendicular) resonance field at the same frequency. When the perpendicular biasing field at resonance is close to the saturation field, the spins near the boundary are not fully saturated, which might produce an inhomogeneous linewidth broadening at lower frequencies and mask small Gilbert contributions from wavenumber effect. From the data in Fig. 4 inset the high- pSWR modes is more affected by this inhomo- geneous broadening and complicate the extraction of k2 damping. Second, high- pmodes in thick films are close to the anomalous conductivity regime, kλM∼1, where λMis the electronic mean free path. The Rado-type model such as that applied in Fig. 4 is no longer valid in this limit[35], beyond which Gilbert damping has been shown to decrease significantly in Ni and Co[36]. Based on published ρλMproducts for Py[37] and our experimental value of ρc= 16.7µΩ·cm, we find λM∼8 nm andkλM∼1 for the p= 6 mode in Py 150 nm. For the 1st SWR mode in Py 25 nm, on the other hand, eddy currents are negligible and the anomalous behavior is likely suppressed due to surface scattering, which reducesλM. An important conclusion of our work is that the intralayer spin pumping, as measured classically through PSSWR, is indeed present but more than 10 times smaller than estimated in single nanoscale ellipses[24]. The advantages of the PSSWR measurements presented in this manuscript are that the one-dimensional mode profile is well-defined, two-magnon effects are reduced, if not absent[39], and there are no lithographic edges to complicate the analysis. The lower estimates of A∗ kfrom PSSWR aresensible, basedonphysicalparametersofPy, Co, and CoFeB. The polarization of continuum-pumped spins in a nearly uniformly magnetized film, like that of pumped spin current in a parallel-magnetized F/N/F structure, is transverse to the magnetization[14]. Fromthe measured transverse spin conductance σ⊥we extract that the relaxation lengths of pumping intralayer spin current are 0.8-1.9 nm for the three ferromagnets[27], in good agreement with the small transverse spin coherence lengths found in these same ferromagneticmetals[23, 40]. Finally, we show that the magnitude of the intralayer mT p=1, tFM =25-200 nm p=0-6, t FM =150 nm (pπ/t FM )2 (×1016 m -2 ) p* * FIG.4. Mode-dependentdamping αpfor Py(150nm), 0 ≤p≤ 6. Crosses are EM+LLG calculated values with and without the wavenumber-dependent damping term. Inset: Inhomoge- neous broadening ∆ H0vs 0≤p≤6, 150nm film. Larger, k-dependent values are evident, compared with those in the thickness series ( tFM=25-200 nm). spin pumping identified here is consistent with the damping size effect notattributable to interlayer spin pumping, in layers without obvious spin sinks. For the p= 0 mode, a small but finite wavenumber is set by the surface anisotropy through[30, 41] Aexk2= 2Ks/tFM. The damping enhancement due to intralayer spin pumping will, like the interlayer spin pumping, be inverse in thickness, leading to an ’interfacial’ term as α= 2Ks(A∗ k/Aex)t−1 FM. This contribution is indicated by the grey shadow in Fig. 3 insetand provides a good account of the additional size effect in the SiO 2- capped film. Here we use Ks=0.11 mJ/m2extracted by fitting the thickness-dependent magnetization to µ0Meff=µ0Ms−4Ks/MstFM. While alternate contributions to the observed damping size effect for the SiO2-capped film cannot be ruled out, the data in Fig. 3insetplace an upper bound on A∗ k. In summary, we have identified a wavenumber- dependent, Gilbert-type damping contribution to spin waves in nearly uniformly magnetized, continuous films of the metallic ferromagnets Py, Co and CoFeB using classical spin wave resonance. The term varies quadratically with wavenumber, ∆ α∼A∗ kk2, with the magnitude, A∗ k∼0.08-0.10 nm2, amounting to ∼20% of the bulk damping in the first excited mode of a 25 nm film of Py or Co, roughly an order of magnitude smaller than previously identified in patterned elements. The measurements quantify this texture-related contribution5 to magnetization dynamics in the limit of nearly homo- geneous magnetization. µ0Ms(T)α0Aex(J/m) A∗ k(nm2)∆α∗ 0 Py 1.00 0.0073 1.2×10−110.11±0.02 -0.0008 Co 1.47 0.0070 3.1×10−110.08±0.01 -0.0002 CoFeB 1.53 0.0051 1.8×10−110.09±0.02 -0.0011 TABLE I. Fit parameters extracted from resonance fields and linewidths of uniform and 1st SWR modes. Values of A∗ k and ∆α∗ 0for Co and CoFeB are calculated using the spin mixing conductances measured in FM/Cu/Pt[17]. See the Supplemental Material for details. [1] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [2] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [3] L. Berger, Phys. Rev. B 54, 9353 (1996). [4] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). [5] S. O. Valenzuela and M. Tinkham, Nature442, 176 (2006). [6] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel and P. Gambardella, Nature Mater.9, 230 (2010). [7] Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [8] S. Zhang, P. M. Levy and A. Fert, Phys. Rev. Lett. 88, 236601 (2002). [9] P. M. Haney, H. W. Lee, K. J. Lee, A. Manchon and M. D. Stiles, Phys. Rev. B 87, 174411 (2013). [10] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas and G. E. W. Bauer, Phys. Rev. B 71, 064420 (2005). [11] C. Kittel, Phys. Rev. 81, 869 (1951). [12] E. M. Hankiewicz, G. Vignale and Y. Tserkovnyak, Phys. Rev. B78, 020404(R) (2008). [13] J. Foros, A. Brataas, Y. Tserkovnyak and G. E. W. Bauer,Phys. Rev. B 78, 140402(R) (2008). [14] Y. Tserkovnyak, E. M. Hankiewicz and G. Vignale, Phys. Rev. B79, 094415 (2009).[15] S. Zhang and Steven S.-L. Zhang, Phys. Rev. Lett. 102, 086601 (2009). [16] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002). [17] A. Ghosh, J. F. Sierra, S. Auffret, U. Ebels and W. E. Bailey,Appl. Phys. Lett. 98, 052508 (2011). [18] P. E. Wigen, Phys. Rev. 133, A1557 (1964). [19] T. G. Phillips and H. M. Rosenberg, Phys. Lett. 8, 298 (1964). [20] G. C. Bailey, J. Appl. Phys. 41, 5232 (1970). [21] F. Schreiber and Z. Frait, Phys. Rev. B 54, 6473 (1996). [22] P. Pincus, Phys. Rev. 118, 658 (1960). [23] A. Ghosh, S. Auffret, U. Ebels and W. E. Bailey, Phys. Rev. Lett. 109, 127202 (2012). [24] H. T. Nembach, J. M. Shaw, C. T. Boone and T. J. Silva, Phys. Rev. Lett. 110, 117201 (2013). [25] Y. Li, Y. Lu and W. E. Bailey, J. Appl. Phys. 113, 17B506 (2013). [26] M. H. Seavey and P. E. Tannenwald, Phys. Rev. Lett. 1, 168 (1958). [27] See the Supplemental Information for details. [28] W. S. Ament and G. T. Rado, Phys. Rev. 97, 1558 (1955). [29] C. Scheck, L. Cheng and W. E. Bailey, Appl. Phys. Lett. 88, 252510 (2006). [30] M. Jirsa, phys. stat. sol. (b) 113, 679 (1982). [31] A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111, 097602 (2013). [32] O. Mosendz, J. E. Pearson, F. Y. Fradin, S. D. Bader and A. Hoffmann, Appl. Phys. Lett. 96, 022502 (2009). [33] K. Gilmore, Y. U. Idzerda and M. D. Stiles, Phys. Rev. Lett.99, 027204 (2007). [34] S. M. Bhagat andP.Lubitz, Phys. Rev. B 10, 179(1974). [35] G. T. Rado, J. Appl. Phys. 29, 330 (1958). [36] V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769 (1972). [37] B. A. Gurney, V. S. Speriosu, J.-P. Nozieres, H. Lefakis , D. R. Wilhoit and O. U. Need, Phys. Rev. Lett. 71, 4023 (1993). [38] J. N. Lloyd and S. M. Bhagat, Solid State Commun. 8, 2009 (1970). [39] R. D. McMichael, M. D. Stiles, P. J. Chen and W. F. Egelhoff Jr., J. Appl. Phys. 83, 7037 (1998). [40] J. Zhang, P. M. Levy, S. F. Zhang and V. Antropov, Phys. Rev. Lett. 93, 256602 (2004). [41] R. F. Soohoo, Phys. Rev. 131, 594 (1963).

2014-01-24

New terms to the dynamical equation of magnetization motion, associated with spin transport, have been reported over the past several years. Each newly identified term is thought to possess both a real and an imaginary effective field leading to fieldlike and dampinglike torques on magnetization. Here we show that three metallic ferromagnets possess an imaginary effective-field term which mirrors the well-known real effective-field term associated with exchange in spin waves. Using perpendicular standing spin wave resonance between 2-26 GHz, we evaluate the magnitude of the finite-wavenumber ($k$) dependent Gilbert damping $\alpha$ in three typical device ferromagnets, Ni$_{79}$Fe$_{21}$, Co, and Co$_{40}$Fe$_{40}$B$_{20}$, and demonstrate for the first time the presence of a $k^2$ term as $\Delta\alpha=\Delta\alpha_0+A_{k}\cdot k^2$ in all three metals. We interpret the new term as the continuum analog of spin pumping, predicted recently, and show that its magnitude, $A_{k}$=0.07-0.1 nm$^2$, is consistent with transverse spin relaxation lengths as measured by conventional (interlayer) spin pumping.

Wavenumber-dependent Gilbert damping in metallic ferromagnets

1401.6467v2

Sub-micrometer yttrium iron garnet LPE lms with low ferromagnetic resonance losses Carsten Dubs,1Oleksii Surzhenko,1Ralf Linke,1Andreas Danilewsky,2Uwe Br uckner,3and Jan Dellith3 1)INNOVENT e.V., Technologieentwicklung, Pr ussingstr. 27B, 07745 Jena, Germany 2)Kristallographie, Albert-Ludwigs-Universit at Freiburg, Hermann-Herder-Str. 5, 79104 Freiburg, Germany 3)Leibniz-Institut f ur Photonische Technologien (IPHT), Albert-Einstein-Str. 9, 07745 Jena, Germany (Dated: 30 August 2016) Using liquid phase epitaxy (LPE) technique (111) yttrium iron garnet (YIG) lms with thicknesses of 100 nm and surface roughnesses as low as 0.3 nm have been grown as a basic material for spin-wave propagation experiments in microstructured waveguides. The continuously strained lms exhibit nearly perfect crys- tallinity without signicant mosaicity and with eective lattice mists of
a?=as104and below. The lm/substrate interface is extremely sharp without broad interdiusion layer formation. All LPE lms ex- hibit a nearly bulk-like saturation magnetization of (1800 20) Gs and an `easy cone' anisotropy type with extremely small in-plane coercive elds <0.2 Oe. There is a rather weak in-plane magnetic anisotropy with a pronounced six-fold symmetry observed for saturation eld <1.5 Oe. No signicant out-of-plane anisotropy is observed, but a weak dependence of the eective magnetization on the lattice mist is detected. The narrowest ferromagnetic resonance linewidth is determined to be 1.4 Oe @ 6.5 GHz which is the lowest values reported so far for YIG lms of 100 nm thicknesses and below. The Gilbert damping coecient for investigated LPE lms is estimated to be close to 1 104. PACS numbers: 81.15.Lm, 75.50.Gg, 76.50.+g I. INTRODUCTION Magnonics is an increasingly growing new branch of spin-wave physics, specically addressing the use of magnons for information transport and processing1{4. Single crystalline yttrium iron garnet (YIG), which is a ferrimagnetic insulator with the smallest known magnetic relaxation parameter5, appears to be a superior candi- date for this purpose6{8. As bulk or as thick lm mate- rial, which is commonly grown by liquid phase epitaxy (LPE)9, it has a very low damping coecient and allows magnons to propagate over distances exceeding several centimeters6. However YIG functional layers for practi- cal magnonics should be nanometer-thin with extremely smooth surfaces in order to achieve optimum eciency in data processing and dramatic reduction in energy con- sumption of sophisticated spin-wave devices. Therefore, high-quality thin and ultra-thin YIG lms were grown us- ing dierent growth techniques such as LPE, pulsed laser deposition (PLD) and rf-magnetron sputtering to inves- tigate diverse spin-wave eects and to design YIG waveg- uides as well as nanostructures for spin wave excitation, manipulation and detection in prospective magnonic cir- cuits. From previous reports about sub-micrometer YIG lms with thicknesses between 100 and 20 nm10{15avail- able microwave and magnetic key parameters were taken and summarized in Table I. Thus, ferromagnetic reso- nance (FMR) data were included which have been ex- tracted from measurements of the absorption curves or absorption derivative curves versus sweeping magnetic in-plane eld Hat a xed frequency for vs. sweep-ing rf-exciting eld hrfwith an applied in-plane static magnetic bias eld. The reported FMR linewidths
H and converted peak to peak linewidths
Hppof the eld derivative values (
H=p 3
Hpp), which will be given during the further paper as full-width at half- maximum
HFWHM , varied between 3 Oe and 13 Oe. The Gilbert damping coecient were found in the range from 2104to 8104. Only the lowest given  value of 0:9104was obtained for a very short t range of about 4 GHz without any given data in the low fre- quency range below 10 GHz13and is therefore not really comparable with the other reported values. From this compilation it is obvious that neither
HFWHM noris signicantly in uenced by the YIG lm thickness down to 20 nm. The dierences are probably resulted from additional ferromagnetic losses due to contributions of homogeneous and/or inhomogeneous broadening by mi- crostructural imperfections or magnetic inhomogeneities. In this report we present microstructural, magnetic and FMR properties of LPE-grown 100 nm thin YIG and Lanthanum substituted (La:YIG) lms with low ferro- magnetic resonance losses. Film thicknesses were deter- mined by X-ray re ectometry (XRR) and surface rough- ness by atomic force microscopy (AFM) measurements. Crystalline perfection and compositional homogeneity were investigated by high-resolution X-ray diraction (HR-XRD) and X-ray photoelectron spectroscopy (XPS) as well as by secondary ion mass spectroscopy (SIMS). Static and dynamic (microwave) magnetic characteriza- tions were carried out by vibrating sample magnetometry (VSM) and by Vector Network Analysis (VNA), respec- tively.arXiv:1608.08043v1 [cond-mat.mtrl-sci] 29 Aug 20162 TABLE I: Key parameters reported for thin/ultrathin YIG lms on (111) GGG substrates Growth method Thick- RMS- 4 MsaHca
Haf0
H0a (Reference) ness roughness FWHM FWHM 104 (nm) (nm) (kGs) (Oe) (Oe) (GHz) (Oe) LPE10100 - 1.81 - 3.0 7 1.6 2.8 LPE (this study) 83{113 0.3{0.8 1.78{1.82 0.2 1.4{1.6 6.5 0.5{0.7 1.2{1.7 PLDb 1179 0.2 1.72 <2 3.0 10 1.4 2.2 PLD1223 - 1.60 <1 3.5c9.6 3.5{7c2{4 Sputtering1322 0.13 1.78 0.4 12c16.5 6.4c0.9 Sputtering1420 0.2 - 0.4 13c9.7 7c8 PLD1520 0.2{0.3 2.10 0.2 3.3c6 2.4c2.3 aMeasurements at RT with the in-plane external magnetic eld H bYIG lms grown on the (100) GGG substrates cPeak-to-peak value
Hppof the derivative of FMR absorption transformed into
HFWHM =
Hppp 3 II. RESULTS A. Microstructural properties Selected microstructural and magnetic properties of liquid phase epitaxial grown YIG (sample A-C) and La:YIG (sample D) lms are given in Table II. The con- sistent magnetic as well as microwave properties obtained for lms deposited during dierent growth runs demon- strate a high reproducibility of the LPE growth tech- nique. Fig. 1a shows XRR plots of lms with thicknesses of about 100 nm which are smaller than the previously re- ported thinnest LPE YIG lms16{18. The smallest root- mean-square (RMS) surface roughness of about 0.25 nm obtained for the sample B in Fig. 1b is nearly compa- rable with epi-polished GGG substrate quality of 0.15 nm and with the best PLD and sputtered YIG lms (see e.g. Table I). Besides, lms with slightly rougher surfaces (see Table II)) were obtained as a result of additional dendritic aftergrowth and/or due to plateau formation, so called \mesas", if any solution droplet adheres to the sample surface. HR-XRD studies of our thin epitaxial LPE lms have been found to be dicult because of the nearly super- imposed diraction pattern of YIG lm and GGG sub- strate. Although the angle distances between lm and substrate Bragg re ections were above the resolution limit of our HR-XRD equipment, the diraction inten- sity of the lm re ection was very low and results only in a broadening of the GGG Bragg re ection. Fig. 2a shows a!-scan (rocking curve) with a Gaussian-like t- ted GGG substrate 444 re ection and a second tted peak at the right shoulder which corresponds to the YIG 444 lm re ection. This indicates a tensile stressed YIG lm because of the smaller lm lattice parameter com- pared to the commercially available Czochralski-grown GGG substrate ( as=1.2382 nm). For La:YIG lms we ob- served a perfect pseudo-Voigt tted substrate peak with- out any additional shoulder (not shown) which indicates /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 /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 /s80/s111/s115/s105/s116/s105/s111/s110/s32 /s32 /s40/s67/s117/s45/s75/s76 /s50/s44/s51/s41/s83/s97/s109/s112/s108/s101/s32/s67 /s83/s97/s109/s112/s108/s101/s32/s68(a) (b) FIG. 1: (a) XRR plots of sub-micrometer-thick YIG LPE lms. (b) 55m2AFM surface topography of sample B with RMS roughness of 0.25 nm. a perfect lattice match between substrate and LPE lm. This is in remarkable contrast to YIG lms deposited by various gas phase techniques such as PLD and rf-3 TABLE II: YIG/La:YIG lm properties grown on (111) GGG substrates by LPE technology Thick- RMS- Relative lattice VSMaFMRa Sample ness roughness mist
a?=as 4MsHc 4Me
HFWHMb
H0 (nm) (nm) 104(kGs) (Oe) (kGs) (Oe) (Oe) 104 A 113 0.8 4.7 1.82 0.10 1.637 1.4 0.5 1.4 B 106 0.3 1.8 1.78 0.20 1.658 1.5 0.7 1.2 C 83 0.6 0.3 1.82 0.16 1.672 1.4 0.5 1.6 Dc97 0.8 0.0 1.78 0.18 1.712 1.6 0.7 1.7 Accuracy 1 0:1 0:3 0:04 0:03 0:010 0:1 0:1 0:1 aVSM and FMR measurements at room temperature with applied in-plane magnetic eld bFMR linewidth value at frequency f=6.5 GHz cLa:YIG LPE lm sputtering11{15,19,20on GGG substrates. For those lms the YIG re ection has always been detected at consider- ably lower Bragg angles compared to the GGG substrate indicating a signicant distortion of the cubic YIG garnet cell with signicantly enlarged lattice parameters (com- pressive stress)19,21. The relative eective mist
a?=as= (asa? YIG)=as obtained from strained lm lattice parameter in growth directiona? YIGand the substrate lattice parameter ascan be used as a measure for epitaxial induced in-plane ten- sion or strain. Due to YIG Poisson's ratio of P= 0:29 pseudomorphously grown, fully strained YIG lms with an ideal YIG bulklattice parameter aYIG= 1:2375 nm22 should have a relative eective mist of
a?=as= +11104(tensile stress). In the case of our sub- micrometer YIG lms
a?=ashas been determined to be in the range between zero and +5 104(see Ta- ble II) compared to PLD-grown YIG lms with up to
a?=as=100104(see e.g. Ref.12). Hence, our LPE lms are under tension but not to the extent which we expected for nominally pure YIG material without additional lattice expansion by lattice defects or impu- rities. To nd the reason for this, high-resolution re- ciprocal space map (HR-RSM) and XPS investigations were performed. Fig. 2b shows a HR-RSM plot around the symmetrical 444 Bragg re ection with symmetrical diracted intensity for the GGG substrate and asym- metric diracted intensity toward higher scattering an- gles along Qz(2-!-Scan) which we attribute to the YIG 444 lm re ection. Broadening of the lm re ec- tion along Qzis due to the nite coherence lenght of the sub-micrometer thin lm in growth direction and other broadening mechanisms as for example heteroge- neous strain. The extension of the lm re ection up to the substrate peak position suggests that the lm is continuously strained due to an existing compositional and/or strain gradient. No peak broadening along the Qxdirection (!-scan) indicates single crystalline perfec- tion parallel to the lm plane without signicant mosaic- ity due to tilts of epitaxial regions with respect to one another. To evaluate the compositional homogeneity along the 25.34 25.36 25.38 25.401000200030004000 YIG 444 FWHM=0.011□Intensity (cps) Angle ω (deg) Sample B Gaussian fit GGG Gaussian fit YIG Cumulative fit FWHM=0.0058□GGG 444(a) /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 /s81 /s88/s32/s40/s49/s47/s110/s109/s41/s81 /s90/s32/s40/s49/s47/s110/s109/s41 /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 /s52/s52/s52 /s83/s97/s109/s112/s108/s101/s32/s65 (b) FIG. 2: (a) HR-XRD !-scan around substrate/lm 444 Bragg re ection of sample B. By tting procedures the YIG lm peak has been extracted. (b) HR-RSM scans around substrate/lm 444 reciprocal point reveal asymmetric diracted intensities towards higher Q z values for sample A. growth direction of the lms and to detect expected impurities (e.g. Pb from solvent) depth prole analy-4 0 500 1000 1500 2000 2500 30000246810 substrate Intensity (a.u.) Etch time (s) Y 3p3 Fe 2p O 1s Gd 3d5 Ga 2p3 Pb 4f< 5 nm YIG/GGG interfaceYIG film (a) 0 250 500 750 1000 1250 15001234 GasubstrateLa:YIG film La:YIG/GGG interfaceIntensity 139La, 69Ga (a.u.) Etch time (s) La< 11 nm x 100 (b) FIG. 3: (a) XPS depth prole of sample B reveals a very narrow interface between lm and substrate. The Pb 4f signal could not be detected within the detection limit of about 0.1 at-%. (b) SIMS depth prole analysis detects the139La signal of the lm as well as the69Ga signal of the substrate (sample D) and their changes at the lm/substrate interface. ses were carried out by XPS. Fig. 3a shows a homo- geneous distribution of the YIG matrix elements along the lm growth direction and a sharp transition at the lm/substrate interface. The obtained width of the tran- sition layer for sample B is below 5 nm. But the obtained depth prole consists of a convolution of the true concen- tration prole with the depth resolution of the XPS sys- tem under the concrete measuring conditions and should be narrower. Therefore, these proles demonstrate that no broad interdiusion layer is formed by element in- termixing at the interface at an early state of epitaxial growth or by diusion of substrate ions into the epitaxial layer and vice versa during the subsequent growth pro- cess. Whereas XPS surface analysis of the very rst atomic layers (not shown) gives a Pb content of about 0.2 at-%, no Pb signal could be observed during the depth proleanalyses within the detection limit of 0.1 at-%23. There- fore, it is assumed that the Pb signal corresponds to a surface contamination of condensed PbO vapor from high temperature solution and this contamination is com- pletely removed by the rst argon-ion etching step. For YIG lms grown in La 2O3containing solution no La sig- nal could be detected by XPS that give indicates that the La content must be below 0.5 at-%24. In order to improve the detection capability additional qualitative SIMS measurements were carried out. Due to the result- ing sputtering eect and by time-dependent detection of the sputtered sample ions one obtains depth proles of the lm elements as shown for139La in Fig. 3b. Here, the counts of two separate measurements taken under identi- cal measuring conditions at neighboring sample positions were added up in order to enhance the statistical signi- cance. It is clearly visible that the lanthanum signal de- creases at the lm/substrate interface whereas substrate signals like69:7Ga simultaneously increase. B. Static magnetic measurements The vibrating sample magnetometry was used to mea- sure the net magnetic moment mof the YIG/GGG sam- ples at room temperature. As a thickness of GGG sub- strates5000 times exceeded these of the studied YIG lms, a proper calculation of the YIG parameters re- quired us (i) to extract the GGG contribution that lin- early increased with the external eld Hand (ii) to prefer the in-plane sample orientation that ensured considerably lower elds Hsfor the YIG lms to attain the saturation. Fig. 4a presents a typical dependence of the total mag- netic moment mvs the in-plane magnetic eld Hand illustrates the method allowing us to separate the m' components produced by the YIG lm and the GGG substrate. Being subsequently normalized to the lm volume, the YIG component loops yield the following material parameters { a saturation magnetization Ms, a coercivityHcand a saturation eld Hs, i.e. the eld (av- eraged over ascending H"and descending H#branches of hysteresis loops) where the YIG lm magnetization approaches 0.9Ms. In order to estimate the in-plane anisotropy, we have repeated this procedure for the sam- ples rotated around the h111iaxis perpendicular to lm surfaces. Fig. 4b demonstrates such results as polar semi-log plots vs the azimuthal angle '. A saturation magnetization Msin Fig. 4b seems independent of '. The obtained 4 Msvalues cluster around 1800 Gs usu- ally reported25for bulk YIG single crystals. Within an experimental error (mostly dened by the YIG volume uncertainity of2%), the same is valid for the 4 Msval- ues in other LPE lms listed in Table II. The obtained coercivity ( Hc0:2 Oe) in studied LPE lms is among the best values reported for gas phase epitaxial lms (see Table I). No distinct in uence of the crystallographic ori- entation on the Hcvalues is also registered. In contrast, the azimuthal dependence of the saturation eld Hsob-5 -150 -100 -50 0 50 100 150-800-600-400-2000200400600800 -2 0 2-2000200m (µemu) H (Oe) YIG+GGG YIG GGG m (µemu) H (Oe) (a) /s48/s46/s49/s49/s54/s48 /s49/s50/s48 /s49/s56/s48 /s50/s52/s48 /s51/s48/s48/s48/s46/s49 /s49/s32/s72 /s67/s32/s44/s32/s72 /s83/s32/s40/s79/s101/s41/s32/s32/s32/s32/s32/s32/s32/s32/s52 /s77 /s83/s32/s40/s107/s71/s115/s41 /s32/s32/s52 /s77 /s115 /s32/s72 /s115 /s32/s72 /s99 (b) FIG. 4: (a) The net VSM magnetic moment mof the sample D as well as its components induced by the YIG lm and the GGG substrate vs the in-plane magnetic eldHparallel to theh110idirection. (b) Azimuthal angle dependencies for the VSM loop parameters of sample D, i.e. a saturation magnetization Ms, a saturation eld Hsand a coercivity Hc. TheHssix-fold symmetry with the mimima along the h112i`easy axes' and the maxima along the h110i`hard axes' indicates the cubic magnetocrystalline anisotropy. viously reveals the six-fold symmetry which matches the crystallographic symmetry of YIGs. The Hsmaxima co- incide with the in-plane h110iprojections of the hard magnetization axes, whereas the Hsminima correspond to theh112icrystallographic directions. The h112i`easy axes' orientation suggests an `easy cone' anisotropy after Ubizskii26. He has also demonstrated27that relatively small in-plane magnetic elds lead to single-domain YIG lms, although a deviation of magnetization vector from the lm plane still remains due to nite values of the cubic anisotropy constants. In conclusion, as the demagnetizing factor at the out- of-plane YIG lm orientation is 1, the out-of-plane satu- ration eld has to be close to the in-plane 4 Msvalues. This fact is qualitatively conrmed by our out-of-plane measurements. Unfortunately, the GGG component ofthe total VSM signal at elds H?1:8 kOe is much larger than magnetic moments of YIG lms with a thick- ness of100 nm (see, for instance, Fig. 4a) and, hence, a reasonable accuracy of 0.5 % at the GGG signal elim- ination inevitably results in too large errors for the YIG parameters. One may conclude that the out-of-plane con- guration may provide reliable results when the ratio of YIG to GGG thickness exceeds, at least, 103. C. FMR absorption FMR absorption spectra for each of studied YIG lms were recorded at several values ( H5 kOe) of the in- plane magnetic eld. The inset in Fig. 5 shows such a spectrum at H= 1:6 kOe that looks like the Lorentz function with a linewidth
fFWHM4 MHz centered near the FMR frequency f6:5 GHz. Since the FMR linewidth is mostly expressed in units of magnetic eld, we, at rst, used the centers fof measured spectra and the corresponding in-plane elds Hto estimate the gy- romagnetic ratio and the eective magnetization Me in the Kittel formula28 f= p H(H+ 4Me): (1) Then, the best tting pair of andMeallowed us (i) to convert every frequency spectrum into the magnetic eld scale, (ii) to t rescaled spectra with the Lorentz function and (iii) to evaluate, thereby, the corresponding linewidth
HFWHM . The selected results of the described proce- dure { namely, 4 Meand
HFWHM at the reference frequencyf= 6:5 GHz { are listed in Table II, while the whole summary of the obtained
HFWHM values is 0 5 10 15012 012 6.48 6.49 6.500.0000.0050.0100.015 A: 1.4 x10-4; 0.5 Oe B: 1.2 x10-4; 0.7 Oe C: 1.6 x10-4; 0.5 Oe D: 1.7 x10-4; 0.7 Oe ∆HFWHM (Oe) f (GHz)Sample: α ; ∆H0 f (GHz)1-S21 FIG. 5: Frequency dependence of FMR absorption linewidth
HFWHM for YIG LPE lms A{D at various values of the in-plane magnetic eld ( H5 kOe). Straight lines are linear ts that the Gilbert damping factorsare obtained from. Inset shows an example of FMR absorption spectrum measured for the sample A atH= 1:6 kOe.6 0 5 10 15 200.00.51.01.52.02.5 α = 1.2×10-4 α = 0.7×10-4 α = 0.4×10-4 d = 106 nm d = 410 nm d = 3.0 µm d = 300 µm∆HFWHM (Oe) f (GHz)α = 0.5×10-4 FIG. 6: Frequency dependencies of the FMR linewidth
HFWHM for YIG LPE lms of various thickness dand the YIG sphere with diameter d= 300m. The Gilbert damping factors are calculated from slopes of the best linear ts according to Eq. (2). presented in Fig. 5 vs the FMR frequency. The plots in Fig. 5 are known10{14to provide data about the Gilbert damping coecient and the inhomogeneous contribu- tion
H0to the FMR linewidth that are mutually related by
HFWHM =
H0+2f (2) As the FMR performance of thin YIG lms strongly depends on the working frequency of future magnonic applications, we have included various quality parame- ters in Table II, viz. i) the Gilbert damping coecient which is mostly responsible for the FMR losses at high magnetic elds ( H4Me), ii) the inhomoge- neous contribution
H0that dominates at small elds (H4Me) as well as iii) the FMR linewidth at the reference frequency f=6.5 GHz which approximately cor- responds to the case H4Me. The latter is estimated down to
HFWHM =1.4 Oe that is to our knowledge the narrowest value reported so far for YIG lms with a thick- ness of about 100 nm and smaller. The Gilbert damping coecients are estimated to be close to 1104 which is comparable to the best values reported so far (compare with Table I). The zero frequency term
H0is found almost the same for all YIG lms including the La substituted one. The obtained value
H00:50:7 Oe appears as well appreciably lower than that for gas phase epitaxial lms (see Table I). In summary, optimized LPE growth and post- processing conditions improve FMR linewidths and Gilbert damping coecients (compare this study and Ref.29 with Ref.10). However, the improved values are still far from these in bulk YIGs and relatively thick YIG lms (see Fig. 6) due to the decreasing volume to inter- face ratio in sub-micrometer lms. For example, imper- fections at the lm interface of thin lms should have astronger in uence on the magnetic losses in contrast to the dominating volume properties of perfect thick lms. It requires us to undertake further attempts to mini- mize the FMR performance deterioration with a decrease of the YIG lm thickness. These attempts will be fo- cused on avoiding the most probable sources of FMR losses such as contributions due to homogeneous broad- ening (interface roughness, homogeneously distributed defects and impurities) and inhomogeneous broadening (geometric and magnetic mosaicity, single surface de- fects) and, thus, on approaching the \target" parameters of
HFWHM = 0:3 Oe at 6.5 GHz and = 0:4104 reported by R oschmann and Tolksdorf30for bulk discs made of single YIG crystals. III. OUTLOOK AND CONCLUSIONS Besides the eorts to avoid growth defects as well as interface roughness and to reduce impurity incorpora- tion during the LPE deposition process further high- resolution investigations are necessary to gain more in- sight into the YIG microstructure and to identify the properties which play an essential role for its FMR per- formance. Therefore, in future studies we will carry out HR-RSM scans with asymmetrical re ections to deter- mine in-plane and axial strain, respectively, the Time- of-Flight (ToF) SIMS analysis technique using element standards to precisely quantify the La substitution con- centration as well as to detect impurity elements from the high-temperature solutions in our sub-micrometer LPE lms. Furthermore, angular dependent measurements of the resonance eld and of the FMR linewidth will be in- tended to determine the in uence of uniaxial magnetic anisotropies on the ferromagnetic resonance losses. In conclusion, liquid phase epitaxy has the potential to provide sub-micrometer YIG lms with outstanding crys- talline and magnetic properties to meet the requirements for future magnon spintronics with ultra-low eective losses if a drastic miniaturization down to the nanometer scale is possible. First sub-100 nm lateral sized structures have presently been prepared31which could be the next step to LPE-based microscaled spintronic circuits. The development of YIG LPE lms with thicknesses below 100 nm is now in progress and remains a big challenge for the classical thick-lm LPE technique. IV. METHODS A. Sample fabrication YIG lms were grown from PbO-B 2O3based high- temperature solutions resistively-heated in a platinum crucible at about 900C using standard dipping LPE technique. During dierent growth runs nominally pure YIG lms were grown on one-inch (111) gadolinium gal- lium garnet (GGG) substrates to check the reproducibil-7 ity of the sub-micrometer liquid phase epitaxial growth. For La substituted lms La 2O3was added to the al- ready used high-temperature solution. To remove solu- tion remnants from the sample surfaces the holder had to be stored in a hot acidic solution after room tem- perature cooling. Afterwards the reverse side layer was removed by mechanical polishing from the double-side grown samples. Chips of dierent sizes were prepared by a diamond wire saw and sample surfaces were cleaned using ethanol, distilled water and acetone. The LPE lm thickness was determined by X-ray re ectometry using a PANanalytical/X-Pert Pro system. B. Microstructural properties The root-mean-square surface roughness was deter- mined by AFM measurements for each sample at three dierent regions over 25 m2ranges using a Park Scien- tic Instruments, M5. HR-XRD studies were performed by a ve-crystal diraction spectrometer of Seifert (3003 PTS HR) equipped with a four-fold Ge 440 asymmetric monochromator using CuK radiation. The resolution limit was 1104deg. GGG substrate lattice param- eters were obtained by the Bond method. Depth pro- le analyses were carried out by an Axis UltraDLDXPS system (Kratos Analytical Ltd.) using a mono-atomic argon-ion etching technique. Qualitative SIMS (Hiden Analytical) measurements were carried out. Here, a lm area of 500500m2is irradiated by 5 keV oxygen ions. C. Magnetic properties The vibrating sample magnetometer (MicroSense LLC, EZ-9) was used to register the in-plane hystere- sis loops of the YIG/GGG samples at room tempera- ture. The external magnetic eld Hwas controlled with an error of0.01 Oe. To estimate the magnetization of the YIG lms we removed a contribution of the GGG substrates from the total VSM signal. To monitor the in-plane anisotropy as a function of the crystallographic orientation, the hysteresis loops at the azimuthal angles 0'360were measured with an angular step of 3. The FMR absorption spectra were registered with a vector network analyzer (Rohde & Schwarz GmbH, ZVA 67) attached to a broadband stripline. The sample was disposed face-down over a stripline and the transmission signals (S21&S12) were recorded. During the measure- ments, a frequency of microwave signals with the input power of10 dBm (0.1 mW) was swept across the res- onance frequency, while the in-plane magnetic eld H was constant and measured with an accuracy of 1 Oe. Each recorded spectrum was tted by the Lorentz func- tion and allowed us to dene the resonance frequency and the FMR linewidth
HFWHM corresponding to the applied eld H.1V. V. Kruglyak and R. J. Hicken, \Magnonics: Experiment to prove the concept," J. Magn. Magn. Mater. 306, 191{194 (2006), cond-mat/0511290. 2S. Neusser, B. Botters, and D. Grundler, \Localization, conne- ment, and eld-controlled propagation of spin waves in Ni 80Fe20 antidot lattices," Phys. Rev. B 78, 054406 (2008). 3V. V. Kruglyak, S. O. Demokritov, and D. Grundler, \Magnon- ics," J. Phys. D: Appl. Phys. 43, 264001 (2010). 4R. L. Stamps, S. Breitkreutz, J. Akerman, A. V. Chumak, Y. Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen, S. A. Majetich, M. Kl aui, I. Lucian Prejbeanu, B. Dieny, N. M. Dempsey, and B. Hillebrands, \The 2014 Magnetism Roadmap," J. Phys. D: Appl. Phys. 47, 333001 (2014), arXiv:1410.6404 [cond-mat.mtrl-sci]. 5R. C. LeCraw, E. G. Spencer, and C. S. Porter, \Ferromagnetic Resonance Line Width in Yttrium Iron Garnet Single Crystals," Phys. Rev. 110, 1311{1313 (1958). 6A. A. Serga, A. V. Chumak, and B. Hillebrands, \YIG magnon- ics," J. Phys. D: Appl. Phys. 43, 264002 (2010). 7A. V. Chumak, A. A. Serga, and B. Hillebrands, \Magnon tran- sistor for all-magnon data processing," Nat. Commun. 5, 4700 (2014). 8A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, \Magnon spintronics," Nat. Phys. 11, 453{461 (2015). 9E. A. Giess, J. D. Kuptsis, and E. A. D. White, \Liquid phase epitaxial growth of magnetic garnet lms by isothermal dipping in a horizontal plane with axial rotation," J. Cryst. Growth 16, 36{42 (1972). 10P. Pirro, T. Br acher, A. V. Chumak, B. L agel, C. Dubs, O. Surzhenko, P. G ornert, B. Leven, and B. Hillebrands, \Spin- wave excitation and propagation in microstructured waveguides of yttrium iron garnet/Pt bilayers," Appl. Phys. Lett. 104, 012402 (2014), arXiv:1311.6305 [cond-mat.mes-hall]. 11M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M. Kl aui, A. V. Chumak, B. Hillebrands, and C. A. Ross, \Pulsed laser deposition of epitaxial yttrium iron garnet lms with low Gilbert damping and bulk-like magnetization," APL Mater. 2, 106102 (2014). 12B. M. Howe, S. Emori, H.-M. Jeon, T. Oxhol, J. G. Jones, K. Ma- halingam, Y. Zhuang, N. X. Sun, and G. J. Brown, \Pseudomor- phic yttrium iron garnet thin lms with low damping and inho- mogeneous linewidth broadening," IEEE Magn. Lett. 8, 3500504 (2015). 13H. Chang, P. Li, W. Zhang, T. Liu, A. Homann, L. Deng, and M. Wu, \Nanometer-thick yttrium iron garnet lms with extremely low damping," IEEE Magn. Lett. 5, 6700104 (2014). 14H. Wang, Understanding of Pure Spin Transport in a Broad Range of Y3Fe5O12-based Heterostructures , Ph.D. thesis, The Ohio State University (2015). 15O. d'Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef, C. Hahn, A. H. Molpeceres, C. Carr et ero, E. Jacquet, C. Der- anlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loubens, O. Klein, V. Cros, and A. Fert, \Inverse spin Hall eect in nanometer-thick yttrium iron garnet/Pt system," Appl. Phys. Lett. 103, 082408 (2013), arXiv:1308.0192 [cond-mat.mtrl-sci]. 16V. Castel, N. Vlietstra, B. J. van Wees, and J. B. Youssef, \Fre- quency and power dependence of spin-current emission by spin pumping in a thin-lm YIG/Pt system," Phys. Rev. B 86, 134419 (2012), arXiv:1206.6671 [cond-mat.mtrl-sci]. 17C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, \Comparative measurements of inverse spin Hall eects and magnetoresistance in YIG/Pt and YIG/Ta," Phys. Rev. B 87, 174417 (2013), arXiv:1302.4416 [cond-mat.mes-hall]. 18L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, \Long-distance transport of magnon spin information in a magnetic insulator at room temperature," Nat. Phys. 11, 1022{1026 (2015), arXiv:1505.06325 [cond-mat.mes-hall]. 19S. A. Manuilov, R. Fors, S. I. Khartsev, and A. M. Grishin, \Sub- micron Y 3Fe5O12Film Magnetostatic Wave Band Pass Filters," J. Appl. Phys. 105, 033917{033917 (2009).8 20Y. Sun, Y.-Y. Song, H. Chang, M. Kabatek, M. Jantz, W. Schnei- der, M. Wu, H. Schultheiss, and A. Homann, \Growth and ferromagnetic resonance properties of nanometer-thick yttrium iron garnet lms," Appl. Phys. Lett. 101, 152405 (2012). 21S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, \Pulsed laser deposited Y 3Fe5O12lms: Nature of magnetic anisotropy I," J. Appl. Phys. 106, 123917{123917 (2009). 22R. Hergt, H. Pfeier, P. G ornert, M. Wendt, B. Keszei, and J. Vandlik, \Kinetic Segregation of Lead Impurities in Garnet LPE Films," Phys. Stat. Sol. (a) 104, 769{776 (1987). 230.1 at-% Pb corresponds to xPb0:02 formular units in stoi- chiometric Y 3xPbxFe5O12. 240.5 at-% La corresponds to yLa0:1 formular units in stoichio- metric Y 3yLayFe5O12. 25G. Winkler, \Magnetic garnets," in Vieweg tracts in pure and applied physics; Volume 5 (Friedrich Vieweg & Sohn Verlag, Braunschweig, Wiesbaden, 1981) Chap. 2, pp. 75{79. 26S. B. Ubizskii, \Orientational states of magnetization in epitaxial (111)-oriented iron garnet lms," J. Magn. Magn. Mater. 195, 575{582 (1999). 27S. B. Ubizskii, \Magnetization reversal modelling for (111)- oriented epitaxial lms of iron garnets with mixed anisotropy," J. Magn. Magn. Mater. 219, 127{141 (2000). 28C. Kittel, \On the Theory of Ferromagnetic Resonance Absorp- tion," Phys. Rev. 73, 155{161 (1948). 29V. Lauer, D. A. Bozhko, T. Br acher, P. Pirro, V. I. Vasyuchka, A. A. Serga, M. B. Jung eisch, M. Agrawal, Y. V. Kobljanskyj, G. A. Melkov, C. Dubs, B. Hillebrands, and A. V. Chumak, \Spin-transfer torque based damping control of parametrically excited spin waves in a magnetic insulator," Appl. Phys. Lett. 108, 012402 (2016), arXiv:1508.07517 [cond-mat.mes-hall]. 30P. R oschmann and W. Tolksdorf, \Epitaxial growth and anneal- ing control of FMR properties of thick homogeneous Ga substi- tuted yttrium iron garnet lms," Mat. Res. Bull. 18, 449{459 (1983).31T. L ober, A. V. Chumak, and B. Hillebrands, Unpublished re- sults. V. ACKNOWLEDGEMENTS We acknowledge the partial nancial support by Deutsche Forschungsgemeinschaft (DU 1427/2-1). We thank M. Frigge for EPMA analysis, Ch. Schmidt for XRR measurements and R. Meyer and B. Wenzel for technical support. VI. AUTHOR CONTRIBUTIONS STATEMENT C.D. conceived the experiments, prepared all samples and analyzed the data. O.S. performed VSM and FMR measurements and analyzed the data. R.L. performed the XPS experiments. J.D. and U.B. performed the SIMS experiments. A.D. conducted the XRD experiments and analyzed the data. C.D. and O.S. wrote the manuscript. All authors contributed to scientic discussions and the manuscript review. VII. ADDITIONAL INFORMATION A. Competing nancial interests The authors declare no competing nancial interests.

2016-08-29

Using liquid phase epitaxy (LPE) technique (111) yttrium iron garnet (YIG) films with thicknesses of ~100 nm and surface roughnesses as low as 0.3 nm have been grown as a basic material for spin-wave propagation experiments in microstructured waveguides. The continuously strained films exhibit nearly perfect crystallinity without significant mosaicity and with effective lattice misfits of delta a(perpendicular)/a(substrate) ~10-4 and below. The film/substrate interface is extremely sharp without broad interdiffusion layer formation. All LPE films exhibit a nearly bulk-like saturation magnetization of (1800+-20) Gs and an `easy cone' anisotropy type with extremely small in-plane coercive fields <0.2 Oe. There is a rather weak in-plane magnetic anisotropy with a pronounced six-fold symmetry observed for saturation field <1.5 Oe. No significant out-of-plane anisotropy is observed, but a weak dependence of the effective magnetization on the lattice misfit is detected. The narrowest ferromagnetic resonance linewidth is determined to be 1.4 Oe @ 6.5 GHz which is the lowest value reported so far for YIG films of 100 nm thicknesses and below. The Gilbert damping coefficient for investigated LPE films is estimated to be close to 1 x 10-4.

Sub-micrometer yttrium iron garnet LPE films with low ferromagnetic resonance losses

1608.08043v1

Spin Pumping, Dissipation, and Direct and Alternating Inverse Spin Hall Eects in Magnetic Insulator-Normal Metal Bilayers Andr e Kapelrud and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway We theoretically consider the spin-wave mode- and wavelength-dependent enhancement of the Gilbert damping in magnetic insulatornormal metal bilayers due to spin pumping as well as the enhancement's relation to direct and alternating inverse spin Hall voltages in the normal metal. In the long-wavelength limit, including long-range dipole interactions, the ratio of the enhancement for transverse volume modes to that of the macrospin mode is equal to two. With an out-of- plane magnetization, this ratio decreases with both an increasing surface anisotropic energy and mode number. If the surface anisotropy induces a surface state, the enhancement can be an order of magnitude larger than for to the macrospin. With an in-plane magnetization, the induced dissipation enhancement can be understood by mapping the anisotropy parameter to the out-of-plane case with anisotropy. For shorter wavelengths, we compute the enhancement numerically and nd good agreement with the analytical results in the applicable limits. We also compute the induced direct- and alternating-current inverse spin Hall voltages and relate these to the magnetic energy stored in the ferromagnet. Because the magnitude of the direct spin Hall voltage is a measure of spin dissipation, it is directly proportional to the enhancement of Gilbert damping. The alternating spin Hall voltage exhibits a similar in-plane wave-number dependence, and we demonstrate that it is greatest for surface-localized modes. PACS numbers: 76.50.+g, 75.30.Ds, 75.70.-i, 75.76.+j, 75.78.-n I. INTRODUCTION In magnonics, one goal is to utilize spin-based sys- tems for interconnects and logic circuits1. In previous decades, the focus was to gain control over these systems by exploiting long-range dipole interactions in combina- tion with geometrical shaping. However, the complex nature of the nonlinear magnetization dynamics persis- tently represents a challenge in using geometrical shaping alone to realize a variety of desired properties1. In magnonic systems, a unique class of materials con- sists of magnetic insulators. Magnetic insulators are elec- trically insulating, but localized magnetic moments cou- ple to form a long-range order. The prime example is Yttrium Iron Garnet (YIG). YIG is a complex crystal2 in the Garnet family, where the Fe2+and Fe3+ions at dierent sites in the unit cell contribute to an overall fer- rimagnetic ordering. What dierentiates YIG from other ferromagnetic (ferrimagnetic) systems is its extremely low intrinsic damping. The Gilbert damping parame- ter measured in YIG crystals is typically two orders of magnitude smaller than that measured in conventional metallic ferromagnets (Fe, Co, Ni, and alloys thereof). The recent discovery that the spin waves in mag- netic insulators strongly couple to spin currents in ad- jacent normal metals has re-invigorated the eld of magnonics3{12. Although there are no mobile charge car- riers in magnetic insulators, spin currents ow via spin waves and can be transferred to itinerant spin currents in normal metals via spin transfer and spin pumping13,14. These interfacial eects open new doors with respect to local excitation and detection of spin waves in magnonic structures. Another key element is that we can transfer knowledge from conventional spintronics to magnonics,opening possibilities for novel physics and technologies. Traditionally, spin-wave excitation schemes have focused on the phenomenon of resonance or the use of rsted elds from microstrip antennas. A cornerstone for utilizing these systems is to estab- lish a good understanding of how the itinerant elec- trons in normal metals couple across interfaces with spin-wave dynamics in magnetic insulators. Good mod- els for adressing uniform (macrospin) magnetization that agrees well with experiments have been previously developed13{15. We recently demonstrated that for long- wavelength magnons the enhanced Gilbert damping for the transverse volume modes is twice that of the uniform mode, and for surface modes, the enhancement can be more than ten times stronger. These results are con- sistent with the theory of current-induced excitations of the magnetization dynamics16because spin pump- ing and spin transfer are related by Onsager reciprocity relations17. Moreover, mode- and wave-vector-dependent spin pumping and spin Hall voltages have been clearly observed experimentally4. In this paper, we extend our previous ndings18in the following four aspects. i) We compute the in uence of the spin back ow on the enhanced spin dissipation. ii) We also compute the induced direct and alternating in- verse spin Hall voltages. We then relate these voltages to the enhanced Gilbert damping and the relevant energies for the magnetization dynamics. The induced voltages give additional information about the spin-pumping pro- cess, which can also be directly measured. iii) We also provide additional information on the eects of interfa- cial pinning of dierent types in various eld geometries. iv) Finally, we explain in more detail how the numerical analysis is conducted for a greater number of in-planearXiv:1612.07020v2 [cond-mat.mes-hall] 6 Apr 20172 wave numbers. It was discovered19{23and later quantitatively explained13,15,24,25that if a dynamic ferromagnetic mate- rial is put in contact with a normal metal, the magnetiza- tion dynamics will exert a torque on the spins of electrons in the immediate vicinity of the magnet. This eect is known as spin pumping (SP)13,15,25. As the electrons are carried away from the ferromagnet-normal metal inter- face, the electrons spin with respect to each other, caus- ing an overall loss of angular momentum. The inverse eect, in which a spin-polarized current can aect the magnetization of a ferromagnet, is called spin-transfer torque (STT)26{28. The discovery that a precessing magnetization in mag- netic insulators3, such as YIG, also pumps spins into an adjacent metal layer was made possible by the fact that the mixing conductance in YIG-normal metal systems is of such a size that the extra dissipation of the magneti- zation due to the spin pumping is of the same order of magnitude as the intrinsic Gilbert damping. A conse- quence of this eect is that the dissipation of the magne- tization dynamics is enhanced relative to that of a system in which the normal metal contact is removed. This paper is organized in the following manner. Sec- tion II presents the equation of motion for the magne- tization dynamics and the currents in the normal metal and the appropriate boundary conditions, both for gen- eral nonlinear excitations and in the fully linear response regime. In Section III, we derive approximate solutions to the linearized problem, demonstrating how the mag- netization dissipation is enhanced by the presence of an adjacent metal layer. Section IV presents our numerical method and results. Finally, we summarize our ndings in Section V. II. EQUATIONS OF MOTION The equation of motion for the magnetization is given by the Landau-Lifshitz-Gilbert equation29(presented here in CGS units) @M @t= MHe+ MsM@M @t; (1) where =jgB=~jis the magnitude of the gyromagnetic ratio;g2 is the Land e g-factor for the localized elec- trons in the ferromagnetic insulator (FI); and is the di- mensionless Gilbert damping parameter. In equilibrium, the magnitude of the magnetization is assumed to be close to the saturation magnetization Ms. The magneti- zation is directed along the z-axis in equilibrium. Out of equilibrium, we assume that we have a small transverse dynamic magnetization component, such that M=M(r;t) =Ms+m(r;t) =Ms^z+m(r;t);(2) wherejmjMsandm
^z= 0. Furthermore, we assume that the dynamic magnetization can be described by a x hzx yz fq(a) d L2 -L2NM FI SUBx (b) FIG. 1. a) The coordinate system. ^is the lm normal and ^is the spin-wave propagation direction. form a right-handed coordinate system. The ^zaxis is the direc- tion of the magnetization in equilibrium, such that xyis the magnetization-precession plane. b) The lm stack is in the normal direction. plane wave traveling along the in-plane -axis. In the (;; ) coordinate system (see Figure 1), we have m(r;t) =m(;;t ) =mQ()ei(!tQ); (3) where!is the harmonic angular frequency, Qis the in- plane wave number, and mQ() =XQ()^x+YQ()^y, whereXQandYQare complex functions. Note that m is independent of the coordinate due to translational invariance. Heis the eective eld, given as the functional deriva- tive of the free energy29,30 He(r;t) =U[M(r;t)] M(r;t)=Hi+2A M2sr2M(r;t)+ + 4ZL 2 L 2d0bGxy(0)m(0;;t);(4) where Hiis the internal eld, which is composed of the applied external eld and the static demagnetization eld. The direction of Hidenes the z-axis (see Fig- ure 1). The second term of Eq. (4) is the eld, Hex, induced by to the exchange interaction (assuming cu- bic symmetry), where Ais the exchange stiness pa- rameter. The last term is the dynamic eld, hd(r;t), induced by dipole-dipole interactions, where bGxyis the upper 22 part of the dipole{dipole tensorial Green's functionbGin the magnetostatic approximation31ro- tated to the xyzcoordinate system (see Appendix A for coordinate-transformation matrices).32 The eect of the dipolar interaction on the spin-wave spectrum depends on the orientation of the internal eld with respect to both the interface normals of the thin lm, ^, and the in-plane spin-wave propagation direc- tion, ^. Traditionally, the three main congurations are the out-of-plane conguration ( = 0), in the forward volume magnetostatic wave (FVMSW) geometry (see Fig. 2a); the in-plane and parallel-to- ^conguration, in3 thebackward volume magnetostatic wave (BVMSW) ge- ometry (see Fig. 2b); and the in-plane and perpendicular- to-^conguration, in the magnetostatic surface wave (MSSW) geometry (see Fig. 2c).1,32{36Here, the term \forward volume modes" denotes modes that have posi- tive group velocities for all values of QL, whereas back- ward volume modes can have negative group velocities in the range of QL, where both exchange and dipolar interactions are signicant. Volume modes are modes in which mQ() is distributed across the thickness of the entire lm, whereas the surface modes are localized more closely near an interface. A. Spin-Pumping Torque We consider a ferromagnetic insulator (FI) in contact with a normal metal (NM) (see Figure 1). If the magneti- zation in the FI close to the interface is precessing around the eective eld, electron spins in the NM re ected at the interface will start to precess due to the local ex- change coupling to the magnetization in the FI. The re- ected electrons carry the angular momentum away from the interface, where the spin information can get lost through dephasing of the spins within a typical spin diu- sion length lsf. This loss of angular momentum manifests itself as an increased local damping of the magnetization dynamics in the FI. The magnetization dissipation due to the spin-pumping eect can be taken into account by adding the local dissipation torque15 sp= ~2g? 2e2M2s(L 2)M(r;t)@M(r;t) @t;(5) to the right-hand side (rhs) of Eq. (1). Here, g?is the real part of the spin-mixing conductance per area, and e is the electron charge. We neglect the contribution from the imaginary part of the mixing conductance, because this has been shown to be signicantly smaller than that of the real part, in addition to aecting only the gyro- magnetic ratio.15The spin-current density pumped from the magnetization layer is thus given by j(s) sp=~2g? 2e2M2s M(r;t)@M(r;t) @t =L=2;(6) in units of erg. Next, we will see how the spin pumping aects the boundary conditions. B. Spin-Pumping Boundary Conditions Following the procedure of Rado and Weertman37, we integrate Eq.(1) with the linear expansion of Eq. (2) over a small pill-box volume straddling one of the interfaces of the FI. Upon letting the pill box thickness tend to zero, only the surface torques of the equation survive. Accounting for the direction of the outward normal ofthe lid on the dierent top and bottom interfaces, we arrive at the exchange-pumping boundary condition 2A M2sM@M @+~2 2e2M2sg?M@m @t =L=2= 0:(7) There is no spin current pumped at the interface to the insulating substrate; thus, a similar derivation results in a boundary condition that gives an unpinned magnetiza- tion, @M(r;t) @ =L=2= 0: (8) In the next section, we will generalize the bound- ary conditions of Eqs. (7) by also considering possible surface-anisotropy energies. Including surface anisotropy: In the presence of surface anisotropy at an interface with an easy-axis (EA) pointing along the direction ^n, the surface free energy is Us[M(r;t)] =Z dV Ks" 1M(r;t)
^n Ms2# (i); (9) whereKsis the surface-anisotropy energy density at the interface, which is assumed to be constant; ^nis the direc- tion of the anisotropy easy axis; andiis the transverse coordinate of the interface. The contribution from the EA surface-anisotropy energy to the eective eld is de- termined by Hs=Us[M(r;t)] M(r;t)=2Ks M2s(M
^n)(j)^n: However, if we have an easy-plane (EP) surface anisotropy with, ^nbeing the direction of the hard axis, the eective eld is the same as that for the EA case, except for a change of sign of Ks. We unify both cases by dening Ks>0 to imply that we have an EA surface anisotropy with its easy axis along ^n, whereasKs<0 implies that we have an EP surface anisotropy with its hard axis along ^n. Following the approach from Section II B, the total boundary condition, including exchange, pumping and surface anisotropy, becomes  2A M2sM@M @2Ks M2s(M
^n) (M^n) + +~2 2e2M2sg?M@M @t =L=2= 0;(10) where the positive (negative) sign in front of the exchange term indicates that the bulk FI is located below (above) the interface coordinate.4 x,z z (a) x z,zq (b) x z zfq (c) FIG. 2. Laboratory eld congurations, i.e., directions of ^z(green arrow) in relation to lm normal ^and the spin-wave propagation direction ^, resulting in the dierent geometries: a) FVMSW geometry; b) BVMSW geometry; c) MSSW geometry. C. Linearization We linearize the equation of motion using Eq. (2) with respect to the dynamic magnetization m. The linearized equation of motion for the bulk magnetization Eq. (1) becomes32  i! !M 1 1 +11!H !M+ 8 2A !2 M Q2d2 d2

mQ() =ZL 2 L 2d0bGxy(0)mQ(0);(11) where!H Hi,!M4 Ms, and 11 =1 0 0 1
. Next, we linearize the boundary conditions of Eq. (10). We choose the anisotropy axis to be perpendicular to the lm plane, ^n=^, which in the xyzcoordinate system is given by ^xyz= (sin;0;cos), whereis the angle between the z-axis and the lm normal (see Fig. 1). The nite surface anisotropy forces the magnetization to be either perpendicular or coplanar with the lm surface so that= 0;=2;. Linearizing to 1st order in the dy- namic magnetization, we arrive at the linearized bound- ary conditions for the top interface  L@ @+i! !M+dcos(2) mQ;x() =L 2= 0;(12a)  L@ @+i! !M+dcos2() mQ;y() =L 2= 0;(12b) wheredLKs=Ais the dimensionless surface- pinning parameter that relates the exchange to the surface anisotropy and the lm thickness and  !ML~2g?=4Ae2is a dimensionless constant relating the exchange stiness and the spin-mixing conductance. D. Spin Accumulation in NM and Spin Back ow The pumped spin current induces a spin accumulation, (s)=(s)^s, in the normal metal. Here, ^sis the spin- polarization axis, and (s)= ("#)=2 is half of thedierence between chemical potentials for spin-up and spin-down electrons in the NM. As the spin accumulation is a direct consequence of the spin dynamics in the FI (see Eq. (6)), the spin ac- cumulation cannot change faster than the magnetization dynamics at the interface. Thus, assuming that spin- ip processes in the NM are must faster than the typical pre- cession frequency of the magnetization in the FI25, we can neglect precession of the spin accumulation around the applied eld and any decay in the NM. With this assump- tion, the spin-diusion equation@(s) @t=Dr2(s)(s) sf, whereDis the spin-diusion constant, and sfis the material-specic average spin- ip relaxation time, be- comes (s)l2 sfr2(s); (13) wherelsfpsfDis the average spin- ip relaxation length. The spin accumulation results in a back owing spin- current density, given by j(s) bf(L=2) =~g? e2M2sh M(r;t) M(r;t)(s)(r;t)i =L=2; (14) where the positive sign indicates ow from the NM into the FI. This spin current creates an additional spin- transfer torque on the magnetization at the interface bf= ~g? e2M2s L 2 M(r;t) M(r;t)(s) : (15) Because the spin accumulation is a direct result of the pumped spin current, it must have the same orientation as the M(r;t)@tM(r;t) term in Eq. (5). That term is comprised of two orthogonal components: the 1st-order termMs^z_min thexyplane, and the 2nd-order term m_moriented along ^z. Because the magnetization is a real quantity, care must be taken when evaluating the 2nd-order term. Using Eq. (3), the 2nd-order pumped5 spin current is proportional to Refmg@tRefmg =L=2=e2Imf!gtRef!g ^zh ImXQReYQReXQImYQi ;(16) which is a decaying direct-current (DC) term. This is in contrast to the 1st-order term, which is an alternating- current (AC) term. Thus, we write the spin accumula- tion as (s)=(s) AC(^z^mt) +(s) DC^z; (17) where we have used the shorthand notation mt=_m(= L=2), such that ^mt=mt=jmtj, which in general is not parallel to mbut guaranteed to lie in the xyplane. In- serting Eq. (17) into Eq. (13) gives one equation each for the AC and DC components of the spin accumulation, @2(s) j @2=l2 sf;j(s) j; (18) wherejdenotes either the AC or DC case and lsf,DC =lsf whilelsf,AC =lsf(1+l2 sfQ2)1=2because mt/exp(i(!t Q)). Eq. (18) can be solved by demanding spin-current conservation at the NM boundaries: at the free surface of the NM, there can be no crossing spin current; thus, the  component of the spin-current density must vanish there, @(s) jj=L=2+d= 0. Similarly, by applying conservation of angular momentum at the FI-NM interface, the net spin-current density crossing the interface, due to spin pumping and back ow, must equal the spin current in the NM layer, giving  ~2g? 2e2M2sM@M @t+~g? e2M2sM M(s) =L=2 =~ 2e2@(s)j=L=2;(19) whereis the conductivity of the NM. Using these boundary conditions, we recover the solutions (see, e.g.,25,38) (s) j=(s) j;0sinh l1 sf;j (L=2 +d) sinh d lsf;j; (20) where(s) j;0is time dependent, and depends on the co- ordinate only in the AC case. We nd that the AC and DC spin accumulations (s) j;0are given by (s) AC;0=~ 2mt Ms 1 + 2g?lsf;ACcothd lsf;AC1 ; (21) (s) DC;0=lsf~ M2s~g?tanhd lsf ^z
[m_m]=L=2; (22)TABLE I. Typical values for the parameters used in the calculations.6,7,11,39{41 Parameter Value Unit A 3:66
107erg cm1  3
104{ Ks 0:05 erg cm2 g? 8:18
1022cm1s1 1:76
107G1s1 4M s 1750 G  8:45
1016s1 d 50 nm lsf 7.7 nm  0.1 { where ~g?is a renormalized mixing conductance, which is given by ~g?=g?( 1 1 + 2g?lsf;ACcothd lsf;AC1) : (23) This scaling of g?occurring in the DC spin accumulation originates from the second-order spin back ow due to the AC spin accumulation that is generated in the normal metal. Adding both the spin-pumping and the back ow torques to Eq. (1) and repeating the linearization pro- cedure from Sec. II C, we nd that the AC spin accumu- lation renormalizes the pure spin-mixing conductance. Thus, the addition of the back ow torque can be ac- counted for by replacing g?with ~g?in the boundary con- ditions of Eqs. (12), making the boundary conditions Q- dependent in the process. Using the values from Table I, which are based on typical values for a YIG-Pt bilayer system, we obtain ~ g?=g?0:4 forQL1, whereas ~g?=g?!1 for large values of QL. Thus, AC back ow is signicant for long-wavelength modes and should be con- sidered when estimating g?from the linewidth broaden- ing in ferromagnetic resonance (FMR) experiments.11 Inverse Spin Hall Eect The inverse spin Hall eect (ISHE) converts a spin current in the NM to an electric potential through the spin-orbit coupling in the NM. For a spin current in the^direction, the ISHE electric eld in the NM layer isEISHE =e1h(@(s))^i, where  is the di- mensionless spin-Hall angle, and h
iis a spatial average across the NM layer, i.e., for 2(L=2;L=2 +d). Using the previously calculated spin accumulation, we nd that6 the AC electric eld is EAC ISHE =~ 2deMs 1 + 2g?lsf;ACcothd lsf;AC1   ^(mt;ycoscos+mt;xsin)+ +^(mt;xcosmt;ycossin) ;(24) where mt;i=[Im!Remi+Re!Immi]=L=2; (25) andi=x;y. For BVMSW ( ==2;= 0) modes, the AC eld points along ^, whereas for MSSW ( = ==2) modes, it points along ^(i.e., in plane, but transverse to ; see Fig. 1). Notice that for both BVMSW and MSSW mode geometries, only the xcomponent of mtcontributes to the eld. In contrast, for FVMSW (= 0) modes, the eld points somewhere in the  plane, depending on the ratio of mt;xtomt;y. Similarly to the AC eld, the DC ISHE electric eld is given by EDC ISHE = (s) DC;0 desin(^cos^sin);(26) which is perpendicular to the AC electric eld and zero for the FVMSW mode geometry. The total time-averaged energy in the ferromagnet Etotal(see Morgenthaler42) is given by hEtotaliT=Z ferriteRe i! !M(mm)^z dV; (27) where the integral is taken over the volume of the ferro- magnet. Because the DC ISHE eld is in-plane, the voltage measured per unit distance along the eld direction, ^=^cos^sin, can be used to construct an esti- mate of the mode eciency. Taking the one-period time average of Eq. (26) using Eq. (22) and normalizing it by Eq. (27) divided by the in-plane surface area, A, we nd an amplitude-independent measure of the DC ISHE: DC=he^
EDC ISHEiT hEtotaliT=A=2 lsf~ dMs~g?tanhd lsf sin Reh i! !M(mm)^zi =L=2RL=2 L=2Reh i! !M(mm)^zi d;(28) given in units of cm, and where f
gdenotes complex conjugation. Similarly, the AC ISHE electric eld, being time- varying, will contribute a power density that, when nor- malized by the power density in the ferromagnet, be-comes AC=h EAC ISHE
2iT Ref!g 2ALhEtotaliT= Ref!g~ 2deMs2   1 + 2g?lsf;ACcothd lsf;AC2  jmt;xj2+ cos2jmt;yj2 1 LRL=2 L=2Reh i! !M(mm)^zi d:(29) To be able to calculate explicit realizations of the mode- dependent equations Eqs. (28) and (29), one will need to rst calculate the dispersion relation and mode proles in the ferromagnet. III. SPIN-PUMPING THEORY FOR TRAVELLING SPIN WAVES Because, the linearized boundary conditions (see Eqs. (12)) explicitly depend on the eigenfrequency !, we cannot apply the method of expansion in the set of pure exchange spin waves, as was performed by Kalinikos and Slavin32. Instead, we analyze and solve the system di- rectly for small values of QL, whereas the dipole-dipole regime ofQL1 is explored using numerical computa- tions in Sec. IV. A. Long-Wavelength Magnetostatic Modes WhenQL1 Eq. (11) is simplied to ( sin20 0 0! +i! !M 1 1! + +11!H !M8 2A !2 Md2 d2
mQ() = 0;(30) where the 1st-order matrix term describe the dipole- induced shape anisotropy and stems from bGxy(see32). We make the ansatz that the magnetization vector in Eq. (3) is composed of plane waves, e.g., mQ()/eik. Inserting this ansatz into Eq. (30) produces the disper- sion relation ! !M2 =!H !M+2 exk2+i! !M  !H !M+2 exk2+ sin2+i! !M ;(31) whereexp 8 2A=!2 Mis the exchange length . Keep- ing only terms to rst order in the small parameter , we arrive at !(k) !M=r!H !M+2exk2!H !M+2exk2+ sin2 + +i!H !M+2 exk2+sin2 2 : (32)7 The boundary conditions in Eq. (12) depend explicitly on !andkand give another equation k=k(!) to be solved simultaneously with Eq. (32). However, in the absence of spin pumping, i.e., when the spin-mixing conductance vanishesg?!0, it is sucient to insert the constant k solutions from the boundary conditions into Eq. (32) to nd the eigenfrequencies. Dierent wave vectors can give the same eigenfre- quency. It turns out that this is possible when !(k) = !(i), which has a non-trivial solution relating tok: 2 ex2= sin2+2 exk2+ 2!H !Mi2!(k)=!M:(33) With these ndings, a general form of the magnetiza- tion is mQ() = 1 r(k)!h C1cos k(+L 2)
+C2sin k(+L 2)
i + + 1 r(i)!h C3cosh (+L 2)
+C4sinh (+L 2)
i ; (34) wherefCigare complex coecients to be determined from the boundary conditions, and where =(k) is given by Eq. (33). The ratio between the transverse com- ponents of the magnetization, r(k) =YQ=XQ, is deter- mined from the bulk equation of motion (see Eq. (30)) and is in linearized form r(k) =sin22ir !H !M+2exk2 !H !M+2exk2+ sin2 2 !H !M+2exk2 ; (35) implying elliptical polarization of mQwhen6= 0. Inserting Eq. (34) into Eq. (8) only leads to a solution whenk= 0, such that C2=C4= 0 in the general case. By solving Eq. (12b) for C3, we nd C3 C1=!H !M+2 exk2+ sin2+i! !M !H !M2ex2+ sin2+i! !M (i! !M~+dcos2) cos(kL)kLsin(kL) (i! !M~+dcos2) cosh(L) +Lsinh(L);(36) where ~jg?!~g?is the pumping parameter altered by the AC spin back ow from the NM (see Section II D). C1 is chosen to be the free parameter that parameterizes the dynamic magnetization amplitude, which can be deter- mined given a particular excitation scheme. Lineariza- tion of Eq. (36) with respect to is straightforward, but the expression is lengthy; we will therefore not show it here. Inserting the ansatz with C2=C4= 0 andC3given by Eq. (36) into Eq. (12a) gives the second equation for kand!(the rst is Eq. (32)). In the general case, thenumber of terms in this equation is very large; thus, we describe it as f(k;!;; ~) = 0; (37) i.e., an equation that depends on the wave vector k, fre- quency!, Gilbert damping constant and spin-pumping parameter ~. Because both the bulk and interface-induced dissipa- tion are weak, 1, ~1, the wavevector is only slightly perturbed with respect to a system without dis- sipation, i.e., k!k+kwhereexk1. It is therefore sucient to expand fup to 1storder in these small quan- tities: f(k;!; 0;0) + (~)@f @~ 0+@f @ 0+ + (exk)@f @(exk) 00;(38) where the sub-index 0 means evaluation in a system with- out dissipation, i.e., when ( ;~;k) = (0;0;0). By solv- ing the system of equations in the absence of dissipation, f(k;!; 0;0) = 0, the dissipation-induced change in the wave vector kis given by k~@f @~ 0+@f @ 0 ex@f @(exk) 0: (39) In turn, this change in the wave vector should be in- serted into the dispersion relation of Eq. (31) to nd the dissipation. Inspecting Eq. (31), we note that k- induced additional terms proportional to !are of the form (k+k)2k22kkwhich renormalize the Gilbert- damping term i! !M. Thus, in Eq. (39), there are terms proportional to the frequency in both terms in the numer- ator. We extract these terms /i! !Mby dierentiating with respect to !and dene the renormalization of the Gilbert damping, i.e., !+
, from spin pumping as
=i2exk!M@! exkj=0
i2exk!M@! exkj~=0
1; (40) where@!represents the derivative with respect to !and kis the solution to the 0th-order equation. Note that in performing a further local analysis around some point k0 in thek-space of Eq. (37), a series expansion of faround k0must be performed before evaluating Eqs. (39) and (40). Eq. (40) is generally valid, except when d= 0 and kL!0, which we discuss below. In the following sec- tion, we will determine explicit solutions of the 0th-order equation for some key cases, and mapping out the spin- wave dispersion relations and dissipation in the process.8 B. No Surface Anisotropy ( d= 0) Let us rst investigate the case of a vanishing sur- face anisotropy. In this case, the 0th-order expansion of Eq. (37) has a simple form and is independent of the magnetization angle . The equation to determine kis given by kLtan(kL) = 0; (41) with solutions k=n=L , wheren2Z. Similarly, the expression for kis greatly simplied, kn=i! !M~ nex L, n6= 0, such that the mode-dependent Gilbert damping is
n= 2~ex L2 ; n6= 0: (42) For the macrospin mode, when n= 0, the linear ex- pansion in kbecomes insucient. This is because kLtan(kL)(kL)2forkL!0; thus, we must expand the function fto second order in the deviation karound kL= 0. Ford= 0, we nd that the boundary condi- tion becomes k2L2=i! !M~2 ex, and when inserted into Eq. (31), it immediately gives
0= ~ex L2 =1 2
n; (43) which is the macrospin renormalization factor found in Ref. 15. Using a dierent approach, our results in this section reproduce our previous result that the renormal- ization of the Gilbert damping for standing waves is twice the renormalization of the Gilbert damping of the macrospin.18Next, we will obtain analytical results be- yond the description in Ref. 18 for the enhancement of the Gilbert damping in the presence of surface anisotropy. C. Including Surface Anisotropy ( d6= 0) In the presence of surface anisotropy, the out-of-plane and in-plane eld congurations must be treated sepa- rately. This distinction is because the boundary condi- tion Eq. (37) has dierent forms for the two congura- tions in this scenario. 1. Out-of-plane Magnetization When the magnetization is out of plane, i.e., = 0, the spin-wave excitations are circular and have a high degree of symmetry. A simplication in this geometry is that the coecient C3= 0. In the absence of dissipation, the boundary condition Eq. (37) determining the wave vectors becomes kLtan(kL) =d: (44) Let us consider the eects of the two dierent anisotropies in this geometry. 0510152025300.00.51.01.52.02.5 LKsADaEA,nDa0 n=0n=5FIG. 3. The ratio of enhanced Gilbert damping
EA,n=
0 in a system with easy-axis surface anisotropy versus the en- hanced Gilbert damping of macrospin modes in systems with no surface anisotropy as a function of surface-anisotropy en- ergy.nrefers to the mode number, where n= 0 is the uniform-like mode. The dashed line represents the ratio
n=
0in the case of no surface anisotropy (see Eq. (42)). a. Easy-Axis Surface Anisotropy ( d > 0):When d1 or larger, the solutions of Eq. (44) are displaced from the zeroes of tan( kL), i.e., the solutions we found in the case of no surface anisotropy, and towards the upper poles located at kuL= (2n+1)=2, wheren= 0;1;2;:::. We therefore expand fin Eq. (37) (and thus also in Eq. (44)) into a Laurent series around the poles from the rst negative order up to the rst positive order in kLto solve the boundary condition for kL, giving kLex L3(1 +d) + 2(kuL)2p 12(kuL)2+ 9(1 +d)2 2kuL: (45) Using this result and the Laurent-series expansion for fin Eq. (39) and Eq. (40), we nd the Gilbert-damping renormalization term ( !+
(oop) EA,n) and the ratio between the modes
(oop) EA,n
03 3(1 +d) + 2(kuL)2p 12(kuL)2+ 9(1 +d)2
 p 4(kuL)2+ 3(1 +d)2p 3(1 +d)
2(kuL)2p 4(kuL)2+ 3(1 +d)2: (46) This ratio is plotted in Figure 3 for n5. We see that the ratio vanishes for large values of d. For small values of the anisotropy energy d, the approximate ratio exceeds the exact result of the ratio we found in the limiting case of no surface anisotropy (see Eq. (42)). For moderate values ofd5, the expansion around the upper poles is sucient, but only for the rst few modes. This im- plies that moderate-strength easy-axis surface anisotropy quenches spin pumping for the lowest excited modes but does not aect modes with higher transverse exchange energy.9 05101520253001234 LÈKsÈADaEP,nDa0 n=1n=5 FIG. 4. Plot of
(oop) EP,n=
0. The dashed line represents the ratio
n=
0in the case of no surface anisotropy (see Eq. (42)). b. Easy-Plane Surface Anisotropy ( d < 0):Easy- plane surface anisotropy is represented by a negative sur- face anisotropy din Eq. (44). In this case, the boundary condition must be treated separately for the uniform- like (n= 0) mode and the higher excitations. When jdj>1, we can obtain a solution by expanding along the imaginary axis of kL. This corresponds to expressing the boundary condition in the form ikLtanh(ikL) =jdj, with the asymptotic behavior kL ijdj. Using the asymptotic form of the boundary condition in Eqs. (39) and calculating the renormalization of the Gilbert damp- ing using Eq. (40), we nd that the renormalization is !+
(oop) EP,0, where
(oop) EP,0
0= 2jdj: (47) Thus, the Gilbert damping of the lowest mode is much enhanced by increasing surface anisotropy. The surface- anisotropy mode is localized at the surface because it decays from the spin-active interface and into the lm. Because the eective volume of the mode is reduced, spin pumping more strongly causes dissipation out of the mode and into the normal metal. For the higher modes ( n > 0), the negative term on the rhs of Eq. (44) forces the kLsolutions closer to thenegative, lower poles of tan( kL), located at k(l) nL= (2n 1)=2, wheren= 1;2;3;:::. We repeat the procedure used for the EA case by expanding finto a Laurent series around these lower poles, arriving at kL3(1jdj) + 2(k(l) nL)2+q 12(k(l) nL)2+ 9(1jdj)2 2k(l) nL: (48) Using this relation and the new lower-pole Laurent ex- pansion for f, Eqs. (39) and (40) give us the renormal- ization of the Gilbert damping ( !+
(oop) EP,n) and the ratio
(oop) EP,n
03 3(1jdj) + 2(kuL)2+p 12(kuL)2+ 9(1jdj)2
 p 4(kuL)2+ 3(1jdj)2+p 3(1jdj)
2(kuL)2p 4(kuL)2+ 3(1jdj)2: (49) This ratio is plotted in Figure 4 from n= 1 up ton= 5. We see that the ratio vanishes for large values of jdj. Similar to the case of EA surface anisotropy, the approx- imation breaks down for large nand/or small values of jdj. Whereas the n= 0 mode exhibits a strong spin- pumping enhanced dissipation in this eld conguration, the DC ISHE eld vanishes when = 0 (see Eq. (26)). This is one of the reasons why this conguration is sel- dom used in experiments. However, this conguration can lead to a signicant AC ISHE, and a similar AC sig- nal was recently detected12. Because of the strong dissi- pation enhancement, the EP surface anisotropy induced localized mode in perpendicular magnetization geometry could be important in future experimental work. 2. In-plane Magnetization We will now complete the discussion of the spin- pumping enhanced Gilbert damping by treating the case in which the magnetization is in plane ( ==2). For such systems, the coecient C36= 0, and the 0th-order expansion of Eq. (37) becomes kLtankL=d (exk)2+!H !M
q 1 + (exk)2+ 2!H !Mq 1 + (exk)2+ 2!H !M 1 + 2(exk)2+ 2!H !M
dex L 1 + (exk)2+!H !M
coth L exq 1 + (exk)2+ 2!H !M: (50) For typical lm thicknesses, of some hundred nanome- ters, we have L= ex1 and (exk)21 for the lowest eigenmodes. Thus, we take the asymptotic coth 1 and neglect the ( exk)2terms, ridding the rhs of Eq. (50) ofanykdependence. Eq (50) now becomes similar to the out-of-plane case kLtan(kL) =de; (51)10 where de=d!H !Mq 1 + 2!H !M 1 + 2!H !M
3=2dex L 1 +!H !M
: (52) deis positive if d<0 and negative for d>0 up to a crit- ical valuedex=L=exKs=A= 1 + 2!H !M
3=2= 1 +!H !M
, where the denominator becomes zero. For negative d, jdej<jdj, whereas for positive d,jdejis initially smaller than that ofjdjbut quickly approaches the critical value. With the value Ksfrom Tab. I, we have jdej<jdj, in- dependent of the sign of d. With this relation, we can calculate an approximate Gilbert damping renormalization in both the EA and EP cases using the EP and EA relations, respectively, obtained in the out-of-plane conguration. Thus,
ip EA,0
oop EP,0jd!deff= 2jdej; (53)
ip EA,n
oop EP,njd!deff; (54)
ip EP,n
oop EA,njd!deff: (55) To summarize this section regarding the enhancement of Gilbert damping, we see that the enhancement can be very strong for the surface modes because their eective sizes are smaller than the thickness of the lm. For all other modes, the enhancement decreases with increasing magnitude of the surface-anisotropy energy. IV. NUMERICAL CALCULATIONS The rst step in the numerical method is to approxi- mate the equation of motion of Eq. (11) into by nite-size matrix eigenvalue problem. We discretize the transverse coordinateon the interval [L=2;L=2] intoNpoints la- beled byj= 1;2;:::;N , and characterize the transverse discrete solutions of the dynamic magnetization vectors mQby (mx;j;my;j) of size 2N. We approximate the 2nd-order derivative arising from the exchange interaction using a nth-order central dif- ference method. For the n2 discretized points next to the boundaries, we also use nth-order methods, using for- ward (backward) dierence schemes for the lower (upper) lm boundary. This strategy avoids the introduction of \ghost" points outside the interval [ L=2;L=2] to satisfy the boundary conditions. Thus, the total operator acting on the magnetiza- tion on the left-hand side of Eq. (11) becomes a sparse 2N2Nmatrix operator. On the right-hand side of Eq. (11), we also represent the convolution integral as a 2N2Ndense matrix operator, where each row is weighted according to the extended integration formu- las for closed integrals to nth order43. The four NN sub-blocks of this integration operator correspond to the four tensor elements of bGxy. In the nal discrete form, we obtained a 2 N2N !-dependent matrix.Next, the 4 boundary conditions (at the left and right boundaries for the two components, mxandmy) are used to reduce the number of equations to 2 N4. This is per- formed by algebraically solving the discretized boundary conditions with respect to the boundary points, i.e., by determining miwherei2f1;N;N + 1;2Ngin terms of the magnetizations at the interior points. Finally, each (2 N4)(2N4) matrix is separated into two parts: a term independent of the frequency ! and a term proportional to !. The dipole interaction causes the eigenvalue problem to be non-Hermitian and therefore computationally more demanding than a gen- eralized eigenvalue problem. We nd the dispersion rela- tion and magnetization vectors by solving this eigenvalue problem. The resulting eigenvectors are used to nd the magnetization at the boundary by back-substitution into the equations for the boundary conditions. We are interested in nding the mode and wave-vector dependence of the spin-pumping enhanced Gilbert damp- ing. To obtain this information numerically, we perform two independent calculations of the (complex) eigenval- ues. First, we calculate the complex eigenvalues !dwhen there is no spin pumping, but dissipation occurs via the conventional bulk Gilbert damping. Second, we calcu- late the complex eigenvalues !spwhen spin pumping is active at the FI-NM interface but there is no bulk Gilbert damping. A mode- and wave-vector-dependent measure of the eective enhanced Gilbert damping enhancement is then given by
=Im!sp Im!d: (56) To ensure that we treat the same modes in the two in- dependent calculations, we check the convergence of the relative dierence in the real part of the eigenvalues. Ta- ble I lists the values for the dierent system parameters that are used throughout this section. Let us rst discuss the renormalization of the Gilbert damping when there is no surface anisotropy. We will present the numerical results for the three main geome- tries described in Sec. I and compare the results to the analytical results of Sec. III A. A. FVMSW ( = 0) Figure 5 shows the wave-vector dependent renormal- ization of the Gilbert damping
due to spin pumping at the FI-NM interface in the FVMSW geometry. In this geometry, waves travelling along ^have the same symmetry; thus, each line is doubly degenerate and cor- responds to two waves of !. The \spikes" in the gure are due to degeneracies, i.e., mode crossings, and upon inspection, these spikes can be observed in the dispersion relation.11 0.010.1110100QL0.050.100.150.200.250.30DaH10-2L 0.11100.60.70.80.91.0Re8wwM<-L2xL2m® HxL¤ FIG. 5.
versus wave vector for the FVMSW geometry of the four smallest eigenvalues. Top inset: Magnitudes of eigenvectors (in arbitrary units) across the lm at QL= 10. Bottom inset: dispersion relation in the dipole-dipole active regime. 1. Easy-Axis Surface Anisotropy ( ^easy axis) 0.010.1110100QL0.20.40.60.81.01.2DaH10-3L 0.11100.60.70.80.91.0Re8wwM< -L2xL2m® HxL¤ FIG. 6.
EAversus wave vector for the FVMSW geometry showing the four smallest eigenvalues. The horizontal dashed lines indicate solutions of Eq. (46). Left inset: Magnitudes of eigenvectors (in arbitrary units) across the lm at QL= 5. Right inset: Dispersion relation in the dipole-dipole active regime. Figure 6 shows
EAfor the FVMSW geometry with an EA surface anisotropy at the spin-active interface. As predicted in Sec. III C 1 a, all modes exhibit a decreased
compared with those in Eqs. (43) and (42). For small QLand the chosen value of Ks(see Tab. I), the 1stfour modes match the analytical result of Eq. (46), which is consistent with the plot in Figure 3. For even higher ex- cited modes, the eect of the EA surface anisotropy be- comes weaker due to the increase in transverse exchange energy. These modes (not shown in the gure) approach the value of
n.2. Easy-Plane Surface Anisotropy ( ^hard axis) aLDaH10-3L45 0.010.11100.00.10.20.3 QL0.11100.50.60.70.80.91.0Re8wwM<bL -L2xL2m® HxL¤cL FIG. 7. a)
EPversus wave vector for the FVMSW geom- etry, showing the four smallest eigenvalues. The dashed lines represent the analytic solutions from Sec. III C 1 b. b) Disper- sion relation in the dipole-dipole active regime. c) Magnitude of eigenvectors (in arbitrary units) across the lm at QL= 5. Figure 7 shows
EPfor the FVMSW geometry with an EP surface anisotropy. We see that the mode corre- sponding to n= 0 has been promoted to a surface mode with a large
, which for small values of QLmatches Eq. (47). For the higher excited modes, we observe a decrease in
compared to the case with no surface anisotropy. B. BVMSW ( ==2and= 0) 0.010.1110100QL0.20.40.60.81.0DaH10-3L 0.11100.70.80.91.01.11.2Re8wwM< -L2xL2m® HxL¤ FIG. 8.
versus wave vector for the BVMSW geometry (==2 and= 0) withKs= 0, plotted for the four small- est eigenvalues. Left inset: magnitudes of normalized eigen- vectors across the lm at QL= 5. Right inset: dispersion relation in the dipole-dipole active regime. Figure 8 shows the QL-dependent renormalization of the Gilbert damping due to spin pumping at the FI- NM interface in the BVMSW geometry. We see that the enhancement
agrees with the analytic limits in12 Eqs. (43) and (42) for small values of QL. For large val- ues ofQL, we are in the strong exchange regime, in which the in-plane exchange energy becomes large compared to all other energy contributions. This in-plane exchange stiness eectively quenches the coupling to the normal metal layer, causing
!0 for large values of QL. Although Figure 8 only appears to show the three rst eigenvalues and eigenvectors, it actually contains dou- ble this amount. Because ^zis parallel to the wave- propagation direction ^in this geometry, there is no change in dipolar energies, regardless of whether the wave travels in the + ^direction or in the ^direction; thus, the Gilbert damping is enhanced equally in both wave directions. A slight oset from this conguration, taking either < = 2 or6= 0, would result in a splitting of each line in Figure 8 into two distinct lines. Including Surface Anisotropy Figure 9 shows both the EA and the EP surface- anisotropy calculations in the BVMSW geometry. In the case of an EA surface anisotropy, the mode correspond- ing ton= 0 gets promoted to a surface mode, similarly to the case in which there is EP surface anisotropy in the FVMSW geometry. The increase in
is much smaller for the same magnitude of Ks, as explained in detail in Sec. III C. The higher modes, corresponding to n > 0, exhibit increased quenching of the Gilbert damping en- hancement. In the case of EP surface anisotropy, all modes exhibit quenched Gilbert damping enhancement. C. MSSW ( ===2) Figure 10 shows the QL-dependent renormalization of the Gilbert damping due to spin pumping at the FI-NM interface in the MSSW geometry. The computed eigen- values agree with Eqs. (43) and (42) for small values of QL. We see in the inset of Figure 10 that in this geom- etry, the macrospin-like mode behaves as predicted by Damon and Eshbach3433, cutting through the dispersion relations of the higher excited modes for increasing val- ues ofQLin the dipole-dipole regime. A prominent fea- ture of this geometry is the manner in which the modes with dierent signs of Ref!gbehave dierently due to the dipole-dipole interaction. This is because the inter- nal eld direction ( ^z) is not parallel to the direction of travel ( ^) of the spin wave. Hence, changing the sign of !is equivalent to inverting the externally applied eld, changing the xyzcoordinate system in Figure 1 from a right-handed coordinate system to a left-handed system. In the middle of the dipole regime, the lack of symme- try with respect to propagation direction has dierent eects on the eigenvectors; e.g., in the dipole-dipole ac- tive region the modes with positive or negative Ref!g experience an increased or decreased magnitude of the dynamic magnetization, depending on the value of QL,as shown in Figure 10e & f. This magnitude dierence creates dierent renormalizations of the Gilbert damp- ing, as the plot of
()in Figure 10b & c shows. Including Surface Anisotropy Figure 11 shows
computed for modes in the MSSW geometry with EA and EP surface anisotropies. We can clearly see that for small QLan exponentially localized mode exists in the EA case, and as predicted in Sec. III C, all the lowest-energy modes have spin pumping quenched by EP surface anisotropy. This is similar to the corre- sponding case in the BVMSW geometry. D. AC and DC ISHE Figure 12 shows the DC and AC ISHE measures for the BVMSW geometry corresponding to the data repre- sented in Figure 8. In this geometry, the angular term, sin, in Eq. (28) is to equal one, ensuring that the DC measure is nonzero. This is not the case for all geometries because the DC electric eld vanishes in the FVMSW ge- ometry. The mode-dependent DC ISHE measure exhibits the sameQL-dependence as the spectrum of the Gilbert damping enhancement in all geometries where sin 6= 0. We have already presented the renormalization of the Gilbert damping in the most general cases above. There- fore, we restrict ourselves to presenting the simple case of the BVMSW geometry with no surface anisotropy here. The AC ISHE measure plotted in Figure 12 exhibits a similarQLdependence to the Gilbert damping renor- malization (and hence the DC ISHE measure), but with a slight variation in the spectrum towards higher values of QL. Note that because Eq. (24) is non-zero for all values of, the AC eect should be detectable in the FVMSW geometry. By comparing the computed renormalization of the Gilbert damping for the dierent geometries in the previous subsections, we see that the strong renor- malization of the n= 0 induced surface mode that oc- curs in the FVMSW geometry with easy-plane surface anisotropy (see Sec. IV A 2 and Fig. 7) can have a pro- portionally strong AC ISHE signal in the normal metal. V. CONCLUSION In conclusion, we have presented analytical and numer- ical results for the spin-pumping-induced Gilbert damp- ing and direct- and alternating terms of the inverse spin- Hall eect. In addition to the measures of the magnitudes of the DC and AC ISHE, the eective Gilbert damp- ing constants strongly depend on the modes through the wave numbers of the excited eigenvectors. In the long-wavelength limit with no substantial sur- face anisotropy, the spectrum is comprised of standing- wave volume modes and a uniform-like (macrospin)13 0.010.1110100QL0.51.01.52.0aLDaH10-3L 0.11100.70.80.91.01.11.2Re8wwM< -L2xL2m® HxL¤ 0.0010.010.1110100QL0.20.40.60.81.0bLDaH10-3L 0.11100.70.80.91.01.11.2Re8wwM< -L2xL2m® HxL¤ FIG. 9. a) Dispersion relation versus wave vector for the BVMSW geometry ( ==2,= 0) for the four lowest eigenvalues in the case of EA surface anisotropy. b) Dispersion relation in the case of EP surface anisotropy. In both gures, the horizontal dashed lines mark the value of
nin the case of no surface anisotropy. 0.010.11101000.00.20.40.60.8Da+H10-3LaL 0.010.11101000.00.20.40.60.81.0 QLDa-H10-3LbL 0.010.11101000.91.1.1ÈReHwwMLÈcL QL -L2xL20.0.51.1.5m® HxL¤dL -L2xL20.0.51.m® HxL¤eL FIG. 10. Gilbert damping renormalization in the MSSW geometry. Subplots a) and b) show Gilbert damping renormalization
for modes with positive (negative) Ref!g. The horizontal dashed lines represent the analytical values
0and
nfor smallQL. c) Dispersion relation versus wave vector for the MSSW geometry ( ===2) for the four smallest eigenvalues, colored pairwise in !. Subplot d (e) shows the magnitude of normalized eigenvectors (in arbitrary units) at QL= 3 across the lm modes with positive (negative) Ref!g. 0.010.11101000.0.51.1.5Da+H10-3LaL 0.010.11101000.0.51.1.5 QLDa-H10-3LbL 0.010.11101000.00.20.40.60.8Da+H10-3LcL 0.010.11101000.00.20.40.60.8 QLDa-H10-3LdL FIG. 11. a) and b) Gilbert damping renormalization from spin pumping in the MSSW geometry ( ===2) for modes with positive (negative) Ref!gin the case of EA surface anisotropy. The four smallest eigenvalues are colored pairwise in !across the plots. c) and d) show the Gilbert damping renormalization in the case of EP surface anisotropy. mode. These results are consistent with our previous ndings18: in the long-wavelength limit, the ratio be- tween the enhanced Gilbert damping for the higher vol-ume modes and that of the macrospin mode is equal to two. When there is signicant surface anisotropy, the uniform mode can be altered to become a pure lo-14 0.010.11101000.00.10.20.30.40.50.6 QLeACH10-4LaL 0.010.11101000.00.51.01.5 QLeDCH10-9cmLbL FIG. 12. ISHE as a function of in-plane wave vector in the BVMSW geometry with Ks= 0. a) AC ISHE measure of Eq. (28); b) DC ISHE measure of Eq. (28). calized surface mode (in the out-of-plane geometry and with EP surface anisotropy), a blend between a uniform mode and a localized mode (in-plane geometries and EA surface anisotropy), or quenched uniform modes (out-of- plane eld conguration and EA surface anisotropy, or in-plane eld conguration and EP surface anisotropy). The eective Gilbert damping is strongly enhanced for the surface modes but decreases with increasing surface- anisotropy energies for all the other modes. The presented measures for both the AC and DC in- verse spin-Hall eects are strongly correlated with the spin-pumping renormalization of the Gilbert damping, with the DC eect exhibiting the same QLdependency, whereas the AC eect exhibits a slighthly dierent vari- ation for higher values of QL. Because the AC eect is nonzero in both in-plane and out-of-plane geometries and because both EP and EA surface anisotropies in- duce surface-localized waves at the spin-active interface, the AC ISHE can be potentially large for these modes. ACKNOWLEDGMENTS We acknowledge support from EU-FET grant no. 612759 (\InSpin"), ERC AdG grant no. 669442 (\In-sulatronics"), and the Research Council of Norway grant no. 239926. Appendix A: Coordinate transforms The transformation for vectors from toxyzcoor- dinates (see Fig. 1) is given by an ane transformation matrixT, so that f(xyz)=T
f(); for some arbitrary vector f. Tensor{vector products are transformed by inserting a unity tensor I=T1Tbe- tween the tensor and vector and by left multiplication by the tensor T, such that the tensor transforms as TbGT1 for some tensor bGwritten in the basis. Tis given by the concatenated rotation matrices T= R2
R1, whereR1is a rotation around the -axis, andR2is a rotation  2around the new -axis/y-axis. Hence, R1=0 B@1 0 0 0 cossin 0 sincos1 CA; (A1) R2=0 B@sin0cos 0 1 0 cos0 sin1 CA; (A2) such that T=0 B@sincossincoscos 0 cossin cossinsin sincos1 CA: (A3) This transformation matrix consists of orthogonal trans- formations; thus, the inverse transformation, which transforms xyz!, is just the transpose, T1=TT. 1A. Serga, A. Chumak, and B. Hillebrands, J. Phys. D 43, 264002 (2010). 2V. Cherepanov, I. Kolokolov, and V. L'vov, Physics Re- ports 229, 81 (1993). 3Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanasahi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 4C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and B. Hillebrands, Appl. Phys. Lett. 97(2010).5C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, M. B. Jung eisch, E. Saitoh, and B. Hille- brands, Phys. Rev. Lett. 106, 216601 (2011). 6B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107, 066604 (2011). 7S. M. Rezende, R. L. Rodriguez-Suarez, M. M. Soares, L. H. Vilela-Leao, D. L. Dominguez, and A. Azevedo, Appl. Phys. Lett. 102, 012402 (2013). 8L. H. Vilela-Leao, A. A. C. Salvador, and S. M. Rezende,15 Appl. Phys. Lett. 99, 102505 (2011). 9K. Ando and E. Saitoh, Phys. Rev. Lett. 109, 026602 (2012). 10C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Appl. Phys. Lett. 100, 092403 (2012). 11M. B. Jung eisch, V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands, Applied Physics Letters 103 (2013), http://dx.doi.org/10.1063/1.4813315. 12C. Hahn, G. de Loubens, M. Viret, O. Klein, V. V. Naletov, and J. Ben Youssef, Phys. Rev. Lett. 111, 217204 (2013). 13Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 14A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012). 15Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 16J. Xiao and G. E. W. Bauer, Phys. Rev. Lett. 108, 217204 (2012). 17A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and P. J. Kelly, in Spin Current , edited by S. Maekawa, S. O. Valen- zuela, E. Saitoh, and T. Kimura (Oxford University Press, 2012). 18A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111, 097602 (2013). 19P. Monod, H. Hurdequint, A. Janossy, J. Obert, and J. Chaumont, Phys. Rev. Lett. 29, 1327 (1972). 20R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19, 4382 (1979). 21A. Janossy and P. Monod, Phys. Rev. Lett. 37, 612 (1976). 22S. Mizukami, Y. Ando, and T. Miyazaki, Japanese journal of applied physics 40, 580 (2001). 23R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 (2001). 24A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I. Halperin, Phys. Rev. B 66, 060404 (2002).25Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002). 26L. Berger, Physical Review B 54, 9353 (1996). 27J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 28A. Brataas, A. D. Kent, and H. Ohno, Nature materials 11, 372 (2012). 29T. Gilbert, Phys. Rev. 100, 1243 (1955). 30T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 31B. A. Kalinikos, Sov. Phys. J. 24, 719 (1981). 32B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013 (1986). 33J. R. Eshbach and R. W. Damon, Phys. Rev. 118, 1208 (1960). 34R. Damon and J. Eshbach, J. Phys. Chem. Solids 19, 308 (1961). 35H. Puszkarski, IEEE Trans. Magn. 9, 22 (1973). 36R. E. D. Wames and T. Wolfram, J. Appl. Phys. 41, 987 (1970). 37G. Rado and J. Weertman, J. Phys. Chem. Solids 11, 315 (1959). 38H. Jiao and G. E. W. Bauer, Phys. Rev. Lett. 110, 217602 (2013). 39M. B. Jung eisch, V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands, ArXiv e-prints (2013), arXiv:1302.6697 [cond-mat.mes-hall]. 40Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi, T. An, Y. Fujikawa, and E. Saitoh, ArXiv e-prints (2013), arXiv:1302.7091 [cond-mat.mes-hall]. 41H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, and F. Y. Yang, Phys. Rev. Lett. 112, 197201 (2014). 42F. R. Morgenthaler, IEEE Trans. Magn. 8, 130 (1972). 43W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipies , 3rd ed. (Cambridge University Press, 2007).

2016-12-21

We theoretically consider the spin-wave mode- and wavelength-dependent enhancement of the Gilbert damping in magnetic insulator--normal metal bilayers due to spin pumping as well as the enhancement's relation to direct and alternating inverse spin Hall voltages in the normal metal. In the long-wavelength limit, including long-range dipole interactions, the ratio of the enhancement for transverse volume modes to that of the macrospin mode is equal to two. With an out-of-plane magnetization, this ratio decreases with both an increasing surface anisotropic energy and mode number. If the surface anisotropy induces a surface state, the enhancement can be an order of magnitude larger than for to the macrospin. With an in-plane magnetization, the induced dissipation enhancement can be understood by mapping the anisotropy parameter to the out-of-plane case with anisotropy. For shorter wavelengths, we compute the enhancement numerically and find good agreement with the analytical results in the applicable limits. We also compute the induced direct- and alternating-current inverse spin Hall voltages and relate these to the magnetic energy stored in the ferromagnet. Because the magnitude of the direct spin Hall voltage is a measure of spin dissipation, it is directly proportional to the enhancement of Gilbert damping. The alternating spin Hall voltage exhibits a similar in-plane wave-number dependence, and we demonstrate that it is greatest for surface-localized modes.

Spin Pumping, Dissipation, and Direct and Alternating Inverse Spin Hall Effects in Magnetic Insulator-Normal Metal Bilayers

1612.07020v2

arXiv:1805.01776v2 [cond-mat.stat-mech] 7 May 2018Superparamagnetic Relaxation Driven by Colored Noise J. G. McHugh,1R. W. Chantrell,1I. Klik,2and C. R. Chang2 1Department of Physics, The University of York, York, YO10 5D D, UK 2Department of Physics, National Taiwan University, Taipei , Taiwan A theoretical investigation of magnetic relaxation proces ses in single domain particles driven by colored noise is presented. Two approaches are considere d; the Landau-Lifshitz-Miyazaki-Seki equation, which is a Langevin dynamics model based on the int roduction of an Ornstein-Uhlenbeck correlated noise into the Landau-Lifshitz-Gilbert equati on and a Generalized Master Equation ap- proach whereby the ordinary Master Equation is modified thro ugh the introduction of an explicit memory kernel. It is found that colored noise is likely to bec ome important for high anisotropy materials where the characteristic system time, in this cas e the inverse Larmor precession frequency, becomes comparable to the correlation time. When the escape time is much longer than the corre- lation time, the relaxation profile of the spin has a similar e xponential form to the ordinary LLG equation, while for low barrier heights and intermediate da mping, for which the correlation time is a sizable fraction of the escape time, an unusual bi-exponen tial decay is predicted as a characteristic of colored noise. At very high damping and correlation times , the time profile of the spins exhibits a more complicated, noisy trajectory. I. INTRODUCTION Thermally-activated magnetization reversal over an anisotropic energy barrier is the driving force for switch- ing in magnetic materials. Theoretical understanding wasfirst developed by N´ eel1basedon the transition state theory (TST) leading to an Arrhenius-like relaxation time proportional to exp( EB/kBT) whereEBis the en- ergy barrier, kBthe Boltzmann constant and Tthe tem- perature. Brown2provided further insight through the construction ofthe Langevinequation for the problem by the introduction of white-noise fields into the Landau- Lifshitz equation with Gilbert damping, leading to the stochastic Landau-Lifshitz-Gilbert (LLG) equation, An expression for the relaxation time of thermally-driven escape over the energy barrier is then found through the lowest eigenvalue of the correspondingFokker-Planck equation(FPE)governingthetime-evolutionoftheprob- ability density function of the magnetization orientation. The routeto the Arrhenius-likerelaxationtime expres- sion is one of two directions leading from the Langevin equation. The second, Langevin Dynamics (LD) ap- proach is the direct numerical solution of the Langevin equation3–6. There is a natural separation of timescales, with LD used for high frequency applications such as magnetic recording and the Arrhenius-like relaxation timeusedforslowdynamicbehaviorarisingfromthermal activationoverenergybarriers. The twoapproacheshave been compared by Kalmykov et. al.,7who calculated es- cape times for both cases giving excellent agreement for the variation of escape time with damping constant and demonstrating the importance of starting the LD calcu- lations from the correct thermal equilibrium distribution within the energy minimum.The LLG equation for a single spin takes the well- known form dS dt=−γ 1+α2/parenleftbig S×H+αS×(S×H)/parenrightbig ,(1) whereαis the phenomenological damping constant, γ= 1.7611T−1s−1andSisaunitvectorinthedirectionofthe spin,S=µ/µs. The local magnetic field, H, is derived from the first derivative of the spin Hamiltonian Hwith respect to the spin degree of freedom, H=−1 µs∂H ∂S. (2) Thermal fluctuations are necessary to incorporate the deviations of a particular spin from the average tra- jectory. This is done via the formal inclusion of ran- dom fields in the LLG equation. In order to realize the Fluctuation-Dissipation theorem for this system, these thermal fields must also be proportionalto the same phe- nomenological damping constant, αthat occurs in the damping. The moments of the thermal field are then given by ∝angbracketleftHth,i(t)∝angbracketright= 0 (3) ∝angbracketleftHth,i(t)Hth,j(t′)∝angbracketright=2αkBT γµsδ(t−t′)δij(4) wherei,jlabel the spin components. In all numerical simulations, we interpret the stochas- tic equation in the Stratonovich sense and employ the Heun method An implicit assumption of this approach is the presence of white noise, which exists in the zero cor- relation time limit for some physical noise process with a2 well-defined correlation time. Such a colored noise may be implemented for a magnetic system through the use of the Landau-Lifshitz-Miyazaki-Seki pair of Langevin equations, which take the form dS dt=γS×/parenleftbig H+η/parenrightbig , (5) dη dt=−1 τc(η−χS)+R, (6) whereτcis the correlation time and χis a spin-bath cou- pling which is related to the phenomenological damping parameter as α=γχτcin the limit of small correlation times. The autocorrelation of the white noise field, R, is given by ∝angbracketleftRi(t)Rj(t′)∝angbracketright=2χkBT τcµsδijδ(t−t′).(7) This pair of Langevin equations leads to a frequency- dependent damping of the spin together with an expo- nentially correlated noise term in the spin-only space, ∝angbracketleftˆηi(t)ˆηj(t′)∝angbracketright=χkBT µse−(t−t′) τcδij=χkBT µsK(t−t′)δij (8) whereK(t) = exp−(t−t′) τcis the exponential memory ker- nel. For completeness, additional background on the LLMS Langevin equation and colored noise is included in Appendix A. An alternative approach to the Langevin equation is the discrete orientation approximation, whereby, in the limit oflarge barriers, the detailed dynamics are replaced by phenomenological rate equations describing transi- tions between the minima of the magnetic potential. We may augment this description by the introduction of a memory kernel into the rates, thus replacing the master equation description with a generalized master equation which explicitly incorporates the retardation effect into the rate equations. Here we investigate the introduction of colored noise into the calculation of escape rates. This leads to sig- nificant effects for materials with large magnetocrys- talline anisotropy energies, including the prediction of bi-exponential behavior at intermediate damping, when the characteristic time of the relaxation process becomes comparable to the heat bath correlationtime. The paper is organized as follows. We first outline thermally acti- vated escape times for single nanoparticles, followed by anintroductionofcolorednoiseintotheLangevinformal- ism via the LLMS equations. We then derive the relax- ation profile from the non-Markovian generalized exten- sion of the rate equation, followed by a systematic inves- tigationoftheeffects ofthebarrierheightandcorrelation times on the relaxation profile from LLMS simulations.A. Thermally-Assisted Magnetization Reversal We will investigate here the effect that colored noise has on the dynamics of the thermal escape problem for a magnetic nanoparticle. The spin Hamiltonian of the sys- tem contains both an applied field and anisotropy term, taking the form H=−KVS2 z−µs/vectorH·S, (9) whereKis the anisotropy constant and Vis the particle volume. For the escape problem we have a spin energy potential of the form V(θ,φ) =σβ−1/parenleftbig sin2θ−2h(cosψcosθ(10) +sinψsinθcosφ)/parenrightbig , whereθ,φare respectively the polar and azimuthal components of the spin in spherical coordinates, σ= KV/k BTis the reduced barrier height parameter, h= H/2σis the reduced field, β= (kBT)−1andψis the angle between the easy-axis and the applied field. This potential has a bistable character under the condi- tion than the critical applied field value, h < h c(ψ) = ((cos2/3ψ+sin2/3ψ)−3/213, in which case there are local and global minima in the north and south polar regions, with an equatorial saddle point between them. We are then interested in the calculation of the characteristic es- cape time of a spin initialized in one such minimum. For the special case of aligned field and easy axis, for whichψ= 0 the potential is V(θ) =σβ−1/parenleftbig sin2θ−2hcosθ/parenrightbig . (11) In this case the escape time takes the Arrhenius form, where the barrier energy, EB, is proportional to the anisotropy energy, leading to an escape time τ∝f−1 0eKV/k BT(12) wheref0is the attempt frequency, the frequency of Lar- mor gyromagnetic precession at the bottom of the well. Weinvestigatetheescapetimeinthecoloredandwhite noise cases through repeated numerical integration of the Langevin equations for a spin initialized in a potential minimum. An important consideration for such simula- tions is the choice of initial and switching condition for the spin. We will initialize the spins with the Boltz- mann distribution at the bottom of the well in order to avoidinconsistenciesat lowdamping, while the switching condition is chosen such that Sz<−0.5, with the spin initialized in the positive z-direction, so that the spin is sufficiently deep in the well such that it has escaped. II. LLMS ESCAPE TIMES & COLORED NOISE A. System time τsvs.τccharacteristic bath time. For the uniaxial escape problem the external field in the LLMS will consist of an external applied part and an3 anisotropy contribution H=Ha+H0 (13) the magnitude of the anisotropic contribution depends on the orientation of the spin and is given by Ha= 2ku µs/vectorSz·/vectorz=Hk/vectorSz·/vectorzwhere/vectorzis the direction of easy magnetization and kuis the anisotropy energy. To gain intuition into the relevant timescales for the relaxation problem, we will assume the uniaxial case in the follow- ing, where the external field is applied along the same direction as the easy axis, such that both fields only have components in the z-direction. We note that the anisotropic field contribution varies with the projection of the spin on to the easy-axis as Ha=Hk(S·z)/vectorz= (Hkcosθ)/vectorz, (14) Thelargest field magnitude and consequently the fastest timescale of the problem is set by the value for which the anisotropic field contribution is at its largest, which is when the spin and the easy-axis precisely coalign. For any other orientation, the field will be smaller and the timescale of oscillation hence slower. We may then take the spin-only Langevin equation, dS dt=γS(t)×/parenleftbig (Hkcos(θ))/vectorz+¯η−χ/integraldisplayt −∞dt′K(t−t′)dS(t′) dt′/parenrightbig , (15) and proceed to scale this equation by the maximum anisotropy field value. Defining the system time for the spin asτs= (γHk)−1then dS dt=1 τsS(t)×/parenleftbig cos(θ)/vectorz+H−1 k¯η −H−1 kχ/integraldisplayt −∞dt′K(t−t′)dS(t′) dt′/parenrightbig .(16) We may scale the time variable in the Langevin equa- tionsothatthesystemtimeisremovedbytaking ζ=τst. Then we have dS dζ=S(ζ)×cos(θ)/vectorz+S(ζ)×/parenleftbigg H−1 k¯η(ζ) +H−1 kχ/integraldisplayζ′ −∞dζ′e−(ζ−ζ′)τs τcdS(ζ′) dζ′/parenrightbigg (17) The autocorrelation of the noise is similarly transformed to become ∝angbracketleft¯η(ζ)¯η(ζ′)∝angbracketright=τs τc¯De−ζ−ζ′)τs τc=¯D τe−(ζ−ζ′)/τ(18) whereτs/τc=τand¯D=D/τs=χτkBT/µs. We can then write the coupling as ¯ χ=χ/Hk, and ab- sorb theHkfactor into the diffusion constant for the thermal field. Since the thermal fields are given by ¯η(ζ) =√ 2D τ/integraltextζ −∞K(ζ−ζ′)Γ(ζ′), the diffusion constant becomes ¯D=χτkBT µsH2 k=¯χτkBT 2ku=¯χτ 2σ(19)11.21.41.61.82 0.01 0.1 1τ/τLLG γ Hk τcσ = 2 σ = 5 σ = 7.5 FIG. 1. Escape time, normalized to the uncorrelated LLG escape time vs correlation time, from LLMS simulations for a Co nanoparticle with α= 0.05 and different reduced barrier heights, σ, whereσ=ku/kBT. The final expression for the Langevin equation is then dS(ζ) dt=S(ζ)×/parenleftbig cos(θ)/vectorz+¯η−¯χ/integraldisplayζ −∞dζ′K(ζ−ζ′)dS(ζ′) dζ′/parenrightbig . (20) In the case that τ≪1 andτc≪τs, the memory kernels appearing in the noise and damping terms are reduced to delta functions and the white noise behav- ior is restored. Additionally the bath coupling and the strength of the thermal fluctuations are reduced by the anisotropy field, so that in the event of a very large anisotropy the precessional dynamics of the spin dom- inate the thermal and damping parts. We then conclude that the condition τc/greaterorsimilar(γHk)−1dictates whether the effect of correlations are relevant in the system dynamics in the high barrier limit. This prediction is borne out in numerical simulations of the LLMS equation. Figure 1 depicts the escape time calculated using the LLMS model for a Co nanoparti- cle of volume V= 8×10−27m3, with anisotropy en- ergyKV= 1.12×1021J, and a magnetic moment µs= 1.12×10−20J/T.wherethe correlationtime is normalized by the inverse of the Larmor precession frequency, and the escape time in the LLMS is normalized by the escape time calculated from the Markovian LLG equation. The escape rate departs from the LLG escape rate only once the correlation time is some significant fraction of the Larmor time, and for increasing barrier height the corre- lation time must be a larger fraction of the gyromagnetic precession before the escape rate departs from the LLG prediction. Figure 2 shows a comparison of the escape time for the Co nanoparticle and a SmCo 5nanoparticle of the same4 110 1.0E-15 1.0E-14 1.0E-13 1.0E-12 1.0E-11 1.0E-10τ/τLLG τcSmCo Co FIG. 2. Comparison of simulation results for systems with pa - rameters chosen tobe similar toSmCo 5and Co nanoparticles, respectively, for large reduced barriers σ= 13.5, and a fixed α= 0.05. The higher anisotropy SmCo 5exhibits departure from LLG behavior at smaller correlation times. volume. The SmCo 5material parameters are taken to be µs= 6.4×10−18J/T, and anisotropy KV= 2.16×10−16, a much higher anisotropy energy density than Co. This higher anisotropy gives the nanoparticle a faster system time, which causes the LLMS to depart from the LLG for smaller bath correlation times, τc, on the order of 50−100fsfor the SmCo 5particle, while it is approxi- mately 1psfor the Co nanoparticle. The fact that the system time is inversely proportional to the magnitude of the anisotropy field is exhibited in the simulations by the difference between LLMS and LLG escape rates at smaller values of the bath correlation time for the mate- rial with higher magnetic anisotropy. B. Arrhenius Behavior Crucially, it is found that the Arrhenius behavior of the escape rate is recovered from LLMS simulations in the limit of large barrier height. In figure 3 we show the temperature- dependence of the escape time vs reduced barrier height. In the high damping case, we see that the escape rates begin to convergeas the temperature tends towardszero. As the escape time between the wells becomes much longer than the bath correlation time, the detailed dy- namics of the spin within the well becomes less relevant. At low damping, the LLMS and LLG appear not to converge even at the larger barrier heights considered here. We attribute this difference to the difference in damping regimes and the physically distinct mechanisms involved in the escape process between the two regimes. Escape at high damping is mediated by thermal fluctu-0.010.1110100100010000100000 0246810121416τkr γ Hk σLLMS, h=0.2, α = 0.01 LLG, h=0.2, α = 0.01 Ψ = 1/4 0.010.1110100100010000 0246810121416τkr γ Hk σLLMS, h=0.3, α = 1, Ψ = 1/4 LLG, h=0.3, α = 0.01, Ψ = 1/4 0.010.1110100100010000 0246810121416τkr γ Hk σLLMS, h=0.2, α = 1 LLG, h=0.2, α = 1 Ψ = 1/4 0.010.11101001000 0246810121416τkr γ Hk σLLMS, h=0.3, α = 1, Ψ = 1/4 LLG, h=0.3, α = 1, Ψ = 1/4 FIG. 3. Escape time, τγHkvs reduced barrier height, σ, from LLMS and LLG simulations, for different values of the applied field h=µsH/σand damping, α, with a fixed angle of Ψ = π/4 between the applied field and the easy axis of magnetization. 1:Low damping, h= 0.2,2:Low damping, h= 0.3.3:High damping, h= 0.2,4:High damping, h= 0.3. ations, which liberate the bound spin. In the limit of vanishing temperature the infrequency of thermal oscil- lations of sufficient energy dominate the escape behavior and the escape rates converge. In contrast, the energy-controlled diffusion regime is characterized by the almost-free precessional motion of the spin in the well. In the highly correlated case, the5 simple damping is replaced with a frequency-dependent damping, an effect which increases the overall effective damping. In the limit T→0, this inhibits the escape ratebetweenthewellsbydecreasingtherateatwhichthe spin is able to attain a trajectory with sufficient escape energy. III. RATE EQUATIONS FOR THERMALLY-ACTIVATED MAGNETIZATION REVERSAL A. Master Equation The master equation is a phenomenological set of first- order differential rate equations for a multi-level system, which takes the form dni dt= Γij(t)nj(t), (21) whereniis a probability vector representing the proba- bility that the system is in one of a discrete set of states, andi,jlabel those discrete states, while the matrix of coefficients Γ i,jdictates the transition rate from state i to the state jof the system. The dynamics of the thermally-assisted escape prob- lem in a magnetic system may be approximated by such a master equation under the condition that the energy barrier is large compared to the thermal energy, σ >1, but not too large such that it would inhibit inter-well transitions. This approximation to the Langevin dynam- ics is called the discrete orientation approximation . The spin orientations are assumed to be restricted only to the 2 minima of the potential energy dictated by the spin Hamiltonian. The time evolution of the occupation of each state follows from Eq. 21, where i,j= 1,2. The transition matrix elements follow from the applied field, anisotropy and temperature. In particular, we will as- sume a fixed applied field, such that the transition rates are constant in time and the matrix takes the form Γij=/parenleftbigg −κ12κ21 κ12−κ21/parenrightbigg , (22) In the uniaxial case these rates are given by κ1→2= κ12=f0exp(−σ(1+h)2) andκ2→1=κ21= f0exp(−σ(1−h)2), whereσandhare the reduced bar- rier height and applied field, respectively. The time evo- lution of the population of the state n1is then explicitly given by dn1 dt=−κ12n1+κ21n2= (κ12+κ21)n1+κ21.(23) The time-evolution of the magnetization follows from the individual rates for the two wells, where the magne- tization is given by m(t) =n1(t)−n2(t) and is subject to the normalization condition n1(t) +n2(t) = 1. The differential equation for the magnetization is then dm dt=−Γ1m(t)−Γ2, (24)where Γ 1=κ12+κ21and Γ 2=κ12−κ21. This is the same form as the rate for the individual wells, Eq. 23. For an initial magnetization m0=n1(t= 0)−n2(t= 0), the magnetization as a function of time is a simple exponential, m(t) =e−Γ1t(Γ1m0+Γ2) Γ1−Γ2 Γ1,(25) which tends to the value −Γ1 Γ2=κ21−κ12 κ12+κ21. (26) In the long-time limit, the steady state magnetization corresponding to the difference in the transition rates between the wells, if κ2→1> κ1→2, the transition rate into well 1 is greater than the rate out, and we have a positive magnetization, as expected. B. Generalized Master Equation The non-Markovian extension of the master equation formalismiswhat iscalledageneralizedmasterequation. Under this model, the set of i×jrates represented in the transition matrix in Eq. 21 are promoted to a set of i×j memory kernels for the transitions between the wells i,j, replacing the set of first-order differential equations with a set of integro-differential equations for the population of each well, dni dt=/integraldisplay∞ 0Mij(t−τ)n(τ)dτ. (27) We will consider the simplified case Mij(t) =e−t/Θ ΘAij=K(t)Γij, (28) where Γ ijare the same constant transition rates consid- ered in the Markovian master equations, now modified by a simple exponential kernel over the recent popula- tion of the well. The integro-differential expression for the magnetization then becomes dm dt=−Γ1/integraldisplay∞ 0K(t−τ)m(τ)dτ−Γ2/integraldisplay∞ 0K(t−τ)dτ. (29) Where we note that for the exponential kernel, K(t) = e−t/Θ Θ, the uncorrelated form of the master equation is recovered in the limit of vanishing correlation time, limΘ→0K(t) =δ(t). The Laplace transform of this equation is ωm(ω)−m0=−Γ1K(ω)m(ω)−Γ2 ωK(ω),(30) whereK(ω) =L(K(t)) is the Laplace transform of the memory kernel, K(ω) =Θ−1 ω+Θ−1=1 1+Θω, (31)6 we then have m(ω) =−Γ2 ωK(ω)+m0 ω+Γ1K(ω). (32) After inserting the expression for the Laplace transform of the kernel we find m(ω) =−Γ2 ω+m0(1+Θω) Θω2+ω+Γ1. (33) Finally we solve for the time-dependence of the mag- netization by taking the inverse Laplace transform, m(t) =L−1[(1+Θω) Θω2+ω+Γ1] =φ(t)(Γ1m0+Γ2) Γ1−Γ2 Γ1, (34) we note that this bears a strong resemblance to the Markovian expression, Eq. 25, with the exponential be- ing replaced by the function φ(t), which is φ(t) =1 2β/parenleftBig (β−1)e−t(1+β)/2Θ+(β+1)e−t(1−β)/2Θ/parenrightBig , (35) whereβ=√1−4Γ1Θ. In the limit t→ ∞, the value of the magnetization again tends to−Γ2 Γ1. To see that this agrees with the uncorrelated solution for small correla- tion times, we may expand βin Θ for small Θ, hence β= 1−2Γ1Θ, inserting into the magnetization it be- comes m(t) =β−1 2βe−t/2ΘeΓ1t+(β+1) 2βe−Γ1t.(36) As Θ→0,β→1, and only the second term in the expression for the magnetization remains, m(t) =e−Γ1t, so the small correlation time limit of the spin evolution agrees with the non Markovian master equation. 00.10.20.30.40.50.60.70.80.91 00.511.522.53m(t) Γ1tR < 0.01 R=0.1 R = 0.24 FIG. 4. m(t) vst, forR= 0,0.1,0.2, under the initial condi- tionm= 1, with transition rates κ12= 1,κ21= 0Finally, we note that the solution for the magnetiza- tion breaks down into two regimes. First, we note that the expression for βdepends only on the product of the correlation time, Θ, and the rate Γ 1, and not on their specific individual values. We may then discuss the be- havior of the model in terms of only the ratio parameter R= Γ1Θ = Θ/Γ−1 1, which gives the ratio of the well correlation time to the escape time. Rewriting the Eq.35 for the spin vs time, m(t) =(Γ1m0+Γ2) Γ1/parenleftBig (e−t/2Θ([eβt/2Θ(37) −e−βt/2Θ]/2β+[e−βt/2Θ+eβt/2Θ]/2)/parenrightBig −Γ2 Γ1, which may be simplified in terms of hyperbolic trigono- metric functions, m(t) =e−t/2Θ/parenleftBigsinh(βt/2Θ) β+cosh(βt/2Θ)/parenrightBig .(38) For smaller R <1 4, we have a real value of β=√1−4R, and the time-dependence of the spin corre- sponds to Eq. 38. In Figure 4, we plot the time-evolution for values of R <1 4. Once the correlation time is some sizable fraction of the escape time, the behavior begins to depart from the simple exponential behaviorpredicted in the Markovian system. At early times the magnetiza- tion decays more slowly than the exponential decay and at later times it decays more quickly, while the timescale over which the decay occurs (Γ 1) remains the same. The effect of the increasing correlation time between the pop- ulations of the wells is then to shift the process to differ- ent, lower frequencies. In the case that R >1 4, we have an imaginary argu- ment to sinh and cosh, we then have an expression for m(t) m(t) =e−t/2Θ(sin(bt/2Θ) b+cos(bt/2Θ)) (39) whereb=√ 4R−1. We note that the solutions take the form of damped oscillations which tends toward the equilibrium value of the magnetization. However, these solutions are unphysical as the occupation in individual wells maybecome lessthan 0 for these values. This is not surprising, as for longer correlation times the generalized master equation will overestimate the population in each well and generate a time evolution which will continue to reduce the population of a well, even when that well is presentlyempty. Itisalsounclearwhatitwouldmeanfor the correlation time of the well population to exceed or be on the order of the overall escape time, as this would implythatthetimescaleoverwhichthespinpopulationis correlatedexceeds the overallescape time for the system, which is itself determined by changes in the individual well populations.7 IV. COMPARISON We may now directly compare the magnetic relaxation profiles calculated from explicit numerical integration of Eqs. 5, 6 at various barrier heights, damping and cor- relation times, to the biexponential decay predicted by the generalized master equation. In all of the present simulations we again use simulation parameters compa- rable to the Co nanoparticle of volume V= 8×10−27m3, anisotropy energy density K= 4.2×105J/m3giving an anisotropy energy KV= 1.12×1021J, and a magnetic momentµs= 1.12×10−20J/T, while no external applied field is assumed, Hext= 0. 00.20.40.60.81 0e+002e-094e-096e-098e-091e-08M(t)/Mr t (s)σ = 2, α = 0.01 e-t/τ 00.20.40.60.81 0e+002e-094e-096e-098e-091e-08M(t)/Mr t (s)σ = 6, α = 0.5 e-t/τ FIG. 5. Spin relaxation profiles from LLG simulations for TOP:σ= 2,α= 0.01, giving an exponential decay with characteristic escape time τ= 5×10−9sandBOTTOM : σ= 6,α= 0.5,τ= 4.5×10−9s . The spins are initialized in the equilibrium Boltz- mann distribution in one of the minima of the po- tential energy, according to the distribution P(θ)∝ sin(θ)exp(−ku/kBTsin2(θ)). To ensure that the noise is equilibrated with the spin at the correct temperature, the noise is initially set to ηi,j,k= 0, and is then evolved in the presence of the equilibrium distribution in the well untiltheycomeintothermalequilibrium. Theinitialcon- dition of the noise is important, as, for example, a choice ofη(t= 0) = 0, will result in a field which quickly alignswith the spins in the potential minimum and give an un- physical increase in the well population from equilibrium at short times. The time-evolution of the magnetization, M(t) = ∝angbracketleftSz(i)∝angbracketrightis then plotted, normalized by the initial rema- nent magnetization inside of the well, Mr=M(0). 00.20.40.60.81 0e+002e-094e-096e-098e-091e-08M(t)/Mr t (s)τc = 1, σ = 2, α = 0.01 e-t/τ 00.20.40.60.81 0e+002e-094e-096e-098e-091e-08M(t)/Mr t (s)τc = 1, σ = 6, α = 0.5 e-t/τ FIG. 6. Exponential behavior from LLMS simulations, for TOP:τc= 1,σ= 2.α= 0.01 we have an exponential decay with escape time τ= 5.5×10−9, andBOTTOM :τc= 1, σ= 6.α= 0.5τ= 5.3×0−9s For low damping and large barrier heights, the correlation time is much smaller than t he escape time. In Figure 5, we depict the numerical calculation of the relaxation profile from the LLG. This gives rise to an ex- ponential behavior with a single relaxation time, which is directly comparable to the exponential decay of the master equation. In general, the relaxation profile from the LLG may be non-exponential, with both the inte- gral relaxation time and the decay profile depending on the higher-order eigenvalues of the Fokker-Planck opera- tor and the equilibrium correlation functions of the spin, τi int=/summationtext kτi kλk. However, the relaxation is dominated by the first eigenvalue in the high-barrier limit and for small applied fields , for σ>1, with good agreement be- tween the LLG and exponential decay for σas low as 2, as is shown in Figure 5. Figure 6 shows the relaxation from LLMS simulations of the Co nanoparticle, where the correlation time is cho-8 0.10.20.30.40.50.60.70.80.91 0e+005e-111e-101e-102e-103e-10M(t)/Mr t (s)τc = 1, σ = 2, α = 0.5 e-t/τ FIG. 7. Biexponential behavior from LLMS simulations for τ= 1,σ= 2,α= 0.5 andτ= 1.48x10−10 sen to be of the order of the inverse Larmor precession time such that τc≈(γHk)−1. In both the cases of low damping and higher barriers, we see that the ordinary exponential behavior of the LLG is retained. In this case the escape time is much larger than the correlation time ofthenoise,andtherelaxationaldynamicsareunaffected by the intra-well dynamics of the spin which occur on a much faster timescale than the relaxation, τc/τ≈0.01 for both simulations. In the intermediate-to-high damping and high damp- ing regimes, the behavior of the magnetization becomes muchmoreinterestinganddepartsfromthe LLG. In par- ticular,forarelativelysmallbarrierof σ= 2,α= 0.5and acorrelationtimeagainoftheorderoftheinverseLarmor frequency. In this case the ratio of the escape to the cor- relationtime is τc/τ= 9.4×10−12s/1.48×10−10s≈0.06. The influence of the spin correlation is now visible in the relaxation profile of the escape, as shown in Figure 7, which is similar to the biexponential deviation predicted by the generalized master equation, with the relaxation proceeding more slowly at earlier times and speeding up at later times. Finally, for very long correlationtimes and high damp- ing, the correlation time remains a sizable fraction of the escape time. However the biexponential behavior is no longer evident as shown in figure 8. The decay remains approximately exponential with a highly noisy path, a possible indication that the precise decay profile is extremely dependent on the initial conditions for such strong coupling between the spin and bath. V. CONCLUSIONS We have investigated thermal relaxation in magnetic nanoparticles introducing colored noise. Two models00.20.40.60.81 0e+002e-104e-106e-108e-101e-09M(t)/Mr t (s)τc = 5, σ = 2, α = 0.5 e-t/τ 00.20.40.60.81 0e+001e-102e-103e-104e-105e-10M(t)/Mr t (s)τc = 5, σ = 2, α = 5 e-t/τ FIG. 8. LLMS simulations at high damping and long corre- lation times. The behavior continues to depart from a purely exponential decay, but now exhibits a noisy, more compli- cated time-dependence. TOP:τc= 5,σ= 2 ,α= 0.5 and τ= 4.5×10−10,BOTTOM :τc= 5,σ= 2 ,α= 5 and τ= 1.8×10−10 . are considered. The first is an approach based on the numerical solution of the Landau-Lifshitz-Miyazaki-Seki (LLMS) model, which replaces the white noise approx- imation associated with the use of LLB-equation based models. Due to computational requirements the LLMS approach is useful for relatively short timescales, conse- quently a second approach is derived based on a general- izedmasterequationapproachinvolvingtheintroduction of a memory kernel. We find that the importance of col- ored noise is determined by the ratio of the correlation timeτcto the characteristic system time τs= (γHk)−1, which is essentially the Larmor precession time. Con- sequently correlated noise should become important for materials with large magnetic anisotropy such as SmCo 5 where the characteristic time approaches femtoseconds. Both models, the LLMS-based approach and the mas- ter equation, although derived for different timescales, exhibit an unusual bi-exponential decay of the magneti- zation, which represents an interesting signature of the presence of colored noise.9 Appendix A: Colored Noise In this appendix we present some relevant background material on the LLMS equation and colored noise. 1. Landau-Lifshitz-Miyazaki-Seki The LLMS equations constitute an implementation of a colored noise in a system with a thermalization con- dition represented through the Fluctuation-Dissipation theorem. We reproduce here the original derivation by Miyazaki and Seki10, of the spin-only expression of the LLMS, which allows us to compare the LLMS thermal fluctuations directly to the Ornstein-Uhlenbeck. The time evolution of the LLMS noise term is similar to the OU, with an additional term which couples explicitly to the spin, dη dt=−1 τc/parenleftBig η(t)−χS(t)/parenrightBig +R. (A1) TakingD=χkBT µs, then the autocorrelation of the field Ris∝angbracketleftR(t)R(t′)∝angbracketright= 2D τcδ(t−t′), and proceeding to solve as a first-order linear differential equation in the same manner as the OU noise, we have η(t) =χ τc/integraldisplayt −∞dt′K(t−t′)S(t′) (A2) +/radicalbigg 2D τc/integraldisplayt −∞dt′K(t−t′)Γ(t). After integrating the first term by parts, we have η(t) =/radicalbigg 2D τc/integraldisplayt −∞dt′K(t−t′)Γ(t) (A3) −χ/integraldisplayt −∞dt′K(t−t′)dS(t′) dt′, and by inserting this into the precessional equation for the spin, we get the spin-only form for the LLMS equa- tion, dS dt=γS(t)×/parenleftBig H+¯η−χ/integraldisplayt −∞dt′K(t−t′)dS(t′) dt′/parenrightBig ,(A4) where we now label the thermal fluctuations by ¯η(t), ¯η(t) =/radicalbigg 2D τc/integraldisplayt −∞dt′K(t−t′)Γ(t′).(A5) The autocorrelation of this thermal field is ∝angbracketleft¯η(t)¯η(t′)∝angbracketright=DK(t−t′) (A6) =χkBT µsK(t−t′) =β−1 µsχK(t−t′), Recognizing χK(t−t′) as the damping term, we see that this is a representation of the Fluctuation-Dissipationtheorem for the colored noise, where the additional fac- tor ofµsarises from the spin normalization. Taking the zero correlation time limit, lim τc→0∝angbracketleft¯η(t)¯η(t′)∝angbracketright= 2Dτcδ(t−t′).(A7) 00.20.40.60.81 00.511.522.533.5P(θ) θLLMS, σ = 1 Analytical 00.20.40.60.81 00.511.522.533.5P(θ) θLLMS, σ = 10 Analytical FIG. 9. P(θ) vsθ, from numerical simulations of the LLMS equation for TOP:σ= 1 and BOTTOM :σ= 10, with τcγHk= 2. . We note that the LLMS thus derived from the physi- cal consideration of the spin-field interaction is not im- mediately comparable with the typical expression for the Ornstein-Uhlenbeck colored noise, owing to the fact that the 1/τcterm has been implicitly absorbed in the white noise term. If we rescale the driving noise such that Q(t) =τcR(t), we then have a pair of Langevin equa- tions dS dt=γ(S×(H+η)), (A8) while the noise evolves as, dη dt=−1 τc/parenleftBig η(t)−χS(t)+Q/parenrightBig . (A9) The autocorrelation of the white noise is ∝angbracketleftQ(t)Q(t′)∝angbracketright=2χτckBT µsδ(t−t′) = 2Dδ(t−t′),(A10)10 withD=χτckBT µs, while the limit of the autocorrelation of the thermal term in the spin-only expression is now, lim τc→0∝angbracketleft¯Q(t)¯Q(t′)∝angbracketright=D τcδ(t−t′),(A11) which is directly comparable to the Ornstein-Uhlenbeck form of the colored noise. The expression of the LLMS in terms of the bath variable Qhas the additional benefit that/bracketleftbig Q/bracketrightbig =Tand so we can interpret Qas the thermal magnetic field contribution to the evolution of the bath field. Finally, we may see that the limit of the LLMS equa- tion for vanishing correlation time is the LLG equation. For small correlation times we can then take the Taylor expansion about the time tint′, so that the damping term becomes, /integraldisplayt −∞K(t−t′)dS(t′) dt′dt′=/bracketleftBig/integraldisplayt −∞K(t′)dt′/bracketrightBigdS(t) dt+... (A12) Hence the spin and memory kernel decouple in the small correlation time limit, and the Langevin equation be- comes dS dt=γS(t)×/parenleftBig H+¯η−/bracketleftBig χ/integraldisplayt −∞dt′K(t−t′)/bracketrightBigdS(t) dt/parenrightBig , (A13) After performing the integration over t′, the damping is χ/integraldisplayt −∞dt′e−(t−t′)/τc=χτc. (A14) and by direct comparison of the damping terms in this expression and in Gilbert’s equation we have the rela- tionship of the phenomenological damping to the LLMS parameters α=χγτc. We note also that this expression can be seen if we identify the driving white noise in the bath field of the LLMS with the thermal magnetic fieldsof the LLG. ∝angbracketleftQ(t)Q(t′)∝angbracketright=2χτckBT µsδ(t−t′) =2αkBT γµsδ(t−t′) =∝angbracketleftHth(t)Hth(t′)∝angbracketright(A15) under the assumption that α=γχτc. 2. Thermalization As a quantitative evaluation of the LLMS model and our implementation thereof, we compare the equilibrium behavior to the appropriate analytical Boltzmann distri- bution, which the Markovian LLG equation also satis- fies. We simulate a single spin under the influence of anisotropy only. The Boltzmann distribution for such a system is P(θ)∝sinθexp(−kusin2θ kBT) (A16) whereθis the angle between the spin and the easy- axis and the factor of sin θarises from normalizing the probability distribution on the sphere. WE initialize the spin along the easy-axis direction, then allow the spin to evolve for 108steps after equilibration and evaluate the probability distribution by recording the number of steps the spin spends at each angle to the easy-axis. In Figure 9, we compare the numerical results to the analytical expression for both the LLMS model and the standard LLG augmented by Ornstein-Uhlenbeck fields of the type generated by the Langevin equation in Eq. 4. The simulations using the LLMS model agree with the anticipated Boltzmann distribution at equilibrium, while the LLG with Ornstein-Uhlenbeck fails to reproduce the correct distribution. This is because, as we have argued, this does not comprise a correct implementation of the Fluctuation-Dissipation theorem, with deviations corre- sponding to the missing high-frequency components of the damping. 1L. N´ eel, Ann. G´ eophys. C.N.R.S. 5, 99 (1949). 2W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963). 3A. Lyberatos, D.V. Berkov and R.W. Chantrell, J Phys: Condens. Matter 5, 8911 (1993) 4A. Lyberatos and R. W. Chantrell, J. Appl. Phys. 73, 6501 (1993). 5J. L. Garc´ ıa-Palacios and F. J. L´ azaro, Phys. Rev. B 58, 14937 (1998). 6D. V. Berkov, IEEE Trans. Magn. 38, 2489 (2002).7Y. P. Kalmykov, W. T. Coffey, U. Atxitia, O. Chubykalo- Fesenko, P. M. D´ ejardin, and R. W. Chantrell, Phys. Rev. B82, 024412 (2010). 8R. Street and J. C. Woolley, Proc. Phys. Soc. A 62, 562 (1949). 9U Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U Nowak and A. Rebei, Phys. Rev. Lett. 102, 057203 (2009). 10K. Miyazaki and K. Seki, J. Appl. Phys. 112, 121301 (2012).11 11U. Atxitia and O. Chubykalo-Fesenkoo, Phys. Rev. B 84, 144414 .(2011) 12P. H¨ anggi, P. Jung, Adv. Chem. Phys. 89, 239 (1995). 13U. Nowak, Annual Reviews of Computational Physics IX, pg. 105-151 , World Scientific (2001). 14U. Nowak, R. W. Chantrell, and E. C. Kennedy, Phys. Rev. Lett. 84, 163 (2000).15W. T. Coffey and Y. P. Kalmykov, J. Appl. Phys. 112, 121301 (2012). 16R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A. Ellis and R. W. Chantrell, J. Phys.: Condens. Matter26, 103202 (2014). 17I. M. Sokolov, Phys. Rev. E 66, 041101 (2002). 18I. M. Sokolov, Phys. Rev. E 63, 056111 (2001).

2018-05-04

A theoretical investigation of magnetic relaxation processes in single domain particles driven by colored noise is presented. Two approaches are considered; the Landau-Lifshitz-Miyazaki-Seki equation, which is a Langevin dynamics model based on the introduction of an Ornstein-Uhlenbeck correlated noise into the Landau-Lifshitz-Gilbert equation and a Generalized Master Equation approach whereby the ordinary Master Equation is modified through the introduction of an explicit memory kernel. It is found that colored noise is likely to become important for high anisotropy materials where the characteristic system time, in this case the inverse Larmor precession frequency, becomes comparable to the correlation time. When the escape time is much longer than the correlation time, the relaxation profile of the spin has a similar exponential form to the ordinary LLG equation, while for low barrier heights and intermediate damping, for which the correlation time is a sizable fraction of the escape time, an unusual bi-exponential decay is predicted as a characteristic of colored noise. At very high damping and correlation times, the time profile of the spins exhibits a more complicated, noisy trajectory.

Superparamagnetic Relaxation Driven by Colored Noise

1805.01776v2

The General Solution to Vlasov Equation and Linear Landau Damping Deng Zhou Institute of Plasma Physics, Chinese Academy of Sciences Hefei 230031, P . R. China ABSTRACT A general solution to linearized Vlasov equation for an electron electrostatic wave in a homogeneous unmagnetized plasma is derived. The quasi -linear diffusion coefficien t resulting from this solution is a continuous function of 𝜔 at 𝐼𝑚(𝜔)=0 in contrast to that 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. Linear e lectron plasma waves in a collisionless plasma can be obtained by solving the linearized Vlasov equation together with the Poisson equation, which was first treated by Vlasov 1 . The dispersion relation is given by 𝐷(𝜔,𝑘)=1+𝑒2 𝑚𝑘𝜖0∫𝜕𝑓0/𝜕𝑣 𝜔−𝑘𝑣+∞ −∞𝑑𝑣=0 (1) where 𝜔 and 𝑘 are respectively the frequency and the wave number ，other symbols are obvious . 𝐷(𝜔,𝑘) is usually called the plasma dielectric function. From Eq. (1), we can get the 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 𝐷(𝜔,𝑘) is not a continuous function of 𝜔 at 𝐼𝑚(𝜔)=0. To overcome these insufficiencies, Landau solved the Vlasov equation as an initial -value problem and introduc ed the deformed integration route in the plasma dielectric function to make sure that 𝐷(𝜔,𝑘) is an analytical function in the whole complex plane of 𝜔. Landau’s treatment leads to the famous landau damping 2, which was later demonstrated in experiments3. The introduction on the two kinetic treatment s of plasma waves can be found in most plasma textbooks, such as Ref.[4]. An overview by Ryutov summarized the studies of Landau damping by the end of twentieth century5. The Landau damping derived by Landau is a pure achievement of applied mathematics. On it ’s physical explanation different people have different points of view ( see, for example, Drummond6 and references therein ). On a weak nonlinear level, a quasi- linear approach is adopted to describe the interaction between waves and particles7,8. The averaged background particle distribution may experience diffusion in the velocity space due to the wave -particle interaction . Taking the one dimensional electrostatic case as an example, one gets distribution evolution equation 𝜕𝑓0 𝜕𝑡=𝜕 𝜕𝑣𝒟𝜕𝑓0 𝜕𝑣 (2) with the quasi -linear diffusion coefficient 𝒟∝𝛾 (Ω−𝑘𝑣)2+𝛾2 (3) where Ω(𝛾) is the real (imaginary) component of 𝜔. Eq. (3) is derived from the Vlasov’s solution 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, Eq. (3) reduces to 𝒟 ∝𝑠𝑖𝑔𝑛 (𝛾)𝜋𝛿(Ω−𝑘𝑣) (4) where 𝑠𝑖𝑔𝑛 (𝛾) denotes sign of 𝛾. The resonance broadening disappears and a discontinuity appears at 𝛾=0. To overcome this problem, one has to turn to Landau’ s treatment to get the distribution function. It is difficult to get an explicit function of distribution since an inverse Laplace transform is involved. To circumvent this problem9, Hellinger et al recently extended the linear solution to Vlasov equation for the 𝛾<0 case by adding a term involving a complex Dirac delta function. However, these authors didn’t realize that the extended solution is the real solution to Vlasov equation and is related to linear Landau damping, and moreover Dirac delta function is not a we ll-defined function in complex plane. In the following , we derive a general solution to Vlasov equation for the electrostatic Langmuir waves and from this solution the quasi -linear coefficient is derived and shown to be continuous from growing to damped modes. We also show that the general solution is equivalent to the Landau treatment and leads to the famous landau damping. We consider the one dimentional electrostatic oscillation in a homogeneous unmagnetized plasma, the linearized Vlasov equation is 𝜕𝑓1 𝜕𝑡+𝑣𝜕𝑓1 𝜕𝑥=𝑒𝐸1 𝑚𝜕𝑓0 𝜕𝑣 (5) The perturbation of electric field and particle distribution is written in the normal mode form 𝐸 1=𝐸�𝑘𝑒−𝑖(𝜔𝑡−𝑘𝑥) (6) 𝑓 1=𝑓̂𝑘𝑒−𝑖(𝜔𝑡−𝑘𝑥) (7) Eq. ( 6) is then reduced to −𝑖𝜔𝑓̂𝑘+𝑖𝑘𝑣𝑓̂𝑘=𝑒𝐸�𝑘 𝑚𝜕𝑓0 𝜕𝑣 (8) The general form of solution to Eq. (8) is 𝑓̂𝑘=−𝑖𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣�1 𝑣−𝜔𝑘⁄+𝑆𝜔,𝑘(𝑣)𝛥(𝑣,𝜔𝑘⁄)� (9) where 𝑆𝜔,𝑘(𝑣) is any function of 𝜔, 𝑘 and 𝑣, and Δ (𝑣,𝜔𝑘⁄)=𝛿(𝑣−Ω𝑘⁄)+(−𝑖𝛾/𝑘)𝛿′(𝑣−Ω𝑘⁄)+(−𝑖𝛾/𝑘)2 2!𝛿′′(𝑣−Ω𝑘⁄)+⋯ (10) Setting 𝑆𝜔,𝑘(𝑣)=0, we recover the Vlasov treatment and get the dispersion relation Eq. (1). To prove that Eq. (9) is the solution of Eq. (8), one needs to demonstrate (𝑣−𝜔𝑘⁄)Δ(𝑣,𝜔𝑘⁄)=0 (11) This relation is obvious for 𝛾=0. For 𝛾≠0, we have the following integrals for any smooth function 𝐹(𝑣) ∫𝐹(𝑣)+∞ −∞(𝑣−𝜔𝑘⁄)δ(𝑣,Ω𝑘⁄)𝑑𝑣=−(𝑖𝛾/𝑘)𝐹(Ω𝑘⁄) (12a) ∫𝐹(𝑣)+∞ −∞(𝑣−𝜔𝑘⁄)(−𝑖𝛾/𝑘)𝛿′(𝑣−Ω𝑘⁄)𝑑𝑣=(𝑖𝛾/𝑘)𝐹(Ω𝑘⁄)−(𝑖𝛾/𝑘)2𝐹′(Ω𝑘⁄) (12b) ∫1 𝑛!𝐹(𝑣)+∞ −∞(𝑣−𝜔𝑘⁄)(−𝑖𝛾/𝑘)𝑛𝛿(𝑛)(𝑣−Ω𝑘⁄)𝑑𝑣= (𝑖𝛾/𝑘)𝑛 (𝑛−1)!𝐹(𝑛−1)(Ω𝑘⁄)−(𝑖𝛾/𝑘)𝑛+1 𝑛!𝐹(𝑛)(Ω𝑘⁄) (12c) Combining (10) (12a -c), we obtain ∫𝐹(𝑣)+∞ −∞(𝑣−𝜔𝑘⁄)Δ(𝑣,𝜔𝑘⁄)𝑑𝑣=0 (13) Since 𝐹(𝑣) can be any smooth function, then from the basic theorem of function theory, Eq. (11) is verified. The form of Δ (𝑣,𝜔𝑘⁄) is superficially the Taylor expansion of the complex Dirac delta function 𝛿(𝑣−𝜔𝑘⁄). However, it is impossible to define a proper complex Dirac delta function 𝛿(𝑧) which is analytical at 𝑧=0 since we know that a complex function analytical at whole complex plane is only a constant. Now we need to determine 𝑆𝜔,𝑘(𝑣). Without loss of generality, we set 𝑆𝜔,𝑘(𝑣)=0 for 𝛾>0 and derive the value for 𝛾≤0 through the continuity of density perturbation at 𝛾=0. The density perturbation is given by 𝑛�𝑘=−𝑖∫𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣�1 𝑣−𝜔𝑘⁄+𝑆𝜔,𝑘(𝑣)𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞ −∞ (14) The integration route is from −∞ to +∞ in 𝑣-space. In Eq. (14), the principal value of integral is taken if a singularity lies on the integration route. For 𝛾>0, we get from (14) 𝑛�𝑘=𝑖∫𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣1 𝑣−𝜔𝑘⁄𝑑𝑣𝐶𝑅+2𝜋𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣� 𝑣=𝜔𝑘⁄ (15) where 𝐶𝑅 denotes the integration route of the semi -circle on the upper half complex plane from +∞ to −∞, as indi cated in Fig. 1. For 𝛾<0, the density perturbation is given by 𝑛�𝑘=𝑖∫𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣1 𝑣−𝜔𝑘⁄𝑑𝑣𝐶𝑅−𝑖𝑆𝑒𝐸�𝑘 𝑚𝑘�𝜕𝑓0 𝜕𝑣� 𝑣=𝜔 𝑘+(𝑖𝛾)𝜕2𝑓0 𝜕𝑣2� 𝑣=𝜔 𝑘+(𝑖𝛾)2 2!𝜕3𝑓0 𝜕𝑣3� 𝑣=𝜔 𝑘+∙∙∙� (16) One obtains 𝑆=2𝜋𝑖 for 𝛾<0 from (15) and (16) by the requirement of 𝑛�𝑘(𝛾→0+)= 𝑛�𝑘(𝛾→0−). The same procedure yields 𝑆=𝜋𝑖 for 𝛾=0. In summary, we get the general solution to Eq. (8) given by Eq. (9) and (10) with 𝑆 𝜔,𝑘(𝑣)=�0 𝛾 >0 𝜋𝑖 𝛾 =0 2𝜋𝑖 𝛾 <0 (17) In the weak non- linear level the averaged background particle distribution changes slowly due to the wave -particle interaction, which is described through a quasi -linear approach. Keeping to the second order of perturba tion in Vlasov equation and taking a spa ce averag ing in large space scale, one obtains the distribution evolution 𝜕𝑓0 𝜕𝑡=𝑅𝑒〈𝑒𝐸1∗ 𝑚𝜕𝑓1 𝜕𝑣〉 (18) where 𝑅𝑒 ( 𝐼𝑚 ) denotes the real ( imaginary ), bracket denotes sp ace averaging, and the super script star denotes complex conjugate. Inserting (6) (7) and (9) into (18), we get the diffusion equation, Eq. (2), for distribution 𝑓0 with the diffusion coefficient 𝒟 =�𝑒 𝑚�2 �𝐸�𝑘�2𝐼𝑚�1 𝑘𝑣−𝜔+𝑆𝜔,𝑘(𝑣)𝛥(𝑣,𝜔𝑘⁄) 𝑘� = �𝑒 𝑚�2 �𝐸�𝑘�2�𝛾 (Ω−𝑘𝑣)2+𝛾2+𝑆 𝑖𝑘�𝛿(𝑣−Ω𝑘⁄)−�𝛾 𝑘�2 𝛿′′(𝑣−Ω𝑘⁄)+⋯�� (19) For simplicity we have kept only one wave number while the real diffusion is contributed from all the excited modes. It is obvious that the diffusion coefficient given by (19) is continuous as 𝛾→0±, which is 𝒟 (𝛾→0+)=𝒟(𝛾→0−)∝𝜋𝛿(𝑘𝑣−Ω) (20) The general solution, Eq. (9), can lead to linear Landau damping. Substituting the perturbation of the electric field and the distribution into the one dimensional Poisson equation 𝜖0𝜕𝐸1 𝜕𝑥=−𝑒∫𝑓1+∞ −∞𝑑𝑣 (21) we obtain the dispersion relation 1−𝜔𝑝𝑒2 𝑛0𝑘2∫𝜕𝑓0 𝜕𝑣�1 𝑣−𝜔𝑘⁄+𝑆𝜔,𝑘(𝑣)𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞ −∞ (22) where 𝜔𝑝𝑒=�𝑛0𝑒2 𝜖0𝑚 is the usual Langmuir frequency and 𝑛0 is the background plasma density. Substituting (10) in (22) yields 1−𝜔𝑝𝑒2 𝑛0𝑘2∫𝜕𝑓0 𝜕𝑣1 𝑣−𝜔𝑘⁄𝑑𝑣+∞ −∞−𝑆𝜔𝑝𝑒2 𝑛0𝑘2𝜕𝑓0 𝜕𝑣� 𝑣=𝜔 𝑘=0 (23) This is exactly the plasma dielectric function one obtains from Landau’s treatment by adopting a modified Landau integra tion contour in complex 𝑣-plane. Hence, the general solution 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. In a recent work 10, Wesson reexamined the problem of Landau damping. He solved Vlasov equation separately for 𝛾>0 and 𝛾<0 using real variables . The two cases are related through the continuity of density perturbation at 𝛾→0. Our present treatment is equivalent to Wesson’s since we also solve Vlasov equation separately for 𝛾>0 and 𝛾<0 and also use the continuity condition to rel ate the two cases. To show this, we take the case 𝛾<0 as an example. Like Wesson’s treatment, the density perturbation is separated into two parts, 𝑛1𝑏 and 𝑛1𝑤. 𝑛1𝑏 is the contribution from the basic thermal distribution, which is treated as a Dirac delta function, i. e. 𝑓0=𝑛0𝛿(𝑣). Hence 𝑛1𝑏=−𝑖�𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣1 𝑣−𝜔𝑘⁄𝑑𝑣+∞ −∞ = −𝑖∫𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣𝑣−Ω𝑘⁄+𝑖γ𝑘⁄ (𝑣−Ω𝑘⁄)2+(γ𝑘⁄)2𝑑𝑣+∞ −∞ = −𝑖𝑒𝐸�𝑘 𝑚𝑘�∫𝜕𝑓0 𝜕𝑣1 𝑣−Ω𝑘⁄𝑑𝑣+∞ −∞+∫𝜕𝑓0 𝜕𝑣𝑖γ𝑘⁄ (𝑣−Ω𝑘⁄)2𝑑𝑣+∞ −∞� =−𝑖𝑒𝐸�𝑘 𝑚𝑘�𝑛0 (Ω𝑘⁄)2−2𝑖𝑛0γ𝑘⁄ (Ω𝑘⁄)3� (24) where we have neglected the term (γ𝑘⁄)2 in the denominate of the second line since 𝛾≪Ω. 𝑛1𝑤 is contribut ed from the particles in resonance with the wav e. Taking 𝜕𝑓0 𝜕𝑣 to be a constant, we have 𝑛1𝑤=−𝑖�𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣�1 𝑣−𝜔𝑘⁄+𝑆𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞ −∞ = −𝑖𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣� 𝑣=Ω 𝑘∫�𝑣−Ω𝑘⁄+𝑖γ𝑘⁄ (𝑣−Ω𝑘⁄)2+(γ𝑘⁄)2+𝑆𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞ −∞ (25) Changing to the coordinate system travelling with the wave velocity, one obtains 𝑛1𝑤=−𝑖𝑒𝐸�𝑘 𝑚𝑘𝜕𝑓0 𝜕𝑣� 𝑣=𝜔 𝑘∫�𝑣+𝑖γ𝑘⁄ 𝑣2+(γ𝑘⁄)2+𝑆𝛥(𝑣,𝜔𝑘⁄)�𝑑𝑣+∞ −∞ = −𝑖𝑒𝐸�𝑘 𝑚𝑘�𝑖𝜋𝜕𝑓0 𝜕𝑣� 𝑣=𝜔 𝑘−𝛾2𝜋𝜕2𝑓0 𝜕𝑣2� 𝑣=𝜔 𝑘+⋯� (26) Substitution of (24) (26) into the Poisson equation (21) yields respectively from the real and the imaginary parts 𝛾 =𝜋Ω3 2𝑛0𝑘2𝜕𝑓0 𝜕𝑣� 𝑣=𝜔 𝑘 (27) and Ω =𝜔𝑝𝑒=�𝑛0𝑒2 𝜖0𝑚 (28) They are exactly the same as those given by Wesson. In summary, we have presented the general form of solution to Vlasov equation for the electrostatic plasma wave , shown by Eqs. (9) (10) and (17). From this solution, one can get the quasi -linear diffusion coefficients valid for both da mping and growing modes. The solution can also lead to the well -known Landau damping and it is equivalent to Wesson ’s recent explanation of Landau damping using real variables. ACKNOWLEDGEMENT This work is supported by National Natural Science Foundation of China under Grant No. 11175213. 1A. Vlasov, J. Phys. (USSR) 9, 25 (1945). 2L. D. Landau, J. Phys. (USSR) 10, 25 (1946). 3J. Malmberg and C. Wharton, Phys. Rev. Lett. 17, 175 (1966). 4R. Goldston and P . Rutherford, Intro duction to Plasma Physics (Institute of Physics Publishing, 1995). 5D. Ryutov, Plasma phys. Contr. Fusion 41, A1 (1999) . 6W. Drummond, Phys. Plasmas 11, 552 (2004). 7A. Vedenov, E. Velikhov, and R. Sagdeev, Nucl. Fusion 1, 82 (1961). 8W. Drummond and D. Pines, Nucl. Fusion Suppl. 3, 1049 (1962). 9P . Hellinger and P . Travnicek, Phys. Plasmas 19, 062307 (2012). 10J. Wesson, Phys. Plasmas 22, 0 22519 (2015 ). Fig. 1 The integration route used in calculation of density perturbation.

2015-10-14

A general solution to linearized Vlasov equation for an electron electrostatic wave in a homogeneous unmagnetized plasma is derived. The quasi-linear diffusion coefficient resulting from this solution is a continuous function of omega in contrast to that derived from the traditional Vlasov treatment. The general solution is also equivalent to the Landau treatment of the plasma normal oscillations, and hence leads to the well-known Landau damping.

The General Solution to Vlasov Equation and Linear Landau Damping

1510.03949v1

1 Conductivity -Like Gilbert Damping due to Intraband Scattering in Epitaxial Iron Behrouz Khodadadi1, Anish Rai2,3, Arjun Sapkota2,3, Abhishek Srivastava2,3, Bhuwan Nepal2,3, Youngmin Lim1, David A. Smith1, Claudia Mewes2,3, Sujan Budhathoki2,3, Adam J. Hauser2,3, Min Gao4, Jie-Fang Li4, Dwight D. Viehland4, Zijian Jiang1, Jean J. Heremans1, Prasanna V. Balachandran5,6, Tim Mewes2,3, Satoru Emori1* 1 Department of Physics, Virginia Tech , VA 24061, U.S.A 2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA 3 Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa, AL 35487, U .S.A. 4 Department of Material Science and Engineering, Virginia Tech , Blacksburg, VA 24061, U.S.A . 5 Department of Material Science and Engineering, University of Virginia, Charlottesville, VA 22904, U.S.A . 6 Department of Mechanical and Aerospace Engineering , Univer sity of Virginia, Charlottesville, VA 22904, U.S.A. *email: semori@vt.edu Confirming the or igin of Gilbert damping by experiment has remained a challenge for many decades , even for simple ferromagnetic metals . In this Letter, we experimentally identify Gilbert damping that increases with decreasing electronic scattering in epitaxial thin films of pure Fe . This observation of conductivity -like damping, which cannot be accounted for by classical eddy current loss , is in excellent quantitative agreement with theoretical predictions of Gilbert damping due to intraband scatte ring. Our results resolve 2 the longstanding question about a fundamental damping mechanism and offer hints for engineering low -loss magnetic metals for cryogenic spintronic s and quantum devices. Damping determines how fast the magnetization relaxes towards the effective magnetic field and plays a central role in many aspects of magnetization dynamics [1,2] . The magnitude of viscous Gilbert damping governs the threshold current for spin -torque magnetic switching and auto-oscillations [3,4] , mobility of magnetic domain walls [5,6] , and decay leng ths of diffusive spin waves and superfluid -like spin current s [7,8] . To enable spintronic technologies with low power dissipation , there is currently much interest in minimizing Gilbert damping in thin films of magnetic m aterials [9–13], especially ferromagnetic metals [14–23] that are compatible with conventional device fabrication schemes . Despite the fundamental and technological importance of Gilbert damping, its physical mechanisms in various magnetic materials have yet to be confirmed by experiment . Gilbert damping is generally attributed to spin-orbit coupling that ultimately dissipates the energy of the magnetic system to the lattice [1,2] . Kambersky’s torque correlation model [24] qualitatively captures the temperature dependence of damping in some experiments [25–28] by partitioning Gilbert damping into two mechanisms due to spin -orbit coupling, namely interband and intraband scattering mechanisms, each with a distinct dependence on the elect ronic momentum scattering tim e e. For the interband scattering mechanism where magnetization dynamics can excite electron -hole pairs across dif ferent bands, the resulting Gilbert damping is “resistivity -like” as its magnitude scales with e-1, i.e., increased electronic scattering results in higher damping [29,30] . By contrast, the intraband scattering mechanism is typically understood through the breathing Fermi surface mode l [31], where electron -hole pairs are excited in the 3 same band , yielding “conductivity -like” Gilbert damping that scales with e, i.e., reduced electronic scattering results in higher damping. Conductivity -like Gilbert damping was reported experimentally more than 40 years ago in bulk crystals of pure Ni and Co at low temperatures , but surprisingly not in pure Fe [25]. The apparent absence of co nductivity -like damping in Fe has been at odds with many theoretical predictions that intraband scattering should dominate at low temperatures [32–38], although some theoretical studies have suggested that intraband scattering may be absent alt ogether in pure metals [39,40] . To date, no experimental work has conclusively addressed the role of intraband scattering in pure Fe1. There thus remains a significant gap in the fundamental understanding of damping in one of the simplest ferromagnetic metals. Intrinsic conductivity - like Gilbert damping in Fe is also technologically relevant, since minimizing damping in ferromagnetic metals at low temperatures is crucial for cryogenic superconducting spintronic memories [41,42] and quantum information transduction schemes [43,44] . In this Letter, we experimentally demonstrate the presence of conductivity -like Gilbert damping due to intr aband scattering in epitaxial thin films of body -centered -cubic (BCC) Fe. By combining broadband ferromagnetic resonance (FMR) measurements with characterization of structural and transport properties of these model -system thin films, we show that conductivity - like Gilbert damping dominates at lo w temperatures in epitaxial Fe . These experimental results 1 Ref. [36] includes experimental data that suggest the presence of conductivity -like Gilbert damping in an ultrathin Fe film, although no detailed information is given about the sample and t he experimental results deviate considerably from the calculations. An earlier study by Rudd et al. also suggests an increase in Gilbert damping with decreasing temperature [27], but quantification of the Gilbert damping parameter in this experiment is difficult. 4 agree remarkably well with the magnitude of Gilbert damping derived from first -principles calculations [32,33,36] , thereby providing evidence for intraband scatterin g as a key mechanism for Gilbert damping in pure BCC Fe. Our experiment thus resolves the longstanding question regarding the origin of damping in the prototypical ferromagnetic metal . Our results also confirm that – somewhat counterintuitively – disorder can partially suppress intrinsic damping at low temperatures in ferromagnetic metals, such that optimally disordered films may be well suited for cryogenic spintronic and quantum applications [41–44]. Epitaxial BCC Fe thin films were sputter deposited on (001) -oriented MgAl 2O4 (MAO) and MgO single crystal substrates. The choices of substrates were inspired by the recent experiment by Lee et al. [20], where epitaxial growth is enabled with t he [100] axis of a B CC Fe-rich alloy oriented 45o with respect to the [100] axis of MAO or MgO. MAO with a lattice parameter of a MAO /(2√2) = 0.2858 nm exhibits a lattice mismatch of less than 0.4% with Fe (a Fe ≈ 0.287 nm) , whereas the lattice mismatch between MgO ( aMgO/√2 = 0.2978 nm) and Fe is of the order 4%. Here , we focus on 25 -nm-thick Fe films that were grown simultaneously on MAO and MgO by confocal DC magnetron sputtering [45]. In the Supplemental Material [45], we report on additional films depos ited by off -axis magnetron sputtering. We verified the crystalline quality of the epitaxial Fe films by X -ray diffraction, as s hown in Fig. 1( a-c). Only (00X )-type peaks of the substrate and film are found in each 2θ-ω scan, consistent with the single -phase epitaxial growth of the Fe films. The 2θ-ω scans reveal a larger amplitude of film peak for MAO/Fe, suggesting higher crystalline quality than that of MgO/Fe. Pronounced Laue oscillations, indicative of atomically smooth film interfaces, are o bserved around the film peak of MAO/Fe, whereas they are absent for MgO/Fe. The high crystalline quality of MAO/Fe is also evidenced by its narrow film -peak rocking curve with a FWHM of 5 only 0.02o, comparable to the rocking curve F WHM of the substrate2. By contrast, the film -peak rocking curve of MgO/Fe has a FWHM of 1o, which indicates substantial mosaic spread in the film due to the large lattice mismatch with the MgO substrate. Results of 2θ -ω scans for different film thicknesses [45] suggest that the 25 -nm-thick Fe film may be coherently strained to the MAO substrate , consistent with the smooth interfaces and minimal mosaic spread of MAO/Fe . By contrast, i t is likely that 25 -nm-thick Fe on MgO is relaxed to accommodate the large film-substrate lattice mismatch. Static magnetometry provides further evidence that Fe is strained on MAO a nd relaxed on MgO [45]. Since strained MAO/Fe and relaxed MgO/Fe exhibit distinct crystalline quality, as evidenced by an approximately 50 times narrower rocking FWHM for MAO/Fe , we have two model systems that enable experimental investigation of the impact of structural disorder on Gilbert damping. The residual electrical resistivity also reflects the structural quality of metal s. As shown in Fig. 1(d ), the residual resistivity is 20 % lower for MAO/Fe compared to MgO/Fe, which corroborates the lower defect density in MAO/Fe. The resistivity increases by nearly an order of magnitude with increasing temperature, reaching 1.1×10-7 m for both samples at room temperature , consistent with behavior expected for pure metal thin film s. We now examine how the difference in crystalline quality correlates with magnetic damping in MAO/Fe and MgO /Fe. Broadband FMR measurements were performed at room temperature up to 65 GHz with a custom spectrometer that empl oys a coplanar waveguide (center conductor width 0.4 mm ) and an electromagnet (maximum field < 2 T) . For each measurement at a fixed excitat ion frequency, an external bias magnetic field was swept parallel to the film plane along the [110] axis of Fe , unless otherwise noted. I n the Supplemental 2 The angular resolu tion of the diffractometer is 0.0068o. 6 Material [45], we show similar results with the field applied along the [110] and [100] axes of Fe; Gilbert damping is essentially isotropic within the film plane for our epitaxial Fe films , in contrast to a recent report of anisotropic damping in ultrathin epitaxial Fe [22]. Figure 2 shows that the peak -to-peak FMR linewidth Hpp scales linearly with frequency f, enabling a precise determination of the measured Gilbert damping parameter 𝛼𝑚𝑒𝑎𝑠 from the standard equation, 𝜇0∆𝐻𝑝𝑝=𝜇0∆𝐻0+2 √3𝛼𝑚𝑒𝑎𝑠 𝛾′𝑓, (1) where Hpp,0 is the zero -frequency linewidth and 𝛾′=𝛾/2𝜋≈29.5 GHz/T is the reduced gyromagnetic ratio . Despite the difference in crystalline quality , we find essentially the same measured Gilbert damp ing parameter of 𝛼𝑚𝑒𝑎𝑠 ≈ 2.3×10-3 for MAO/Fe and MgO/Fe. We note that t his value of 𝛼𝑚𝑒𝑎𝑠 is comparable to the lowest damping parameters reported for epitaxial Fe at room temperature [15–17]. Our results indicate that Gilbert damping at room temperature is insensitive to the strain state or structural disorder in epitaxial Fe.3 The measured damping parameter 𝛼𝑚𝑒𝑎𝑠 from in-plane FMR can generally include a contribution from non-Gilbert relaxation , namely two -magnon scattering driven by defects [46– 49]. However, two-magnon scattering is suppressed when the film is magnetized out-of- plane [19,48] . To isolate any two -magnon scattering contribution to d amping, we performed out- of-plane FMR measurements under a sufficiently large magnetic field (>4 T) for complete saturation of the Fe film, using a custom W-band shorted waveguide combined with a 3 However, the crystallographic texture of Fe has significant impact on damping; for example, non -epitaxial Fe films deposited directly on amorphous SiO 2 substrates exhibit an order of magnitude wider linewidths, due to much more pronounced non -Gilbert damping (e.g., two -magnon scattering), compared to (001) -oriented epitaxial Fe films. 7 superconducting magnet. As shown in Fig. 2, the out -of-plane and in -plane FMR data yield the same slope and hence 𝛼𝑚𝑒𝑎𝑠 (Eq. 1) to within < 8%. This finding indicates that two -magnon scattering is negligible and that frequency -dependent magnetic relaxation is dominated by Gilbert damping in epitaxial Fe examined here. The insensitivity of Gilbert damping to disorder found in Fig. 2 can be explained by the dominance of the interband (resistivity -like) mechanis m at room temperature, with phonon scattering dominating over defect scattering. Indeed, since MAO/Fe and MgO/Fe have the same room -temperature resistivity (Fig. 1(d )), any contributions to Gilbert damping from electronic scattering should be identical for both samples at room temperature. Moreover, according to our density functional theory calculations [45], the density of states of BCC Fe at the Fermi energy, D(EF), does not depend significantly on the strain state of the crystal. Therefore, i n light of the recent reports that Gilbert damping is proportional to D(EF) [18,50,51] , the different strain states of MAO/Fe and MgO/Fe are not expe cted to cause a significant difference in Gilbert damping. However, since MAO/Fe and MgO/Fe exhibit distinct resistivities (electronic scattering times e) at low temperatures, one might expect to observe distinct temperature dependence in Gilbert damping for these two samples. To this end, we performed variable -temperature FMR measurements using a coplanar -waveguide -based spectrometer (maximum frequency 40 GHz, field < 2 T) equipped with a clos ed-cycle cryostat4. Figure 3(a,b) shows that meas is enhanced for both samples at lower temperatures. Notably, this damping enhancement with decreasing temperature is significantl y greater for MAO/Fe . Thus, at low temperatures, we find a 4 The W -band spectrometer for out -of-plane FMR (Fig. 2) could not be cooled below room temperature due to its large thermal mass , limiting us to in -plane FMR measurements at low temperatures. 8 conductivity -like damping increase that is evidently more pronounced in epitaxial Fe with less structural disorder. While this increased damping at low temperatures is reminiscent of intrinsic Gilbert damping from intraband scattering [31–38], we first consider other possible contributions. One possibility is two -magnon scattering [46–49], which we have ruled out at room temperature (Fig. 2) but could be present in our low -temperature in-plane FMR measurements . From Fig. 3(a,b), the zero -frequency linewidth H0 (Eq. 1 ) – typically attributed to magnetic inhomogeneity – is shown to increase along with meas at low temperatures [45], which might point to the emergence of two -magnon scattering [48,49] . However, our mean -field model calculations (see Supplemental Material [45]) shows that H0 correlates with meas due to interactions among different regions of the inhomogeneous film [52]. The increase of H0 at low temperatures is therefore readily accounted for by increased Gilbert damping , rather than two -magnon scattering . We are also not aware of any mechanism that enhance s two-magnon scattering with decreasing temperature, particularly given that the saturation magnetization (i.e., dipolar interactions) is constant across the measured temperature range [45]. Moreover, the isotropic in - plane damping found in our study is inconsistent with typically anisotropic two-magnon scattering tied to the crystal symmetry of epitaxial films [46,47] , and the film thickness in our study (e.g., 25 nm) rules out t wo-magnon scattering of interfacial origin [49]. As such, we conclude that two -magnon scattering does not play any essential role in our experimental observations. Another possible contribution is dissipation due to classical eddy current s, which increase s proportionally with the increasing conductivity 𝜎 at lower temperatures . We estimate the eddy current contribution to the measured Gilbert damping with [15,53] 9 𝛼𝑒𝑑𝑑𝑦 =𝜎 12𝛾𝜇02𝑀𝑠𝑡𝐹2, (2) where 𝜇0𝑀𝑠≈2.0 T is the saturation magnetization and tF is the film thickness . We find that eddy curr ent damping accounts for only ≈20% (≈ 30%) of the total measured damping of MAO/Fe (MgO/Fe) even at the lowest measured temperature (Fig. 3(c)) . Furthermore, a s shown in the Supplemental Material [45], thinner MAO/Fe film s, e.g., tF = 11 nm , with negligible eddy still exhibit a significant increase in damping with decreasing temperature. Our results thus indicate a substantial contribution to conductivity -like Gilbert damping that is not accounted for by classica l eddy current damping. For further discussion , we subtract the eddy -current damping from the measured damping to denote the Gilbert damping parameter attributed to intrinsic spin-orbit coupling as 𝛼𝑠𝑜= 𝛼𝑚𝑒𝑎𝑠 − 𝛼𝑒𝑑𝑑𝑦. To correlate electronic transport and magnetic damping across the entire meas ured temperature range, we perform a phenomenological fit of the temperature dependence of Gilbert damping with [26] 𝛼𝑠𝑜=𝑐𝜎(𝑇) 𝜎(300 𝐾)+𝑑𝜌(𝑇) 𝜌(300 𝐾), (3) where the conductivity -like (intraband) and resistivity -like (interband) terms are scaled by adjustable parameters c and d, respectively. As shown in Fig. 4(a),(b), t his simple phenomenological model using the experimental transport results (Fig. 1(d)) agrees remarkably well with the temperature dependence of Gilbert damping for both MAO/Fe and MgO/Fe. Our fi nding s that Gilbert damping can be phenomenologically partit ioned into two distinct contributions (Eq. 3 ) are in line with Kambersky’s torque correlation model . We compare our experimental resul ts to first-principles calculations by Gilmore et al. [32,33] that relate electronic momentum scattering rate e-1 and Gilbert damping through Kambersky’s torque correlation model. We use the experimentally measured resistivity ρ (Fig. 1(d)) to convert the 10 temperature to e-1 by assuming the constant conversion factor ρ e = 1.30×10-21 m s [33]. To account for the difference in electronic scattering time for the minority spin and majority spin , we take the calculated curve from Gilmore et al. with / = 4 [33], which is close to the ratio of D(EF) of the spin-split bands for BCC Fe , e.g., derived from our density functional theory calculations [45]. For explicit comparison with Refs. [32,33] , the Gilbert damping parameter in Fig. 4(c) is converted to the magnetic relaxation rate 𝜆= 𝛾𝛼𝑠𝑜𝜇0𝑀𝑠. The calculated prediction is in excellent quantitative agreement with our experimental results for both strained MAO/Fe and relaxed MgO/Fe (Fig. 4(c)) , providing additional experimental evidence that intraband scattering predominately contribute s to Gilber t damping at low temperatures. We also compare our experimental results to a more recent first -principles calculation study by Mankovsky et al., which utilizes the linear response formalism [36]. This approach does not rely on a phenomenological electronic scattering rate and instead allows for explicitly incorporating thermal effects and structural disorder . Figure 4(d ) shows the calculated temperature dependence of the Gilbert damping parameter for BCC Fe with a small density of defects, i.e., 0.1% vacancies , adapted from Ref. [36]. We again find good quantitative agreement between the ca lculations and our experimental results for MAO/Fe. On the other hand, the Gilbert damping parameter s at low temperatures for relaxed MgO/Fe are significantly below the calculated values . This is consistent with the reduction of intraband scattering due to enhanced electronic scattering (enhanced e-1) from defects in relaxed MgO/Fe . Indeed, significant defect -mediated electronic scattering may explain the absence of conductivity -like Gilbert damping for crystalline Fe in prior experiments. For example, Ref. [25] reports an upper limit of only a two -fold increase of the estimated Gilbert damping parameter from T = 300 K to 4 K . This relatively small damping enhancement is similar to that for MgO/Fe 11 in our study (Fig. 4(b)) , suggesting that intraband scattering may have been suppressed in Fe in Ref. [25] due to a similar degree of structural disorder to MgO/Fe. We therefore conclude that conductivity -like Gilbert damping from intraband scattering is highly sensitive to disorder in ferromagnetic metals. More generally , the presence of defects in all real metals – evidenced by finite residual resistivity – ensures that the Gilbert damping parameter is finite even in the zero -temperature limit . This circumvents the theoretical deficiency of Kambersky’s torque correlation model where Gilbert damping would diverge in a perfectly clean ferromagnetic metal at T 0 [39,40] . We also remark that a fully quantum mechanical many -body theory of magnetization dynami cs yields finite Gilbert damping even in the clean, T = 0 limit [54]. In summary, we have demonstrated the dominance of conductiv ity-like Gilbert damping due to intraband scattering at low temperatures in high-quality epitaxial Fe . Our experimental results also validate the longstanding theoretical prediction of intraband scattering as an essential mechanism for Gilbert damping in pure ferromagnetic metals [32–38], thereby advancing the fundamental understanding of magnetic relaxation in real materials . Moreover, we have confirmed that, at low temperatures, a ma gnetic metal with imperfect crystallinity can exhibit lower Gilbert damping (sp in decoherence) than its cleaner counterpart. 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Lock, "Eddy current damping in thin metallic ferromagnetic films," Br. J. Appl. Phys. 17, 1645 –1647 (1966). 54. F. Mahfouzi, J. Kim, and N. Kioussis, "Intrinsic damping phenomena from quantum to classical magnets: An ab initio study of Gilbert damping in a Pt/Co bilayer," Phys. Rev. B 96, 214421 (2017). 19 Figure 1. (a,b) 2θ -ω X-ray diffraction scans of MAO/Fe and MgO/Fe (a) over a wide angle range and (b) near the BCC Fe (002) film peak. (c) Rocking curve scans about the film peak. (d) Temperature dependence of resistivity plotted on a log -log scale. Figure 2. Frequency dependence of FMR linewidth Hpp for MAO/Fe and MgO/Fe at room temperature. Linewidths measured under in -plane field are shown as open symbols, whereas those measured under out -of-plane (OP) field are shown as filled symbols . 62 64 66 68log(intensity) [a.u.] 2q [deg.] 30 40 50 60 70log(intensity) [a.u.] 2q [deg.] -1.0 -0.5 0.0 0.5 1.0intensity [a.u.] w002 [deg.] 10 10010-810-7r [ m] T [K] MgO/Fe MAO/Fe MAO (004)MgO (002) Fe (002)MAO/Fe MgO /Fe ( 4)(a) (b) (c) (d)MAO/Fe MgO /Fe (a) 0 20 40 60 80 100 120024681012 MAO/Fe (OP) MgO/Fe MAO/Fem0Hpp [mT] f [GHz]20 Figure 3. (a,b) Frequency dependence of FMR linewidth for MA O/Fe and MgO/Fe at (a) T = 100 K and (b) T = 10 K. (c) Temperature dependence of measured Gilbert damping parameter meas and estimated eddy -current damping parameter eddy. 0 50 100 150 200 250 3000246810 meas MAO/Fe eddy estimate meas MgO/Fe eddy estimatemeas, eddy [10-3] T [K] 0 10 20 30 40051015 MAO/Fe MgO/Fem0Hpp (mT) f (GHz)T = 100 K 0 10 20 30 40051015m0Hpp (mT) f (GHz)T = 10 K(c)(a) (b)21 Figure 4. (a,b) Temperature dependence of the spin-orbit -induced Gilbert damping parameter so, fit phenomenologically with the experimentally measured resistivity for (a) MAO/Fe and (b) MgO/Fe. The dashed and dotted curves indicate the conductivity -like and resistivity -like contributions, respectively; the solid curve represents the fit curve for the total spin -orbit -induced Gilbert damping parameter. (c,d) Comparison of our experimental results with calculated Gilbert damping parameters by (c) Gilmore et al. [32,33] and (d) Mankovsky et al. [36]. 0 100 200 30002468 r-liker-likeso [10-3] T [K]s-likeMAO/Fe 0 100 200 30002468 MgO/Fe r-likes-likeso [10-3] T [K](a) (b) 0 100 200 30002468 MAO/Fe MgO/Fe calculated [Mankovsky]so [10-3] T [K] 0 50 1000123 r-like MAO/Fe MgO/Fe calculated [Gilmore]l [109 s-1] e-1 [1012 s-1]s-like0.0 0.5 1.0 02468 so [10-3]r [10-7 m] (c) (d)

2019-06-25

Confirming the origin of Gilbert damping by experiment has remained a challenge for many decades, even for simple ferromagnetic metals. In this Letter, we experimentally identify Gilbert damping that increases with decreasing electronic scattering in epitaxial thin films of pure Fe. This observation of conductivity-like damping, which cannot be accounted for by classical eddy current loss, is in excellent quantitative agreement with theoretical predictions of Gilbert damping due to intraband scattering. Our results resolve the longstanding question about a fundamental damping mechanism and offer hints for engineering low-loss magnetic metals for cryogenic spintronics and quantum devices.

Conductivity-Like Gilbert Damping due to Intraband Scattering in Epitaxial Iron

1906.10326v2

Anisotropic superconducting spin transport at magnetic interfaces Yuya Ominato1, Ai Yamakage2, and Mamoru Matsuo1;3;4;5 1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 2Department of Physics, Nagoya University, Nagoya 464-8602, Japan 3CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China 4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan and 5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: October 18, 2022) We present a theoretical investigation of anisotropic superconducting spin transport at a magnetic interface between a p-wave superconductor and a ferromagnetic insulator. Our formulation describes the ferromagnetic resonance modulations due to spin current generation depending on spin-triplet Cooper pair, including the frequency shift and enhanced Gilbert damping, in a unied manner. We nd that the Cooper pair symmetry is detectable from the qualitative behavior of the ferromagnetic resonance modulation. Our theory paves the way toward anisotropic superconducting spintronics. Introduction.| Use of spin-triplet Cooper pairs as car- riers for spin currents in the emergent eld of super- conducting spintronics is challenging1,2. Previous stud- ies have demonstrated spin transport mediated by spin- triplet Cooper pairs that formed at the s-wave supercon- ductor (SC)/ferromagnet interfaces of Josephson junc- tions. The spin-singlet pairs in SCs are converted into spin-triplet pairs in half-metallic CrO 23. However, pre- vious studies on spin-triplet pairs at magnetic interfaces have been limited to cases induced by the proximity ef- fect. One promising candidate material system for investi- gation of spin-triplet currents to enable more active use of spin-triplet pairs is the p-wave SC/ferromagnetic in- sulator (FI) bilayer thin lm system4,5. Tunneling of the spins is driven by the magnetization dynamics excited by ferromagnetic resonance (FMR) in the ferromagnetic material via interfacial exchange coupling between the magnetization in the FI and the electron spins in the p-wave SC, and a spin-triplet current is expected to be generated. Furthermore, as a backaction of spin injec- tion, both the FMR frequency and the Gilbert damping of the FI should be modulated6{8. Although similar sce- narios have already been studied vigorously in s-wave SC/ferromagnet systems, most previous studies have fo- cused on the Gilbert damping modulation due to spin injection9{22. To gain an in-depth understanding of the spin-triplet transport mechanisms, the FMR modulation processes, including both the frequency shift and the en- hanced Gilbert damping, should be formulated micro- scopically in a systematic manner. Determination of the pairing symmetry of the spin- tripletp-wave SCs within the same framework is also desirable. Despite many years of research based on sev- eral experimental techniques that detect the pairing sym- metry, including nuclear magnetic resonance23, polar- ized neutron scattering24{26, and muon-spin resonance techniques27, there are few established candidate systems for spin-triplet SCs28{32. The FMR modulation has been observed in various nanoscale magnetic multilayers. Ac- cordingly, the technique is widely used to investigate a (c) FMR modulation due to the coupling between spin-triplet Cooper pair and magnetization H H0H0+ δH (b) Spin-triplet Cooper pair (i) Chiral p-wave (ii) Helical p-wave α+ δα H H0αH0H0+ δHFISC FIxz Y, yθ(a) System θZ XS-HFISC FIG. 1. Mechanism of FMR modulation due to anisotropic superconducting spin transport at magnetic interfaces. (a) Precession axis located on the x-zplane, where the angle between the precession axis and the zaxis is(where 0 =2). (b) Two types of spin-triplet Cooper pairs considered in this work. (c) FMR signal modulation in the SC/FI bilayer system compared with the signal in the FI monolayer. spin transport property in a variety of nanoscale thin lm systems because it is highly sensitive. Thus one can expect that the FMR measurements in p-wave SC/FI bi- layer systems provide useful information about pairing symmetry. In this Letter, we investigate anisotropic superconduct- ing spin transport at the magnetic interfaces of hybrid systems composed of p-wave SC/FI thin lms theoret- ically, as illustrated in Fig. 1(a). The two-dimensional bulk SC is placed on the FI, where the FMR occurs. The precession axis is rotated by an angle from the direc- tion perpendicular to the interface. Here, we use two coordinate systems: ( x;y;z ) and (X;Y;Z ). Thezaxis is perpendicular to the interface and the xandyaxes arearXiv:2103.05871v3 [cond-mat.supr-con] 15 Oct 20222 along the interface. The ( X;Y;Z ) coordinate is obtained by rotating the angle around the yaxis, so that the precession axis and the Zaxis are parallel. Figure 1(b) shows a schematic image of the spin-triplet Cooper pairs for the chiral and helical p-wave SCs considered in this work. Figure 1(c) shows a schematic image of the FMR signal in the FI monolayer and the SC/FI bilayer. The FMR frequency and linewidth in the SC/FI bilayer are both modulated because of the spin transfer occurring at the interface. Using the nonequilibrium Green's function method, we formulate the FMR modulations due to the back action of the spin-triplet transport process systemati- cally. The main advantage of using the nonequilibrium Green's function is dealing with both a spectral function and a nonequilibrium distribution function. Indeed, the interface spin current is given by the expression using the nonequilibrium distribution function, which shows that the interface spin current by the spin pumping and the enhanced Gilbert damping are proportional to each other. Furthermore, as an advantage of eld theoretical treatment, the frequency shift and the enhanced Gilbert damping are both described in a unied manner. Addi- tionally, it is shown that the symmetry of the spin-triplet pairs can be extracted from the FMR modulations. The results presented here oer a pathway toward develop- ment of anisotropic superconducting spintronics. Model Hamiltonian.| The FMR modulation due to the SC adjacent to the FI is calculated microscopically using the spin tunneling Hamiltonian method9{11,33{38. The eect of the SC on the FI is treated as a perturbation and suppression of ferromagnetism with the onset of su- perconductivity is assumed to be negligible, which is con- sistent with the results of spin pumping experiments in magnetic multilayer thin lms. The details of the model Hamiltonians and the formulations are described in the Supplemental Material39. In the main text, we focus on giving an overview of the model Hamiltonians and the formulations. The total Hamiltonian H(t) comprises three terms H(t) =HFI(t) +HSC+Hex: (1) The rst term HFI(t) describes the bulk FI, HFI(t) =X k~!kby kbkh+ ac(t)by k=0h ac(t)bk=0;(2) whereby kandbkdenote the creation and annihilation op- erators of magnons with the wave vector k= (kx;ky;kz), respectively. We assume the parabolic dispersion ~!k= Dk2~ H, where (<0) is the electron gyromagnetic ra- tio. The coupling between the microwave radiation and the magnons is given by h ac(t) = ~ hacp SN=2ei!t, wherehacand!are the amplitude and the frequency of the microwave radiation, respectively. Sis the magni- tude of the localized spin and Nis the number of sites in the FI. Note that the precession axis for the localized spin is xed along the Zaxis [see Fig. 1(a)].The second term HSCdescribes the two-dimensional bulk SCs, HSC=1 2X kcy kHBdGck; (3) where we use the four-component notations cy k= (cy k";cy k#;ck";ck#); (4) ck= (ck";ck#;cy k";cy k#)T: (5) Here,cy ksandcksdenote creation and annihilation op- erators, respectively, of electrons with the wave vector k= (kx;ky) and thezcomponent of the spin s=";#. The Bogoliubov-de Gennes Hamiltonian HBdGis a 44 matrix given by HBdG= k0
k
 kk0 ; (6) wherekrepresents the energy of the electrons as mea- sured from their chemical potential, 0is a 22 unit matrix, and the pairing potential
kis also a 22 ma- trix. We consider three pairing potential types, including the spin-singlet s-wave pairing
k=
iyand two spin- tripletp-wave pairings
k= (dk
)iy, where their d vectors are given by dk=(
(0;0;eik) : Chiral pwave
(sink;cosk;0) : Helical pwave(7) wherek= arctan(ky=kx) is an azimuth angle. The phenomenological form of the gap function is assumed
= 1:76kBTctanh 1:74p Tc=T1 ; (8) withTcthe superconducting transition temperature. By diagonalizing HBdG, the quasiparticle energy is given by Ek=p 2 k+
2for all SCs considered here. There- fore, one cannot distinguish them by the energy spectrum alone, and they are simple models suitable for studying the dierence of the magnetic responses due to the pair- ing symmetry40. The third term Hexrepresents the proximity exchange coupling that occurs at the interface, which describes the spin transfer between the SC and the FI10,33, Hex=X q;k Jq;k+ qS k+ h:c:
; (9) whereJq;kis the matrix element for the spin transfer pro- cesses, q= (X qiY q)=2 represent the spin- ip opera- tors for the electron spins in the SCs, and S k=p 2Sby k andS+ k=p 2Sbkrepresent the Fourier component of the localized spin in the FI. Note that the precession axis is along theZaxis, so that the Zcomponent of the spin is injected into the SC when the FMR occurs. Using3 the creation and annihilation operators of electrons and magnons,Hexis written as Hex=X q;k;k0;s;s0p 2SJq;k+ ss0cy k0sck0+qs0by k+ h:c: : (10) From the above expression, one can see that Hexde- scribes electron scattering processes with magnon emis- sion and absorption. Modulation of FMR.| The FMR modulation can be read from the retarded component of the magnon Green's function33, which is given by GR k(!) =2S=~ !!k+i!(2S=~)R k(!); (11) where the Gilbert damping constant is introduced phenomenologically41{43. In the second-order perturba- tion calculation with respect to the matrix element Jq;k, the self-energy caused by proximity exchange coupling is given by R k(!) =X qjJq;kj2R q(!); (12) where the dynamic spin susceptibility of the SCs is de- ned as R q(!) :=Z dtei(!+i0)ti ~(t)h[+ q(t); q(0)]i:(13) The pole of GR k(!) indicates the FMR modulation, i.e., the shift of resonance frequency and the enhancement of the Gilbert damping. By solving the equation !!k=0(2S=~)ReR k=0(!) = 0; (14) at a xed microwave frequency !, one obtains the mag- netic eld at which the FMR occurs. The imaginary part of the self-energy gives the enhancement of the Gilbert damping. Consequently, the frequency shift and the en- hanced Gilbert damping are given by H=2S ~ReR k=0(!);  =2S ~!ImR k=0(!):(15) From the above equations and Eq. (12), one can see that the FMR modulation provides information about both the interface coupling properties and the dynamic spin susceptibility of the SCs. The form of matrix element Jq;k=0depends on the details of the interface. In this work, we assume the interface with uncorrelated roughness. jJq;k=0j2is given by jJq;k=0j2=J2 1 Nq;0+J2 2l2 NA; (16) where the rst and second terms describe averaged uni- form contribution and uncorrelated roughness contribu- tion, respectively39.J1andJ2correspond to the meanvalue and variance, respectively. Ais the area of the in- terface, which is equal to the system size of the SC. lis an atomic scale length. Using Eq. (16), the self-energy for the uniform magnon mode is given by R k=0(!) =J2 1 NR uni(!)J2 2l2 NAR loc(!); (17) where the uniform and local spin susceptibilities are de- ned as R uni(!) := lim jqj!0R q(!); R loc(!) :=X qR q(!):(18) The self-energy R k=0(!) consists of two terms originating from the uniform and roughness contributions, so that bothR uni(!) andR loc(!) contribute to Hand. Here, we discuss the FI thickness dependence on the FMR modulation44. From Eqs. (15), and (17), one can see that the FMR modulation is inversely proportional to the FI thickness ( /A=N ) becauseR uni(!)/Aand R loc(!)/A2. This is consistent with the experiments on the spin pumping in Y 3Fe5O12=Pt heterostructures45. In order to observe the FMR modulation experimentally, it is necessary to prepare a sample that is suciently thin, e.g., typically, the thickness of several tens of nanometers. Numerical results.| In the following, we consider a at interface where J2= 0, so that the behavior of the FMR modulation is determined by R uni(!). The roughness contribution proportional to R loc(!) is discussed later. Figure 2 shows the frequency shift Hand the enhanced Gilbert damping as a function of temperature and fre- quency. Here, we set = 0 and =kBTc= 0:05, where is a constant level broadening of the quasiparticle intro- duced phenomenologically39. First, we explain the qualitative properties of Hand for the chiral p-wave SC. In the low frequency re- gion, where ~!=kBTc1,His nite and remains al- most independent of !near the zero temperature and decreases and becomes exponentially small with the decrease of the temperature. In the high frequency re- gion, where ~!=kBTc1, a resonance peak occurs at ~!= 2
for both Hand. The qualitative proper- ties ofHandfor the helical p-wave SC are the same as those of the chiral p-wave SC. Next, we explain the qualitative properties of Hand for thes-wave SC. In the low frequency region, where ~!=kBTc1, bothHanddecrease and become exponentially small with the decrease of the temperature. In the high frequency region, where ~!=kBTc1, both Handvanish. Thep-wave SCs show two characteristic properties that thes-wave SC does not show: a nite HatT= 0 and a resonance peak of Hand. These properties can be understood by the analogy between SCs and band insulators as follows. The uniform dynamic spin suscepti- bility consists of contributions from intraband transitions within particle (hole) bands and interband transitions between particles and holes. In the low temperature or4 (a) (b)Γ/kBTc=0.05 Chiral p-wave (c) (d)Γ/kBTc=0.05 Helical p-wave (e) (f)Γ/kBTc=0.05 s-waveT/T c hω/kBTcδα/δα1 0.00.51.0 0 2 410 5 0 T/T c hω/kBTcδH/δH1 0.00.51.0 0 2 410 5 0 T/T c hω/kBTcδα/δα1 0.00.51.0 0 2 410 5 0 T/T c hω/kBTcδH/δH1 0.00.51.0 0 2 410 5 0 T/T c hω/kBTcδH/δH1 0.00.51.0 0 2 42 1 0 T/T c hω/kBTcδα/δα1 0.00.51.0 0 2 410 5 0 FIG. 2. The frequency shift Hand the enhanced Gilbert dampingas a function of temperature and frequency nor- malized by the characteristic values H1=SJ2 1DF=(N ~) and1=SJ2 1DF=(NkBTc) in the normal state. DF(/A) is the density of states at the Fermi level in the normal state. We set= 0 and =kBTc= 0:05. The sign of Hcorresponds to the sign of Re R uni(!), which can be positive and negative at low and high frequencies, respectively. In contrast, is positive at any frequency. high frequency region, the intraband contribution is neg- ligible and the interband contribution is dominant. In the case of the s-wave SC, the interband transitions are forbidden because the Hamiltonian and the spin operator commute. As a result, there is no spin response in the low-temperature or high-frequency regions. In contrast, the Hamiltonian for the p-wave SCs and the spin operator do not commute. Therefore, Hhas a nite value near- zero temperature due to the interband contribution. In addition, a resonance peak occurs when ~!= 2
because the density of states diverges at the band edge E=
. A detailed proof of the above statement is given in the Supplemental Material39. The angle dependences of Handare distinct for chiral and helical p-wave SCs, as shown in Fig. 3. In both cases, we set ~!=kBTc= 3:0 as the typical values at high frequencies, where the main contribution of the uniform spin susceptibility is the interband transitions. In the chiral p-wave SC,Handtend to decrease and are halved at a xed temperature when increases from 0 to=2. Conversely, in the helical p-wave SC, the qual- 0.4 T/Tc1.0 0.2 0.08 4 0 0.8 0.6δH/δH1 2 -210 60.4 T/Tc1.0 0.2 0.08 4 0 0.8 0.6δH/δH1 2 -210 6 Helical p-waveChiral p-wave θ=0 π/4 π/2 θ=0π/4π/2 0.4 T/Tc1.0 0.2 0.02 0 0.8 0.6δα/δα1 13θ=0 π/4 π/2 θ=0π/4π/20.4 T/Tc1.0 0.2 0.02 0 0.8 0.6δα/δα1 13Γ/kBTc=0.05, hω/kBTc=3.0 (a) (b) (c) (d)Γ/kBTc=0.05, hω/kBTc=3.0FIG. 3. Frequency shift and the enhanced Gilbert damping as a function of temperature at angles of = 0;=4;=2. The upper and lower panels show the characteristics for the chiral and helical p-wave SCs, respectively. itative behavior shows the opposite trend. Hand both tend to increase and become 1 :5 times larger at a xed temperature when increases from 0 to =2. In fact, the angle dependences are approximately obtained to be/1 +cos2and 1 +(sin2)=2 for chiral and helical p-wave SCs, respectively39. Therefore, the spin cong- uration of the Cooper pair can be detected from the  dependence data for the FMR modulation. The FMR modulation properties of the three SCs are summarized in Table I. All SCs considered here can be distinguished based on three properties: the frequency shift in the low temperature limit, the presence of their resonance peak, and their dependence. For the s-wave SC,Hbecomes exponentially small in T!0, while for thep-wave SCs, His nite in T!0. For the s-wave SC,Handshow no resonance and no dependence, while for the chiral and helical p-wave SCs, both Hand exhibit a resonance at ~!= 2
and a dependence. In addition, these two p-wave SCs can be distinguished from theirdependences of Hand, which are char- acterized by @(H) and@(), respectively. Here, it should be emphasized that the pairing symmetry can be characterized by the sign of @(H) and@(). These properties are summarized in the Table I. Spin-triplet current generation.| The relationship be- tween the enhanced Gilbert damping discussed above5 and the spin-triplet current generation must also be dis- cussed. The enhancement of the Gilbert damping is known to originate from the spin current generation at the magnetic interface6,33. The interface spin current in- duced by FMRhISiSPis given by39 hISiSP=N(~ hac)2 2 ImGR k=0(!) : (19) One can see that hISiSPandare proportional to each other. In our setup, the enhanced Gilbert damping  will lead to the generation of both the Cooper pair spin- triplet current and the quasiparticle spin current. Since the angular dependence of re ects the direction of the Cooper pair spins, it is expected that the spin-triplet current can be controlled by varying the magnetization direction of the FI. Discussion.| We have considered a at SC/FI inter- face. In the presence of roughness, the correction term proportional to R loc(!) contributes to the FMR mod- ulation, as shown in Eq. (17). In the rough limit, J2 1J2 2,R loc(!) dominates to make the FMR modu- lation isotropic, due to the angle average by summation overq. Namely, the anisotropy peculiar to p-wave SC is smeared by the roughness. The detailed behavior of R loc(!) is shown in the Supplemental Material39. This result implies that it is crucial to control the interface roughness. In principle, the roughness of the interface can be observed using transmission electron microscopy of interfaces46{48and it is possible to detect whether the interface of the sample is at or rough. More detailed spectroscopy can be obtained from the FMR modulation by using a at interface. Our results show that the pairing symmetry can be detected by the sign of @(H) and@() around the in-plane magnetic eld ( =2), where the vortices are negligible. When the external magnetic eld has a large out-of-plane component, the vortex formation may cause problems in observing the angular dependence. The qual- itative behavior is expected to change when the out-of- plane magnetic eld approaches the upper critical eld (HHc21T). This is because the coherence length of the Cooper pair and the distance between the vor- tices can become comparable. Indeed, it has been exper- imentally reported that the vortex formation suppresses the characteristic properties in the spin pumping into SCs20. Therefore, the out-of-plane magnetic eld should be as small as possible when FMR measurements are per- formed for HHc2. TABLE I. FMR modulation properties for the at SC/FI in- terface where J16= 0 andJ2= 0. Pairing symmetry s Chiral Helical Hin the limit of T!0 0 nite nite Resonance peak of H, { X X @(H),@() 0 negative positiveRecent experiments have reported that UTe 2is a can- didate material for spin-triplet p-wave SCs31, which has attracted a great deal of attention. Various experi- ments, including spectroscopic measurements, are now in progress to investigate the pairing symmetry of UTe 2, and indicated that the superconducting transition tem- perature is about 1K 30 GHz. Therefore, the resonance condition ~!= 2
shown above is accessible to recent broadband FMR measurements. In addition, experiments on spin pumping into d-wave SCs have recently been reported49and a theoretical in- vestigation of the enhancement of the Gilbert damp- ing in ad-wave SC/FI bilayer system has recently been presented50. Thus anisotropic superconducting spintron- ics can be expected to develop as a new research direc- tion. We should emphasize two important aspects of the FMR method presented here: the spectroscopic probe method for the p-wave SC thin lms and the versa- tile spin injection method. First, the FMR measure- ment procedure can provide a new spin-sensitive mea- surement method that will complement other measure- ment methods to enable a breakthrough in the discovery of spin-triplet SCs. Second, the FMR method represents a promising way to generate spin-triplet currents in p- wave SC thin lms. Conclusions.| We have investigated the anisotropic superconducting spin transport at magnetic interfaces composed of a p-wave SC and an FI based on a micro- scopic model Hamiltonian. The FMR signal in these p- wave SC/FI bilayer systems is modulated via spin trans- fer at the interface, which generates spin-triplet currents. We have shown that the pairing symmetry of the SCs can be extracted from the FMR modulation character- istics. Our approach provides a unique way to explore anisotropic superconducting spintronics, which will be useful for application to emerging device technologies. Note added.| After the submission of this manuscript, we became aware of a closely related work, where a way to convert spin-triplet currents to magnon spin currents in SC/FI bilayer systems is discussed51. We thank R. Ohshima, M. Shiraishi, H. Chudo, G. Okano, K. Yamanoi, and Y. Nozaki for helpful discus- sions. This work was supported by the Priority Pro- gram of the Chinese Academy of Sciences under Grant No. XDB28000000, and by JSPS KAKENHI under Grants Nos. JP20K03835, JP20H04635, JP20H01863, JP21H04565, and JP21H01800.6 SUPPLEMENTAL MATERIAL I. MODEL HAMILTONIAN In this section, we describe the derivation and details of the model Hamiltonian used in the main text. A. Ferromagnetic Heisenberg model The ferromagnetic Heisenberg model with the transverse AC magnetic eld due to the microwave radiation is given by HFI(t) =JX hi;jiSi
Sj+~ HX jSZ j~ hacX j SX jcos!tSY jsin!t
; (S.1) whereJ >0 is the exchange coupling constant, hi;jirepresents summation over all nearest-neighbor sites, Sjis the localized spin at site jin the ferromagnetic insulator (FI), (<0) is the gyromagnetic ratio, His a static magnetic eld,hacis an amplitude of an transverse oscillating magnetic eld due to the microwave radiation with a frequency !. The rotated coordinates ( X;Y;Z ) are shown in Fig. 1(a). It is convenient to introduce the boson creation and annihilation operators in order to formulate the problem in terms of the quantum eld theory. In the current problem, we perturbatively treat the excitation of the FI. In this case, the Holstein-Primako transformation is useful, where the localized spin can be described using boson creation and annihilation operators bj;by jin Hilbert space constrained to 2 S+ 1 dimensions. The spin operators are written as S+ j=SX j+iSY j= 2Sby jbj1=2 bj; (S.2) S j=SX jiSY j=by j 2Sby jbj1=2 ; (S.3) SZ j=Sby jbj; (S.4) where we require [ bi;by j] =i;j;in order that the S+ j,S j, andSZ jsatisfy the commutation relation of angular momentum. The deviation of SZ jfrom its ground-state value Sis quantied by the boson particle number. We consider low-energy excitation in the FI, where the deviation of SZ jfrom the ground state is small hby jbji=S1. The ladder operators S jare approximated as S+ j(2S)1=2bj; (S.5) S j(2S)1=2by j; (S.6) which is called spin-wave approximation. Here, we dene the magnon operators bk=1p NX jeik
rjbj; (S.7) by k=1p NX jeik
rjby j; (S.8) whereNis the number of sites and k= (kx;ky;kz). The inverse transformation is then given by bj=1p NX keik
rjbk; (S.9) by j=1p NX keik
rjby k: (S.10) The magnon operators satisfy [ bk;by k0] =k;k0and describe the quantized collective excitations. Using the spin-wave approximation and the magnon operators, the Hamiltonian HFI(t) is written as HFI(t)X k~!kby kbkh+ ac(t)by k=0h ac(t)bk=0; (S.11)7 where ~!k=Dk2~ HwithD= 2JSa2and the lattice constant a,h ac(t) =~ hacp SN=2ei!t, and constant terms are omitted. B. BCS Hamiltonian We derive a mean-eld Hamiltonian, which describes a bulk superconductor (SC), and we diagonalize the mean-eld Hamiltonian with the Bogoliubov transformation. At the end of this section, the spin density operators of the SC are written in terms of the Bogoliubov quasiparticle creation and annihilation operators. We start with the eective Hamiltonian in momentum space HSC=X k;skcy kscks+1 2X k;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy ks1cy ks2ck0s3ck0s4; (S.12) wherekis the band energy measured relative to the chemical potential, and cy ksandcksare the creation and annihilation operators of electrons with the wave vector k= (kx;ky) and thezcomponent of the spin s=";#. The matrix elements satisfy Vs1;s2;s3;s4(k;k0) =Vs2;s1;s3;s4(k;k0); (S.13) Vs1;s2;s3;s4(k;k0) =Vs1;s2;s4;s3(k;k0); (S.14) because of the anticommutation relation of fermions, and Vs1;s2;s3;s4(k;k0) =V s4;s3;s2;s1(k0;k); (S.15) because of the Hermitianity of the Hamiltonian. We consider a mean-eld, which is called a pair potential
k;ss0=X k0;s3;s4Vs0;s;s 3;s4(k;k0)hck0s3ck0s4i; (S.16) and its conjugate
 k;ss0=X k0;s1;s2Vs1;s2;s0;s(k0;k)hcy k0s1cy k0s2i: (S.17) Here, we consider a mean-eld approximation where the interaction term is replaced as follows cy ks1cy ks2ck0s3ck0s4!cy ks1cy ks2hck0s3ck0s4i+hcy ks1cy ks2ick0s3ck0s4hcy ks1cy ks2ihck0s3ck0s4i; (S.18) so that the interaction term is rewritten as X k;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy ks1cy ks2ck0s3ck0s4!X k;s1;s2h
k;s1s2cy ks1cy ks2
 k;s1s2cks1cks2i ; (S.19) where an constant term is omitted. Consequently, we derive a mean-eld Hamiltonian HSC=X k;skcy kscks+1 2X k;s1;s2
k;s1s2cy ks1cy ks2
 k;s1s2cks1cks2 : (S.20) Using a four-component notation cy k= (cy k";cy k#;ck";ck#); (S.21) ck= (ck";ck#;cy k";cy k#)T; (S.22) the mean-eld Hamiltonian is written as HSC=1 2X kcy kHBdGck: (S.23)8 HBdGis the 44 matrix HBdG= k0
k
 kk0! ; (S.24) where0is the 22 unit matrix and
kis the 22 matrix given as
k=
k;""
k;"#
k;#"
k;##! : (S.25) In principle, the pair potential is obtained by solving the gap equation self-consistently for an explicit form of the matrix elements Vs1;s2;s3;s4(k;k0). In this work, we do not solve the gap equation, but instead assume an explicit form of the pair potential and perform calculations using a phenomenological gap function. For the singlet pairing, the pair potential is given by
k= kiy; (S.26) with an even function k= k. For ans-wave SC, the pair potential is given by
k=
0 1 1 0! : (S.27) For the triplet pairing, the pair potential is given by
k= [dk
]iy; (S.28) with an odd vectorial function dk=dk. For a chiral p-wave SC and a helical p-wave SC,dkis given by dk=(
(0;0;eik) : chiral pwave
(sink;cosk;0) : helical pwave(S.29) withk= arctan(ky=kx), so that the pair potential is given by
k=8 >>>>< >>>>:
0eik eik0! : chiralpwave
ieik0 0ieik! : helicalpwave(S.30) The phenomenological gap function is given by
= 1:76kBTctanh 1:74p Tc=T1 : (S.31) The Bogoliubov transformation to diagonalize HBdGis given by Uk= ukvk v ku k! ; (S.32) Uy k= ukvk v ku k! ; (S.33) with the 22 matricesukandvkgiven by uk=s 1 2 1 +k Ek 0; (S.34) vk=s 1 2 1k Ek
k
; (S.35)9 whereEkis the eigenenergy Ek=q 2 k+
2: (S.36) Using the Bogoliubov transformation Uk, the 44 matrixHBdGis diagonalized as Uy kHBdGUk=0 BBB@Ek0 0 0 0Ek0 0 0 0Ek0 0 0 0Ek1 CCCA: (S.37) The excitation of HSCis described by the creation and annihilation operators of the Bogoliubov quasiparticles (y) k y k= ( y k"; y k#; k"; k#); (S.38) k= ( k"; k#; y k"; y k#)T; (S.39) where they are obtained by the Bogoliubov transformation k=Uy kck; (S.40) y k=cy kUk: (S.41) The spin density operators a(r) (a=x;y;z ) is dened as a(r) :=1 AX k;k0;s;s0ei(kk0)
ra ss0cy ksck0s0; (S.42) whereAis the area of the system. a(r) (a=x;y;z ) is expanded in Fourier series a(r) =1 AX qeiq
ra q; (S.43) and the Fourier coecient is given by a q=Z dreiq
ra(r) =X k;s;s0a ss0cy ksck+qs0: (S.44) Using the Bogoliubov transformation Uk, the above expression is rewritten as a q=X k;s;s0" sa(1) k;k+q s;s0 y ks k+qs0+ sa(2) k;k+q s;s0 ks y kqs0+ sa(3) k;k+q s;s0 y ks y kqs0+ sa(4) k;k+q s;s0 ks k+qs0# ; (S.45) with the 22 matricessa(i) k;k+qgiven by sa(1) k;k+q=uy kauk+q; (S.46) sa(2) k;k+q=vy kavk+q; (S.47) sa(3) k;k+q=uy kavk+q; (S.48) sa(4) k;k+q=vy kauk+q: (S.49) The rst and second terms describe the intraband transition from particle-to-particle and from hole-to-hole, respec- tively. The third and fourth terms describe the interband transition from hole-to-particle and from particle-to-hole, respectively.10 C. Proximity exchange coupling at interface We start with a model for the proximity exchange coupling given by Hex=Z drX jJ(r;rj)(r)
Sj: (S.50) We rewrite the above expression in the real space into the expression in the wave space. The proximity exchange coupling is rewritten as Hex=Z drX jJ(r;rj)1 Ap NX q;kei(q
r+k
rj) + qS k+ qS+ k
+Z drX jJ(r;rj)Z(r)SZ j; (S.51) where the Fourier series are given by (r) =1 AX qeiq
rq; (S.52) Sj=1p NX keik
rjSk; (S.53) with the area of the SC, A, and the number of sites in the FI, N, and the ladder operators are given by =1 2(XiY); (S.54) S=SXiSY: (S.55) The matrix element is given by Jq;k=1 Ap NZ drX jJ(r;rj)ei(q
r+k
rj): (S.56) Consequently, the exchange coupling which we use in the main text is derived as Hex=X q;k Jq;k+ qS k+J q;k qS+ k
; (S.57) where we use a relation Jq;k=J q;k, and we omit the last term Z drX jJ(r;rj)Z(r)SZ j; (S.58) in order to focus on the spin transfer at the interface. For the uniform magnon mode jkj= 0, the matrix element is given by Jq;k=0=1 Ap NZ drX jJ(r;rj)eiq
r: (S.59) II. TIME DEPENDENT QUANTUM AVERAGE In this section, we show that the ferromagnetic resonance (FMR) frequency and linewidth are read from the magnon Green's function. We consider the Hamiltonian H(t) composed of the unperturbed Hamiltonian H0and the perturbation V(t) H(t) =H0+V(t): (S.60) The time-dependent quantum average of a physical quantity Ois calculated as hO(t)i=hSy(t;1)~O(t)S(t;1)i; (S.61)11 where ~O(t) is the interaction picture and the S matrix S(t;t0) is given by S(t;t0) =Texp Zt t0dt0~V(t0) i~! : (S.62) The time-dependent quantum average hO(t)iis written as hO(t)i=hOieq+hO(t)i; (S.63) wherehOieq= Tr (eqO) is the equilibrium value and hO(t)iis deviation from the equilibrium. When the perturbation is written as V(t) =AF(t), the rst order perturbation calculation gives hO(t)i=Zt 1dt01 i~h[~O(t);~A(t0)]iF(t0) =Z1 1dt0GR(t0)F(tt0); (S.64) where we dene the retarded Green's function GR(t) =1 i~(t)h[~O(t);~A(0)]i: (S.65) When the external force is written as F(t) =Fei(!+i0)t,hO(t)iis written as hO(t)i=Fei(!+i0)tZ1 1dt0ei(!+i0)t0GR(t0) =Fei!tGR(!): (S.66) Using the above formula, the dynamics of hS+ k=0(t)iis written as hS+ k=0(t)i=~ hacp N 2ei!tGR k=0(!); (S.67) whereGR k(!) is the Fourier transform of the retarded component of the magnon Green's function GR k(t). They are dened as GR k(t) :=1 i~(t)h[S+ k(t);S k(0)]i; (S.68) GR k(!) :=Z1 1dt0ei(!+i0)t0GR k(t0): (S.69) From Eq. (S.67), one can see that the FMR frequency and linewidth are read from GR k(!). III. MAGNON GREEN'S FUNCTION In this section, we perform perturbative calculation for the magnon Green's function. We treat the proximity exchange coupling as a perturbation. The Hamiltonian is written as H=H0+V; (S.70) whereH0is the unperturbed Hamiltonian H0=X k~!kby kbk+X k;sEk y ks ks; (S.71) andVis the perturbation V=X q;k Jq;k+ qS k+ h:c:
: (S.72)12 (e) Vertex(b) Keldysh contour (d) Self-energytime(a) Magnon Green’s function (c) Dyson equation k/uni2032 +qs/uni2032 −k −k/uni2032 −qs/uni2032 /uni03C3+ qS− k= + + +−k −k −k−k/uni2032 s k/uni2032 s k/uni2032 s −k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 sGk(/uni03C4,/uni03C4/uni2032 )= = + /uni03A3 /uni03A3 = + + + k/uni2032 +qs/uni2032 −k/uni2032 −qs/uni2032 −k/uni2032 s k/uni2032 s k/uni2032 s −k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 s Bogoliubov quasiparticle: Magnon: FIG. 4. (a) The Feynman diagram for the magnon Green's function. (b) Keldysh contour to perform perturbative calculations. (c) The Feynman diagram for the Dyson equation. (d) The self-energy within the second-order perturbation is given by the dynamic spin susceptibility of the SCs. (e) The Feynman diagrams for the vertex + qS k, which represent scattering of a Bogoliubov quasiparticle with magnon emission. The solid and wavy lines represent a Bogoliubov quasiparticle and a magnon, respectively. We dene the magnon Green's function Gk(;0) :=1 i~hTCS+ k()S k(0)i; (S.73) whereTCis the time-ordering operator on the Keldysh contour (see Figs. 4(a) and (b)). To perform the perturbative calculation, we introduce interaction picture. The perturbation is written as ~V(t) =X q;k Jq;k~+ q(t)~S k(t) + h:c: : (S.74) The magnon Green's function is given by Gk(;0) =1 i~hTCSC~S+ k()~S k(0)iconn; (S.75) whereh


i connmeans the connected diagrams and the S matrix is given by SC=TCexp Z Cd~V() i~! : (S.76) The above expressions lead to the Dyson equation (see Fig. 4(c)) Gk(;0) =G(0) k(;0) +Z Cd1Z Cd2G(0) k(;1)k(1;2)Gk(2;0); (S.77) whereG(0) k(;0) is the unperturbed magnon Green's function G(0) k(;0) =1 i~hTC~S+ k()~S k(0)i; (S.78) and  k(1;2) is the self-energy. Within the second-order perturbation, the self-energy is given by (see Fig. 4(d)) k(;0) =1 i~X qjJq;kj2hTC~+ q()~ q(0)i: (S.79) The Feynman diagram for the vertex is shown in Fig. 4(e). Substituting the ladder operators expressed in terms of13 (y) ks, the self-energy is written as k(;0) =i~X qjJq;kj2X k0;s;s0"(s+(1) k0;k0+q)s;s02 (s+(1) k0;k0+q)s;s0(s(2) k0q;k0) s0;s gk0;s(0;)gk0+q;s0(;0) +(s+(2) k0;k0+q)s;s02 (s+(2) k0;k0+q)s;s0(s(1) k0q;k0) s0;s gk0;s(;0)gk0q;s0(0;) (s+(3) k0;k0+q)s;s02 (s+(3) k0;k0+q)s;s0(s(3) k0q;k0) s0;s gk0;s(0;)gk0q;s0(0;) (s+(4) k0;k0+q)s;s02 (s+(4) k0;k0+q)s;s0(s(4) k0q;k0) s0;s gk0;s(;0)gk0+q;s0(;0)# ; (S.80) where the quasiparticle Green's function is dened as gk;s(;0) :=1 i~hTC~ ks()~ y ks(0)i: (S.81) The rst and second terms give the intraband contribution, and the third and fourth terms give the interband contribution. Evaluating the Dyson equation, the retarded component of the magnon Green's function is given by GR k(!) =1 h G(0)R k(!)i1 R k(!); (S.82) where the unperturbed Green's function is written as G(0)R k(!) =2S=~ !!k+i!: (S.83) Here, we introduce the phenomenological dimensionless damping parameter . Using Eq.(S.83), the retarded Green's function is written as GR k(!) =2S=~ !!k+i!(2S=~)R k(!): (S.84) From the above expression, the frequency shift at a xed !is given by H=2S ~ReR k(!); (S.85) and the enhanced Gilbert damping is given by =2S ~!ImR k(!): (S.86) The Fourier transform of the self-energy is given as R k(!) =Z dtei(!+i0)tR k(t) =X qjJq;kj2R q(!); (S.87) where the dynamic spin susceptibility of the SC is dened as R q(!) :=Z dtei(!+i0)ti ~(t)h[~+ q(t);~ q(0)]i: (S.88) Evaluating the self-energy Eq. (S.87), one can obtain the information of the FMR modulation, Hand. Using the system's symmetry, the dynamic spin susceptibility R q(!) can be written as R q(!) = cos2xx q(!) +yy q(!) + sin2zz q(!); (S.89) which means that both Handshow a dependence on when the dynamic spin susceptibility is anisotropic.14 IV. SPIN CURRENT AT THE INTERFACE In this section, we derive the general expression of spin current at the interface. We treat the tunneling Hamiltonian as a perturbation and the other terms as the unperturbed Hamiltonian H(t) =H0(t) +Hex; (S.90) H0(t) =HFI(t) +HSC: (S.91) The operator of spin current owing from the SC to the FI at the interface is dened by IS:=~ 2_Z tot=~ 21 i~[Z tot;Hex] =i 2[Z tot;Hex]; (S.92) whereZ totis given by Z tot=Z drZ(r): (S.93) Calculating the commutation relation, we obtain the following expression IS=iX q;k Jq;k+ qS kh:c:
: (S.94) The time-dependent quantum average of ISis written as hIS(t)i= Re2 42iX q;kJq;kh+ q(t)S k(t)i3 5; (S.95) whereh


i = Tr[0


] denotes the statistical average with an initial density matrix 0. In order to develop the perturbation expansion, we introduce the interaction picture hIS(1;2)i= Re2 42iX q;kJq;khTCSC~+ q(1)~S k(2)i3 5: (S.96) SCand ~O(t) are given by SC=TCexp Z Cd~Hex() i~! ; (S.97) and ~O(t) =Uy 0(t;t0)OU0(t;t0); (S.98) where U0(t;t0) =TexpZt t0dt0H0(t0) i~ : (S.99) ExpandingSCas SC1 +Z CdTC~Hex() i~; (S.100) the spin current is given by hIS(1;2)i=X q;kjJq;kj2Re" 2 ~Z CdhTC~+ q(1)~ q()ihTC~S+ k()~S k(2)i# : (S.101)15 Using the contour ordered Green's functions q(1;) =1 i~hTC~+ q(1)~ q()i; (S.102) Gk(;2) =1 i~hTC~S+ k()~S k(2)i; (S.103) the above equation is rewritten as hIS(1;2)i=X q;kjJq;kj2Re" 2~Z Cdq(1;)Gk(;2)# : (S.104) We put2on the forward contour and 1on the backward contour to describe spin transfer at the interface in appropriate time order. Assuming a steady state, the spin current is written as hISi= 2~X q;kjJq;kj2Re"Z1 1d!0 2 R q(!0)G< k(!0) +< q(!0)GA k(!0)# : (S.105) We introduce the distribution functions as < q(!) =fSC q(!) 2iImR q(!) ; (S.106) G< k(!) =fFI k(!) 2iImGR k(!) : (S.107) The formula of the spin current at the interface is derived as hISi= 4~X q;kjJq;kj2Z1 1d!0 2ImR q(!0) ImGR k(!0) fFI k(!0)fSC q(!0) : (S.108) When both the SC and the FI are in equilibrium, the dierence of the distribution functions is zero (i.e. fFI k(!0) fSC q(!0) = 0), so that no spin current is generated. Under the microwave irradiation, the distribution function of the FI deviates from equilibrium, which generates the interface spin current. Performing a second-order perturbation calculation, the deviation of the distribution function of the FI, fFI k(!0), is given by fFI k(!0) =2NS ( hac=2)2 !0k;0(!0!): (S.109) Consequently, the interface spin current is written as hISiSP= 4~X q;kjJq;kj2Z1 1d!0 2ImR q(!0) ImGR k(!0) fFI k(!0): (S.110) Finally, one can show that the spin current is proportional to the enhanced Gilbert damping hISiSP= 4~NS( hac=2)2 ! ImGR k=0(!)X qjJq;k=0j2ImR q(!); =N(~ hac)2 2 ImGR k=0(!) : (S.111) V. MODEL FOR INTERFACE CONFIGURATIONS In order to calculate Eq. (S.87), one needs to set up an explicit expression for jJq;k=0j2. We consider an interface with uncorrelated roughness. To model this interface, we assume that J(r;rj) satises DX jJ(r;rj)E ave=J1; (S.112) DX j;j0J(r;rj)J(r0;rj0)E ave=J2 1+J2 2l2(rr0); (S.113)16 whereh


i avemeans interface conguration average. The spatially averaged J(r;rj) is given by a constant J1as shown in Eq. (S.112). Equation (S.113) means that the interface roughness is uncorrelated and J2 2l2is a variance. J1andJ2are coupling constants with dimension of energy, and are independent of the system size. lis introduced because the Hamiltonian of the SCs is treated as a continuum model. Performing the interface conguration average, and using Eq. (S.112) and (S.113), one can obtain the expression for jJq;k=0j2in the main text. VI. DYNAMIC SPIN SUSCEPTIBILITY OF SC Evaluating the retarded component of the self-energy Eq. (S.80), the dynamic spin susceptibility of the SC is given by R q(!) =Z1 1dEf(E)X ;k( M;(a) k;k+q 1 ImgR ;k(E)gR ;k+q(E+~!)1 ImgR ;k+q(E)gA ;k(E~!) +M;(a) k;k+q 1 ImgR ;k(E)gR ;k+q(E+~!)1 ImgR ;k+q(E)gA ;k(E~!)) ; (S.114) whereM;0(a) k;k+qwitha=s;c;andhare given by M;0(s) k;k+q=(k+Ek)(k+q+0Ek+q) 4Ek0Ek+q+
2 4Ek0Ek+q; (S.115) M;0(c) k;k+q=(k+Ek)(k+q+0Ek+q) 4Ek0Ek+q
2ei(kk+q) 4Ek0Ek+qcos2; (S.116) M;0(h) k;k+q=(k+Ek)(k+q+0Ek+q) 4Ek0Ek+q
2sinksink+q 4Ek0Ek+qsin2: (S.117) ;0=give a sign, and a=s;c;andhcorrespond to matrix elements for s-wave, chiral p-wave, and helical p-wave SCs, respectively. In Eq. (S.114), the terms multiplied by M;(a) k;k+qdescribe the intraband transition processes, i.e., transition processes from particle to particle and from hole to hole, and the terms multiplied by M;(a) k;k+qdescribe the interband transition processes, i.e., transition processes from particle to hole and vice versa. The retarded and advanced Green's functions of the quasiparticles gR=A ;k(E) are given by gR ;k(E) =1 EEk+i; (S.118) gA ;k(E) =1 EEki; (S.119) where is a constant level broadening introduced phenomenologically. is introduced to incorporate the intraband contribution in the calculation of the uniform spin susceptibility. The details are explained in the next section. The sum over kis replaced by the integral near the Fermi energy X kF(k)!DFZ1 0dEDs(E)X =F(E); (S.120) X kF(k) sin2k!DFZ1 0dEDs(E)X =1 2F(E); (S.121) whereDFis the density of states near the Fermi energy in the normal state and Ds(E) is the density of states of quasiparticles Ds(E) =jEjp E2
2(jEj
): (S.122) F(E) means to assign p E2
2tocontained in F(k).17 VII. UNIFORM SPIN SUSCEPTIBILITY In this section, we explain three properties related to the calculation of the uniform spin susceptibility. First, the matrix element's properties are explained, which is essential to understand the qualitative dierence between spin- singlets-wave and spin-triplet p-wave SCs. Second, the reason to introduce the constant level broadening . Third, the analytical expression for the uniform spin susceptibility of the p-wave SCs is given. Performing the angular integral and replacing the sum over kby theEintegral, the matrix elements are replaced by M;0(s) k;k!1 +0 40; (S.123) M;0(c) k;k!(1 +0)E2(1 + cos2)
2 40E2; (S.124) M;0(h) k;k!(1 +0)E2(1 +1 2sin2)
2 40E2: (S.125) Here, the rst-order terms in kare omitted because they vanish in the Eintegral. From the above expressions, the intraband matrix elements become nite for all SCs considered here, while the interband matrix elements vanish in thes-wave SC and becomes nite in the p-wave SCs. The above properties of the intraband and interband matrix elements can be understood using the commutation relation between the Hamiltonian and the spin operators. We introduce the BdG form of the spin operators a BdG(a=x;y;z ) as below a BdG= a0 0(a)T! : (S.126) The commutation relation of HBdGanda BdGis given by [HBdG;a BdG] = 0 :swave; (S.127) [HBdG;a BdG]6= 0 :pwave: (S.128) Equation (S.127) means that both the Hamiltonian and the spin operator are diagonalized simultaneously, so that the matrix elements of the spin operator between a particle and a hole with the same wave-number vanish. This is because thes-wave SC is spin singlet. Therefore, the interband matrix elements vanishes in the s-wave SC. In contrast, in thep-wave SCs, the commutation relation between the Hamiltonian and the spin operator is nite as shown in Eq. (S.128), so that the matrix elements of the spin operator between a particle and a hole with the same wave-number is nite. This is because the p-wave SCs are spin triplet. As a result, the interband matrix elements are nite. Here, we explain the reason to introduce the constant level broadening for gR=A ;k(E). The intraband and interband transitions are schematically shown in Fig. 5. The quasiparticles are scattered due to the magnon emission or absorption. The scattering process conserves the wave-number. Consequently, in the case of the intraband transition, the transition process is forbidden when = 0. In order to incorporate the intraband processes, one needs to introduce , otherwise the intraband contribution vanishes, which can be directly shown by calculating Eq. (S.114). When = 0, the uniform spin susceptibility for the chiral p-wave SCs is given by ReR uni(!) =2DFZ1
dEEp E2
2(1 + cos2)
2 4E2(f(E)f(E))1 2E+~!+1 2E~! ; (S.129) and ImR uni(!) =2DFj~!=2jp (~!=2)2
2(1 + cos2)
2 (~!)2(f(~!=2)f(~!=2)): (S.130) From the above expressions, one can show that both the real part and imaginary part of the uniform spin susceptibility diverge at ~!= 2
, leading a resonance peak. The expressions for the helical p-wave SC can be obtained by replacing cos2with1 2sin2. Therefore, dependence of R uni(!) explained in the main text is obtained from the above expressions.18 Γintra inter k+Ek −EkE E Spectral function 2/uni0394/uni210F/uni03C9 /uni210F/uni03C9Γ FIG. 5. Schematic image of intraband transition and interband transitions. The intraband transition gives contribution to the uniform spin susceptibility when the excitation energy is comparable to or smaller than the level broadening, ~!.. The interband contribution is dominant when the excitation energy is comparable to the superconucting gap, ~!2
. VIII. LOCAL SPIN SUSCEPTIBILITY Performing the angular integral and replacing the sum over k;qby theE;E0integral, the matrix elements are replaced by M0(s) k;q!1 4+
2 4E0E0; (S.131) M0(c) k;q!1 4; (S.132) M0(h) k;q!1 4: (S.133) The matrix elements for the chiral and helical p-wave SCs are identical. From the above expressions, one can see that the interband contribution in the s-wave SC is suppressed. Unlike the uniform spin susceptibility, the intraband contribution for the local spin susceptibility is nite even when = 0. This is because the transition processes considered here leads to momentum transfer and the intraband transition is not forbidden. Therefore, we calculate the local spin susceptibility at = 0. The local spin susceptibility for the s-wave SC is given by R loc(!) =D2 FZ1 1dEZ1 1dE0Ds(E)Ds(E0) 1 +
2 EE0f(E)f(E0) EE0+~!+i0; (S.134) and the local spin susceptibility for the p-wave SCs is given by R loc(!) =D2 FZ1 1dEZ1 1dE0Ds(E)Ds(E0)f(E)f(E0) EE0+~!+i0: (S.135) IX. FMR MODULATION: ROUGH INTERFACE In this section, we show the numerical results and summarize the characteristic properties of the FMR modulation for the rough interface limit. In the following calculations, we set J1= 0 and assume that only R loc(!) contributes to Hand. Figures 6 show (a) Hand (b)for the chiral and helical p-wave SCs as a function of frequency and temperature. His nite inT!0 and has a resonance peak at ~!= 2
.exhibits a coherence peak just below the transition temperature in the suciently low frequency region, where ~!=kBTc1.drops abruptly at ~!= 2
.is almost independent of both frequency and temperature when ~!>2
. Figures 6 show (c) Hand (d)for thes-wave SC as a function of frequency and temperature. In the low frequency region, where ~!=kBTc1,Hat a xed frequency decreases by about thirty percent with the decrease of the temperature, and His nite inT!0. As the frequency increases, His almost independent of the temperature. shows a coherence peak just below the transition temperature in the suciently low frequency, where ~!=kBTc1.19 The coherence peak in the s-wave SC is larger than the corresponding coherence peak in the p-wave SCs. has a kink structure at ~!= 2
. Note that the cuto energy Ecwas introduced here to cause the integral for Re R loc(!) to converge. Although ReR loc(!) is approximately proportional to Ec, the qualitative properties explained above are independent of Ec. The FMR modulation properties of the three SCs are summarized in Table II. In the case of the rough interface limit, the pairing symmetry can be detected from either the absence or the existence of the resonance peak of H. The pairing symmetry may also be detected from the properties of , the height of the coherence peak, and the structure at~!= 2
. When compared with the resonance peak for H, however, the properties of are too ambiguous to allow the pairing symmetry to be distinguished clearly. (c) (d)s-wave(a) (b)Chiral & Helical p-wave T/T chω/kBTcδH/δH2 0.00.51.00 2 410 8 6 T/T chω/kBTcδα/δα2 0.00.51.00 2 42 1 0 T/T chω/kBTcδH/δH2 0.00.51.00 2 410 8 6 T/T chω/kBTcδα/δα2 0.00.51.00 2 42 1 0 FIG. 6. (a) The frequency shift and (b) the enhanced Gilbert damping as a function of both frequency and temperature for the p-wave SCs. (c) The frequency shift and (d) the enhanced Gilbert damping as a function of both frequency and temperature for thes-wave SC. The terms H2and2are given by H2=2SJ2 2l2D2 FkBTc=(NA ~) and2= 2SJ2 2l2D2 F=(NA), where they are characteristic values in the normal state. The cuto energy is set to be Ec=kBTc= 10. TABLE II. FMR modulation properties for the rough SC/FI interface where J1= 0 andJ26= 0. Pairing symmetry s Chiral Helical Resonance peak of H { X X Coherence peak of  large small small at~!= 2
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2021-03-10

We present a theoretical investigation of anisotropic superconducting spin transport at a magnetic interface between a p-wave superconductor and a ferromagnetic insulator. Our formulation describes the ferromagnetic resonance modulations due to spin current generation depending on spin-triplet Cooper pair, including the frequency shift and enhanced Gilbert damping, in a unified manner. We find that the Cooper pair symmetry is detectable from the qualitative behavior of the ferromagnetic resonance modulation. Our theory paves the way toward anisotropic superconducting spintronics.

Anisotropic superconducting spin transport at magnetic interfaces

2103.05871v3

Damping Separation of Finite Open Systems in Gravity-Related Experiments in the Free Molecular Flow Regime Hou-Qiang Teng,1Jia-Qi Dong,1,∗Yisen Wang,1Liang Huang,1,†and Peng Xu1, 2, 3 1Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, and Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou, Gansu 730000, China 2Center for Gravitational Wave Experiment, National Microgravity Laboratory, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China 3Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou 310124, China (Dated: January 11, 2024) The residual gas damping of the test mass (TM) in the free molecular flow regime is studied in the finite open systems for high-precision gravity-related experiments. Through strict derivation, we separate the damping coefficients for two finite open systems, i.e., the bi-plate system and the sensor core system, into base damping and diffusion damping. This elucidates the relationship between the free damping in the infinite gas volume and the proximity damping in the constrained volume, unifies them into one microscopic picture, and allows us to point out three pathways of energy dissipation in the bi-plate gap. We also provide the conditions that need to be met to achieve this separation. In applications, for space gravitational wave detection, our results for the residual gas damping coefficient for the 4TM torsion balance experiment is the closest one to the experimental and simulation data compared to previous models. For the LISA mission, our estimation for residual gas acceleration noise at the sensitive axis is consistent with the simulation result, within about 5% difference. In addition, in the test of the gravitational inverse-square law, our results suggest that the constraint on the distance between TM and the conducting membrane can be reduced by about 28%. I. INTRODUCTION Einstein’s general relativity, which still stands as the most successful theory of gravitation, has a far-reaching influence on the vision of nature as geometry. With the establishment of the Dicke framework [1] based on Einstein’s equivalence principle (EP) in the 1960s, ex- perimental gravity or gravitational experiments can be roughly divided into two classes, including the tests of foundations of gravitational theory such as EP and the precision measurements of spacetime curvature ef- fects of the so-called metric theories of gravity [2, 3]. Both classes of gravitational experiments heavily rely on the establishments of high-precision inertial references and the measurements of the relative free-falling mo- tions or geodesic deviations between them, and these include, for example, the MICROSCOPE mission [4– 6] for the test of the weak EP, ground-based or space- borne gravitational wave (GW) antennas like the LIGO- VIRGO collaboration [7–9], the LISA [10, 11] and LISA- like missions [12, 13], satellite gravity recovery missions such as GRACE/GRACE-FO [14, 15] and GOCE [16], and ground-based experiments with high-precision tor- sion balances [17–26] et al. In present days, isolated test masses (TMs) coupled with high-precision readout systems, such as capacity sensors [27], laser interferome- ters [28], or SQUIDs [29], are the main technical imple- mentation methods for high-precision inertial reference ∗dongjq@lzu.edu.cn †huangl@lzu.edu.cnsystems. The TM, as the key unit, is generally suspended (electrically or magnetically in space, or through a wire for ground-based experiments) inside a very stable envi- ronment with extremely weak couplings to external dis- turbances, which ensures the free-falling state of certain degrees of freedom of the TM in the gravitational field. While, the unavoidable fluctuations of the physical fields coupled to the TM, especially the random collisions from residual gas molecules, will limit the acceleration noise of TM and therefore the precisions or sensitivities of the in- ertial reference [30]. Therefore, the accurate assessments of numerous environmental noises become a crucial task in the high-precision gravity-related experiments. Among the various stray forces [30], those associated with residual gases include damping force noise [31, 32], radiometer effect, and outgassing effect [33, 34]. Damp- ing force noise refers to the Brownian motion of TMs due to gas collisions, which typically becomes one of the lim- its in high-precision measurements [31, 32, 35–40]. The damping force noise can be obtained from the damp- ing coefficient through the fluctuation-dissipation theo- rem [31, 41–44] Sf(ω) = 4 kBTRe −∂F(ω) ∂v(ω) = 4kBTRe[Z(ω)],(1) where Sf(ω) is the power spectral density of the fluctua- tion force on the TM, kBis the Boltzmann constant, Tis the temperature, Fis the damping force, vis the TM’s velocity, Z(ω) is the mechanical impedance of the sys- tem. In high-precision gravity-related experiments, to re- duce residual gas noise, the environment pressure is gen- erally less than 10−4Pa [19, 34, 45], and the environmentarXiv:2401.04808v1 [gr-qc] 9 Jan 20242 around the TM is in a free molecular flow regime, where the mean free path is much larger than the distance from the TM to the surrounding walls, and collisions between molecules rarely occur. The collisions between molecules and the surfaces are inelastic, following the Knudsen hy- pothesis [46–49]. So the collisions can be treated as in- dependent impulses, the resulting fluctuation force noise has a frequency independent spectrum [32], so residual gas leads to an impedance Z(ω) =β, with βreferred to here as the gas damping coefficient. The residual gas damping coefficient is dependent on the specific environments surrounding TMs. In high- precision gravity-related experimental setups with TMs, two fundamental structures frequently appear, i.e., bi- plate and sensor core [50, 51]. The bi-plate structure comprises a movable TM and a fixed parallel plate. The sensor core is a cuboid nested structure within gravita- tional reference sensors (GRSs)/inertial sensors, with the TM in the middle and surrounded by an electrode hous- ing. And they constitute the finite open systems. In GW detections, the damping caused by the motion of the TM is called proximity-enhanced damping [32, 38, 39], while in other fields like MEMS/NEMS, it is known as squeeze-film damping [52, 53]. In traditional squeeze-film damping studies, when the pressure is at one atmospheric pressure or less, the gas is treated as a continuous viscous fluid, and damping forces are calculated using the viscos- ity coefficient or effective viscosity coefficient [54–56]. As the pressure decreases until collisions between molecules rarely occur, i.e. the free molecular flow regime, the envi- ronment around TM and the choice of surface boundary conditions and analysis methods significantly influence the damping calculations. Commonly employed meth- ods include the free molecular model [57–61], isothermal piston model [32, 52], shot noise model [32], and non- isothermal models [38, 62]. In traditional analyses, the damping coefficient βin the finite open systems consists of two parts [32, 37, 39, 52], i.e., the free damping/kinetic damping β∞within an in- finite gas volume, and proximity damping ∆ βwithin a constrained volume. And the total damping coefficient isβ=β∞+ ∆β, corresponding to the force noise power spectral density, Sf(ω) =S∞ f+ ∆Sf(ω). However, for this linear separation, there is no consistent understand- ing of the relationship between the two dampings. For example, Ref. [32] gives a simple explanation: there is a continuum of behavior between the free damping limit and proximity damping, which is confirmed by Monte Carlo simulations. Reference [52] compares the magni- tudes of two dampings, and kinetic damping is eventu- ally ignored. And Ref. [39] considers only the proximity damping on the front side, with shear free damping on the lateral side. Therefore, a quantitative and rigorous analysis for the separation of the damping and the cor- responding condition is needed. To address the above issues, some researchers have studied from the perspective of non-isothermal thermo- dynamics. Ref. [62] considers the impact of gas tem-perature increase during the compression process, intro- duces the thermal-squeeze film damping, and highlights the influence of molecular degrees of freedom and surface roughness on damping. However, their use of adiabatic conditions contradicts the Knudsen hypothesis. Subse- quently, Ref. [38] introduces isothermal conditions, and employs effective temperature to compensate for the ex- cess internal energy in the non-equilibrium state com- pared to the equilibrium state. While effective tempera- ture is a macroscopic approximation, and high-precision measurements demand a more microscopically accurate explanation for residual gas noise. In this paper, we introduce finite open models, and pre- cisely decompose the damping coefficient into base damp- ing and diffusion damping for both bi-plate and sensor core finite open systems. This clarifies the relationship between the free damping and the proximity damping, and unifies them into one microscopic picture. In ap- plications, the finite open models effectively correct the constraints on gap spacing in the test of the gravitational inverse-square law (ISL), and emerge as the theoretical model most aligned with the simulation and measure- ment results of the residual gas damping in GRSs for the LISA mission and other LISA-like space GW detection missions. The rest of this paper is organized as follows: Section II calculates the changes in molecular number density. The separation of damping coefficients is carried out for both bi-plate and sensor core finite open systems in Sec. III. Section IV gives separation and sources of damping from an energy perspective. The finite open models are ap- plied to experiments in the test of the ISL and the GW detections in Sec. V. Discussions and conclusions are pro- vided in Sec. VI. Appendices include detailed derivations and expressions for some physical quantities. II. CHANGES IN MOLECULAR NUMBER DENSITY IN THE BI-PLATE FINITE OPEN SYSTEM The distinction between the bi-plate finite open system and the infinite volume lies in: as the TM moves, the molecular number density within the gap changes. When calculating the momentum exchange between molecules and the TM, the number density is required. This will lead to the unique physical processes. The change in number density is addressed in the following discussion. In Fig. 1, as the TM moves, the gap volume V, the total number of molecules Nand the number density nundergo continuous changes over time. Considering the total differential of the molecular number density n(t) =N(t)/V(t), and taking the derivative with respect to time, we have d∆n dt=1 V0d∆N dt−N0 V2 0d∆V dt, (2) where ∆ nis the change in molecular number density,3 GapTM Fixed plate 0+∆ℎL W FIG. 1. Schematic side view of the bi-plate finite open system. The TM moves downward parallel to the x-axis, with a gap spacing denoted as h. Gas molecules can exchange with the environment through the gap boundary. The pure light-gray gas molecules are from the fixed plate surface or the environ- ment, while the gas molecules represented by light-gray and black desorb from the TM surface. The color black repre- sents more or less momentum or energy carried compared to the case when the TM is stationary. The effective molecular number density on the windward and leeward sides is denoted asn′ Wandn′ L, respectively. ∆N(t) and ∆ V(t) are the changes in the molecular num- ber and gap volume relative to the equilibrium point. Considering the diffusion process for the molecules to es- cape into the environment, as shown by the arrows in the left of Fig. 1, we deal with the first and second terms on the right side of Eq. (2) in Appendix A, then under condition ∆ h≪h0, Eq. (2) is rearranged as d∆n dt=−∆n τ0−n0 h0dh dt, (3) where τ0is diffusion time for one molecule to escape from the gap and h(t) =h0+Bcos(ωt) is gap spacing in the frequency domain, and the general solution to Eq. (3) is ∆n=n0Bωτ 0 h0p 1 +ω2τ2 0cos(ωt+φ) +Ce−t τ0,(4) where φ= (arctan(1 /ωτ0)−π)∈(−π,−π/2) is the phase angle, and Cis the integration constant. The first term in Eq. (4) is the steady-state solution, and the second term is the transient solution, which diminishes over time. In the most high-precision gravity-related experiments, the product of detection frequency bands and diffusion times isωτ0∈(10−8,10−2) [7, 10–16, 18–23, 63, 64], and thus φ≈ −π/2. The steady-state part of Eq. (4) is approxi- mated as: ∆n≈n0Bωτ 0 h0cos(ωt−π 2). (5) The system’s finite openness can be characterized by the diffusion time τ0. A small degree of finite openness means that the number of exchanges between the gap and the environment is small per unit time, equivalent to a largediffusion time τ0, i.e., a long relaxation time for the sys- tem, and vice versa. When the gap boundary is closed, the gap spacing change ∆ his in phase opposition with the density change ∆ nclosed, this is consistent with our general understanding. As the gap boundary opens, τ0 decreases, the phase φof ∆nchanges −π→ −π/2. Inter- estingly, when ∆ hcontinues to increase beyond the equi- librium position, ∆ nopenalso increases, which may seem counterintuitive. The explanation is that when the vTM is very small, the difference in number density between the gap and the environment is small. However, when thevTMis very large, due to the reaction time required for diffusion, the number density in the gap will differ significantly from that in the environment. Furthermore, considering the direction of motion, ∆ nopenshould be in phase opposition to vTM. III. SEPARATION OF DAMPING COEFFICIENTS The Knudsen hypothesis posits that molecules, upon impacting a surface, undergo adsorption, remain for a brief period, and forget their velocity and direction be- fore impact. During desorption, the speed of emitted molecules follows the Maxwell-Boltzmann distribution, and the emission angles obey cosine law [40, 65–72]. Ac- cording to the cosine law, the probability for a molecule to leave the surface within the solid angle d ωis expressed as dP= dωcosθ/π, where θdenotes the angle with the surface normal direction. The cosine law is initially discovered in experiments using glass surfaces [48, 49], and subsequent studies on various materials have also verified the same emission rule [35, 40, 65–72]. Especially, Ref. [40] provides a condi- tion for non-elastic collisions in a vacuum ( p0<10−4Pa) with gold-plated surfaces. In realistic applications in the field of MEMS/NEMS, various models considering dif- ferent boundary conditions are available [53, 60, 73, 74], including isothermal and adiabatic, rough and smooth surfaces, diffuse and specular scattering, etc., which are often selected based on experimental data. Despite some deviations observed in individual experi- ments [65, 75], it is assumed that the cosine law remains valid for surfaces of general materials. The cosine law and the parallel arrangement of the bi-plate ensure that the gas within the gap adheres to the Maxwell-Boltzmann velocity distribution [76–78]. Given the extremely short adsorption time at room temperature (on the order of picoseconds or nanoseconds) [32], much shorter than the motion time of molecules in the gap (on the order of microseconds), it is reasonable to assume a negligible ad- sorption time. Next, we will proceed to separate the damping coeffi- cients in the bi-plate and sensor core finite open systems, respectively.4 A. Bi-plate finite open system As shown in Fig. 1, assuming the TM is moving down- ward, the frequency of collisions with molecules increases on the windward side and decreases on the leeward side, impacting the force experienced by the TM. We will an- alyze this effect separately for surfaces perpendicular ( x surface) and parallel ( yorzsurface) to the direction of motion. Damping force of incident molecules on xsurface. — Consider a surface element d Axperpendicular to the x- axis. Assuming the TM has motion solely parallel to the x-axis, the number of molecular per unit time colliding with the element d Axon the windward side, with a ve- locity component vxin the range vx∼vx+ dvxis given by [77–79] dNvx= (n0+ ∆n)(vx−vTM)f(vx)dvxdAx,(6) where fis the Maxwell-Boltzmann velocity distribution function, i.e. f(vx) = (m/2πkBT)1/2exp −mv2 x/2kBT
. The molecules of vx> v TMwill collide with d Ax, so vx∈(vTM,∞). Taking into account momentum conser- vation, the force exerted per unit time by collisions on the windward side is Fin ⊥,W= (n0+ ∆n)AxmZ∞ vTM(vx−vTM)2f(vx)dvx,(7) where mis the molecular mass. Similarly, the force ex- erted per unit time by collisions on the leeward side is Fin ⊥,L=−n0AxmZvTM −∞(vx−vTM)2f(vx)dvx. (8) Note that the leeward side is in the external environ- ment, and the number density remains n0. From the Eq. (5), it can be seen that ∆ n=−n0τ0vTM/h0. Tak- ing into account that in gravity-related experiments, vTM is very small, only the first-order terms are retained in the subsequent derivation. The total damping force from molecules incident on surfaces perpendicular to the di- rection of motion is then approximately given by Fin ⊥≈ −2Axn02mkBT π1/2 vTM−Axn0τ0kBT 2h0vTM. (9) Damping force of molecules desorbed from the xsur- face.— Since molecules lose memory after colliding with the surface, the incident and outgoing processes are in- dependent. Therefore, the force exerted by desorbed molecules on the TM can be calculated separately. The cosine law ensures that the incident distribution is the same as the outgoing distribution when taking the TM as the stationary coordinate system [76–78]. Due to the identical distributions, the outgoing process is equivalent to the incident process, and Eq. (6) can be used directly. As the TM moves, the number of incident molecules will change, so will the number of outgoing molecules. In thisscenario, we refer to the outgoing number as the effective incident number. From Eq. (6), after integrating the velocity vx, only the variables nandTremain. Under isothermal conditions, Tremains constant, so the only variable that can change isn=n0+ ∆n. Therefore, the difference between the real incident number and the effective incident number lies in the effective number densities n′. By setting the original incident number equal to the outgoing number per unit time, we can write the following identity: −n′ WZ0 −∞vxf(vx)dvx= (n0+∆n)Z∞ vTM(vx−vTM)f(vx)dvx, (10) Considering that as vTM→0, Erfhp m/(2kBT)vTMi → 0, and exp[ −mv2 TM/(2kBT)]→1, Eq. (10) can be solved to obtain the effective number density on the windward side n′ W= (n0+ ∆n)" −vTMπm 2kBT1/2 + 1# . (11) Similarly on the leeward side, n′ L=n0" vTMπm 2kBT1/2 + 1# . (12) Thus, the total damping force generated by molecules emitted parallel to the direction of motion is: Fout ⊥=−n0Axπmk BT 21/2 vTM−Axn0τ0kBT 2h0vTM. (13) Damping forces on yorzsurface. — Since the three di- rections of the Maxwell-Boltzmann velocity distribution are completely independent, we can first use the vydis- tribution to get the numberR vydNy,vyhitting the side, and then use this number to calculate the damping coeffi- cient in the x-axis. Since the zdirection is perpendicular to the direction of motion of the TM, it does not need to be calculated, and only the velocity distribution of vxin thexdirection is considered. Such analysis also applies to surface z. Since vy/zis perpendicular to the direction of motion, the motion of the TM does not affect the velocity distri- bution and integration limits during calculation. From this, we can get the force per unit area on the side dFy/z,v x=m(vx−vTM)f(vx)dvxdAy/zZ vy/zdNy/z,v y/z. (14) The outgoing molecule velocities cancel each other out in all directions, so the net force exerted in the xdirection is zero. Thus, we obtain the damping force on a single lateral surface: F∥,y/z=−n0Ay/zmkBT 2π1/2 vTM. (15)5 Finally, combining Eq. (9), (13), and (15), we can get the total damping force and then the damping coefficient βB=(4Ax+πAx+ 2Ay+ 2Az)p0m 2πkBT1/2 +Axp0τ0 h0,(16) where p0=n0kBTis the pressure at equilibrium in the gap. B. Sensor core finite open system As depicted in Fig. 2, the sensor core used in GRSs for space GW detection missions such as LISA and similar projects [10–13] is characterized by a cuboid nested struc- ture. This structure is also employed in global gravity field recovery missions albeit with variations in size [14– 16, 27, 63, 64, 80], and serves to monitor the real-time position of the TM by connecting to the front-end circuit and apply electrostatic forces to control the TM’s behav- ior. The sensor core forms a closed gap structure between the TM and the electrode housing. However, for a gap on one side, molecules can exchange with other gaps, form- ing a finite open structure. In Fig. 2, we decompose it into six bi-plate finite open systems, Gx1,Gx2,Gy1,Gy2, Gz1,Gz2, and handle them separately. Electrode HousingTest Mass 2 12 1 FIG. 2. Schematic side view of the sensor core in GRS. The sensor core consists of a TM in the middle and an electrode housing surrounding it, with a gap between them. The gap can be divided into six bi-plate gaps, i.e., bottom gaps Gx1, Gx2, and lateral gaps Gz1,Gz2,Gy1,Gy2, and the last two are not shown. Damping force of incident molecules on xsurface. — In a manner analogous to the previous subsection III A, the force exerted by all molecules hitting the windward face per unit time remains the same as in Eq. (7). Thechange in the number density in gap Gx1is opposite to the change in number density in gap Gx2, i.e., ∆ nx2= −∆nx1=−∆n. Consequently, the force exerted by all molecules hitting the leeward face per unit time becomes Fin ⊥,L=−(n0−∆n)AxmZvTM −∞(vx−vTM)2f(vx)dvx. (17) The total damping force of molecules incident perpen- dicular to the direction of motion is then approximately given by Fin ⊥≈ −2Axn02mkBT π1/2 vTM−Axn0τ0kBT h0vTM. (18) Damping force of molecules desorbed from xsurface. — The effective molecular number density on the windward face is the same as in Eq. (11), while on the leeward side, the effective number density becomes n′ L= (n0−∆n)" vTMπm 2kBT1/2 + 1# , (19) The total damping force generated by molecules emitted perpendicular to the direction of motion is given by Fout ⊥=−n0Axπmk BT 21/2 vTM−Axn0τ0kBT h0vTM. (20) Damping forces on yorzsurface. — As the TM moves up and down, the number of molecules in the gaps Gx1 andGx2changes due to diffusion, causing changes in the number ∆ Nsideand number density of molecules in the lateral gap ( Gy1,Gy2,Gz1,Gz2). Molecules diffuse into gapGx1while diffusing out of gap Gx2, resulting in a den- sity gradient in the x-axis direction on the lateral gaps. However, according to Eq. (14), the molecular density on the side is linearly superposed, allowing for non-uniform density distribution on the side. The side force can be reexpressed as F∥,y/z=−Ay/zn0mkBT 2π1/2 vTM −∆Nside,y/z hy/zmkBT 2π1/2 vTM.(21) Condsidering the number density changes in the gaps Gx1andGx2are ∆ nx1and ∆ nx2, and the instanta- neous volumes are Vx1=Ax(h0+Bcos(ωt)) and Vx2= Ax(h0−Bcos(ωt)), respectively. Thus, we obtain ∆Nside=−(∆nx1Vx1+ ∆nx2Vx2) =4n0Bτ0Ax h0cos(ωt)vTM,(22)6 Substituting Eq. (22) into Eq. (21), F∥,y/z≈ −Ay/zn0mkBT 2π1/2 vTM. (23) Unlike the gaps Gx1andGx2, the fluctuation of number density does not affect the lateral damping force. Combining Eqs. (18), (20), and (23), we can obtain the total damping force and then the damping coefficient βS=(4Ax+πAx+ 2Ay+ 2Az)p0m 2πkBT1/2 +2Axp0τ0 h0.(24) The first term in Eqs. (16) and (24) is consistent with the free damping in the infinite gas volume [36], while the second term is consistent with the proximity damping in the isothermal piston model [32, 52]. The first term can be regarded as a basic damping that always exists regardless of the environment, whereas the second term represents damping induced by molecular diffusion. We refer to them as base damping and diffusion damping, respectively. The models described above are denoted as finite open models. Finally, we achieve the natural separation of damping coefficients within the finite open systems. IV. SEPARATION AND SOURCES OF DAMPING FROM AN ENERGY PERSPECTIVE As shown in Fig. 1, when the molecules desorb from the moving TM surface, in addition to the velocity given by the Maxwell-Boltzmann distribution, there is also a background translational velocity of the TM. In other words, the TM only affects the energy of the molecules in translational degrees of freedom. Using subscripts u for gas molecules desorbed from the TM surface, and d for those desorbed from the fixed plate below, with the total gas molecules indexed as i, the total energy can be expressed as E=1 2X imv2 i=1 2X dmv2 d+1 2X umv2 u ≈1 2X imv′2 i+1 2X umv2 TM+X umv′ u·vTM,(25) where v′ iandv′ u=vu−vTMrepresent the velocities of molecules in the center-of-mass frame assuming that the TM does not move. The first term in Eq. (25),P imv′2 i/2≈P dmv2 d/2 +P umv′2 u/2, assumes that the number Nuof molecules desorbed from the TM surface and the number Nddesorbed from the bottom fixed plate are approximately equal, Nu≈Nd. Molecules desorb from the TM surface and emit to the fixed plate, and then back to the TM surface, this process takes a very short time, ∆ t∼10−5s, while the TM’s velocity is veryslow, vTM<10−8m/s. In other words, during this time, Nd(t0+ ∆t)≈Nu(t0), so this approximation is reason- able. The total number N(t) =Nu+Ndchanges due to the diffusion of gas molecules. Letv′ uxbe the projection of v′ uin the direction of the TM motion, its probability density distribution is 2f(v′ ux). Therefore, the third term can be re-expressed as X umuv′ u·vTM=Nu2mkBT π1/2 vTM. (26) Considering the Knudsen hypothesis, the Eq. (25) can be re-expressed as E(t) =3 2N(t)kBT+1 4N(t)mv2 TM+N(t)mkBT 2π1/2 vTM. (27) The first term is consistent with the isothermal internal energy of the gas in an equilibrium state, and the second and third terms are related to the translational velocity of the TM, representing additional energy introduced by non-equilibrium processes. This result is different from the K¨ onig’s theorem [81], which states that the energy of the system can be separated into the translational kinetic energy of the center-of-mass and the relative kinetic en- ergy within the center-of-mass system. In the Maxwell- Boltzmann distribution, the overall translational veloc- ity of the gas is zero. In other words, when we refer to temperature, this means that the system is discussed in the center-of-mass system. Thus, it is inaccurate to use isothermal gas or effective temperature assumptions in this system. In the free molecular flow regime, molecules do not collide, thus energy cannot be evenly distributed among the degrees of freedom [82], except for the trans- lational degrees of freedom related to the TM. According to analytical mechanics, the damping co- efficient means the dissipation of energy, and there are three pathways of energy dissipation in the bi-plate gap. When molecules are incident on the TM surfaces, the in- cident damping forces, i.e. Eqs. (9) and (18) correspond to the energy dissipation of the TM. In Fig. 1, due to the motion of the TM, the molecules desorbed from the TM will dissipate more kinetic energy of the TM to the fixed plate. This part of the energy results in the addi- tion of the last two terms in Eq. (27). For the bi-plate size at the bottom gaps of the sensor core for LISA mis- sion, only about 9% molecules desorbed from the TM (see Appendix B) leave the gap directly. Similarly, the ratio from the environment directly to TM is the same. The ratio is proportional to the energy dissipated, so the majority of the dissipated energy flows to the fixed plate and the TM, with a very small amount flowing to the environment. Compared to base damping, diffusion damping ac- counts for changes in number density resulting from the diffusion process. Since the change in number density is7 in counter-phase with TM’s velocity, it is equivalent to increasing the number of incident and outgoing molecules per unit time, which in turn amplifies damping force or energy dissipation. Consequently, the dissipation flow from the gap to the environment as noticed in Ref. [32] is not a leading term to the diffusion damping of the energy dissipation. Instead, as mentioned above, this en- ergy is primarily dissipated into the TM and the fixed plate. In isothermal piston models, the utilization of the ideal gas state equation implies that compression is a quasi-static process, wherein only the diffusion damping resulting from changes in number density is considered. As a result, its internal energy corresponds solely to the first term in Eq. 27, excluding the last two terms associ- ated with the damping of the non-equilibrium part. V. APPLICATIONS The investigation of residual gas noise holds significant importance in gravity-related experiments. Residual gas damping noise typically determines the limit of sensitiv- ities in many high-precision gravity-related experiments. In the subsequent sections, we apply the sensor core and bi-plate finite open models to space gravitational wave detection and the test of the gravitational inverse-square law, respectively. A. Space gravitational wave detection The discovery of GWs has positioned it as a new mes- senger for observing the universe. LISA is a spaceborne GW detection mission proposed by the European Space Agency [10, 11, 83]. To investigate the residual gas damp- ing noise used in such missions, the University of Trento (UTN) employs a torsion balance to measure the decay curves in the GRSs [31]. The 4TMs torsion balance experiment setup is shown in Fig. 3. The experimental temperature is 293 K, and the molecular mass utilized in the simulation is 30 u. By measuring the amplitude decay time during free pendu- lum motion, the torsional damping coefficients can be ob- tained. The experimental results indicate that the resid- ual gas damping coefficient is much larger than the theo- retical prediction of infinite volume damping. However, it is in good agreement with UTN’s simulation predictions, as shown in Fig. 4. Since the torsion balance measures the torsional damp- ing coefficient, the translational damping coefficient βSof the finite open model is needed to convert to the 4TM ro- tational damping coefficient β4TM rot=r2βtran+βrot. The GRS Mirror S2TMTM EH EH(a) (b)(c) FIG. 3. Schematic drawing of the 4TMs torsion balance ex- periment setup (not to scale). (a) A cross-shaped structure is suspended by a wire, with four hollow cubic TMs attached at each end. The side length of each TM is s= 46 mm, and the pendulum arm length is r= 0.1065 m. (b) The gap spacings between the GRS’s TM and the electrode housing (EH) in the x,y, and zaxes are 4 .0 mm, 2 .9 mm, and 3 .5 mm, respec- tively. (c) The gap spacings between the S2’s TM and the EH in the x,y, and zaxes are 8 .0 mm, 6 .0 mm, and 8 .0 mm, respectively [31]. total rotational damping coefficient is given by β4TM rot=r2p0s2" 2τside(hx) hx+32m πkBT1/2 1 +π 8# +1 6p0s4τside(hx) hx+τside(hz) hz +p0s42m πkBT1/2 1 +π 12 . (28) In Fig. 4, the result of the finite open model demon- strates a 62% improvement compared to the constrained volume model [39], and exhibits an 8% difference from the UTN’s simulation data. The latter utilizes the dif- fusion time under bi-plate structures, but the electrode housing alters the diffusion conditions at the bi-plate boundaries, subsequently changing the diffusion time un- der the bi-plates. This alteration increases the diffusion time. Finally, we substitute the simulation result into the finite open model, with τside(hx) = 3 .22×10−4s and τside(hz) = 3 .02×10−4s, while τcuboid = 1.31×10−4s. The 8% discrepancy may come from the gradient of molecular number density in the lateral gap ( Gy1,Gy2, Gz1,Gz2) in the x-axis. This might lead to the deviation, i.e., the N0/τ0term in Eq. (A1) is not accurate. The results in Fig. 4 demonstrate the effectiveness of the finite open model. Ultimately, we apply the finite open model to the analysis of the residual gas damping noise in the GRS’s sensitive axis for the LISA space GW detection mission [31] and other LISA-like space GW de- tection missions, such as Taiji [13], TianQin [12]. This noise largely determines the detection sensitivity of GRSs in the primary frequency range. For the 2 kg TM moving8 0 1 2 3 4010203040Experimental data fit of UTN Simulation of UTN Our model Constrained volume model Infinite volume model FIG. 4. Comparison of different rotational damping coeffi- cients in the 4TM torsion balance experiment. The black dashed line (+) and black dashed line ( ×) represent the ex- perimental measured values and simulation values from the UTN, respectively [31]. The green solid line (+) represents the theoretical values from Eq. (28), i.e., our finite open model in this paper, and the short dashed gray line ( ×) represents the theoretical values from the constrained volume model [39]. The short dashed gray line represents the theoretical result of damping coefficients in an infinite volume [36]. along the sensitive axis ( x-axis), we can use Eqs. (1) and (28) to estimate the residual gas acceleration noise, that is S1/2 a=1.23×10−15m s−2Hz−1/2 ×p 10−6Pa1/2m 30 u1/4T 293 K1/4 .(29) Under the same conditions, compared to Ref. [39], our result is closer to the simulation result S1/2 a= 1.3× 10−15m s−2Hz−1/2from Ref. [31], with only a 5% dif- ference. The validity of the finite open model also corrects the previous view on the diffusion time in the sensor core. Ref. [31] states that the diffusion time is the average time for molecules to diffuse from gap Gx1toGx2. However, as shown in Eq. (22), we found that the molecular number changes from Gx1andGx2have a negligible second-order effect on the y/zsurfaces, and the diffusion time can be approximated as the time to diffuse just from the bottom gap ( Gx1orGx2) to the lateral gap ( Gy1,Gy2,Gz1,Gz2). B. Test of the gravitational inverse-square law To reconcile general relativity and the standard model, string theory or M-theory predicts deviations from the ISL at short distances [84, 85]. Huazhong University of Science and Technology (HUST) designs a torsion bal- ance and a eightfold azimuthal symmetric attractor ex- Conducting membranePendulumVacuum feedthrough Fiber Clamp111 1 2 22 2FIG. 5. Schematic drawing of the test of the ISL experiment setup (not to scale). An I-shaped pendulum is suspended from the bottom of the torsion balance through tungsten wire, and the pendulum faces the position of the attractor. The middle part of the pendulum is a glass block (M), and two glass bases ( G1,G2) are symmetrically installed at both ends. Two glass substrates ( Gt1,Gt2), two tungsten TMs ( Wt1,Wt2), and two gravitational compensation pieces ( Wtc 1,Wtc 2) are glued to the glass bases opposite the attractor. A special glass clamp is installed on the top of the middle glass block, it can fix the suspended tungsten wire to the center of the pendulum. On the left side of the conducting membrane, there is an attractor not shown here. The attractor consists of 8 tungsten source masses and 8 tungsten compensation masses, arranged alternately on a rotatable glass disk [19]. perimental platform, and tests the ISL at short distances by dually modulating the signal of interest and the grav- ity calibration signal [18–23]. The experimental setup of the test of the ISL is illus- trated in Fig. 5. To facilitate the experimental design of the gap spacing between the pendulum and the conduct- ing membrane, it is necessary to assess the influence of residual gas damping noise with varying gap spacings. To obtain the constraints on gap spacing hcunder different gas pressures p0[38], the gas damping noise is equated to the torsion balance’s internal damping thermal noise. The internal thermal noise of the torsion balance is Sth(ω) = 4 kBTk/(ωQ)≈3.61×10−30N2m2Hz−1. Here, Qis the quality factor of the torsion balance, ω is the resonant frequency, and kis the spring constant. According to the bi-plate finite open model, the fluctu- ation torque noise power spectral density Srot,Mof the middle glass block, and Srot,G,Wt of the glass blocks ( G1, G2), TMs ( Wt1,Wt2) and gravitational compensation pieces ( Wtc 1,Wtc 2) at both ends can be obtained in Appendix C. From this, the relation p0=f(hc) =Sth (Srot,M+Srot,G,Wt )/p0, (30)9 is obtained, and consequently hc=f−1(p0). (31) 02040608010020406080100Non-isothermal model Our model FIG. 6. Comparison of gap spacing constraints under dif- ferent pressures between the finite open model and the non- isothermal model. The shaded area below the green solid line represents the constraints given by Eq. (31), i.e., our bi-plate finite open model in this paper, and the shaded area below the gray dashed line represents the constraints given by the non-isothermal model [38]. Given temperature T= 297 K, molecular mass 19 .1 u, and thermal adjustment coefficient σ= 1, the results are shown in Fig. 6. Due to the very small gap spacing, ∼100µm, the diffusion damping is magnified, and the base damping is almost negligible. In the range of 10 to 100µPa, the constraints on the gap spacing hcobtained by the finite open model are lower by about 28% com- pared to the non-isothermal model. This suggests that the additional temperature effect in the non-isothermal model is almost negligible. These results are crucial for reducing the test length scale λin ISL experiments, pro- viding valuable insights into the constraints imposed by residual gas damping noise on experimental parameters. VI. DISCUSSIONS AND CONCLUSIONS We separate the damping coefficients for two finite open systems, i.e., the bi-plate system and the sensor core system, into base damping and diffusion damping through rigorous derivation. This effectively elucidates the relationship between the free damping and the prox- imity damping. This separation needs to meet the con- dition ωτ0→0, which holds for most high-precision gravity-related experiments. In our derivation, the key lies in how to deal with the change in molecular number density. The change is so small compared to the overall density that it can be almost ignored [52], as evidenced in the LISA’s GRSs (∆ n/n 0<10−11). However, detailedderivation shows that the results corresponding to such a small amount can not be ignored, which renders high- precision measurements truly distinctive. We address the impact of changes in number density within the bottom gaps on the lateral gaps, which also reveals the possibil- ities for analyzing cross-talk effects in residual gas noise between the sensitive axis and other axes. The diffusion time τis used to quantitatively charac- terize the degree of finite openness in the system, which is different from the constrained volume. It is general and can account for the effects of different shapes of the structure around the TM. However, in a sensor core with lateral walls, an increase in constrained volume does not necessarily equate to a reduction in diffusion time. Inter- estingly, as τ→0, i.e., the system is completely open, the diffusion damping disappears, and the damping becomes consistent with the free damping in an infinite volume. Conversely, as τ→ ∞ , i.e., the system is completely closed, in contrast to the zero damping of the isother- mal piston model [32, 52], there still exists base damping independent of gap spacing. This result is reflected in the lower limit of the power spectral density shown in Ref. [32]. In brief, the concept of finite openness can characterize damping more comprehensively and accu- rately. In terms of the physical picture, the previous analyses of damping in an infinite volume and a constrained vol- ume are disjointed. The finite open models presented in this paper provide a complete microscopic picture that includes both types of damping. This allows us to point out three pathways of energy dissipation of the TM from the microscopic level in the bi-plate gap, that is: energy absorbed by the TM, assimilated by the fixed plate, and a very small portion directly diffused into the environment. In practical applications, for space GW detection, our model predicts a damping coefficient for the 4TM torsion balance experiment 62% higher than previous analyses, and is the closest to the experimental and simulation data to the best of our knowledge. For the LISA mis- sion, our theoretical estimation of the residual gas accel- eration noise along the sensitive axis aligns most closely with simulation results, differing only by 5%. These re- sults validate the effectiveness of the finite open models. And these results are also applicable to other LISA-like space GW detection missions, such as Taiji [13], Tian- Qin [12]. In the test of the ISL, our model reduces the constraints on the gap spacing between the TM and the conducting membrane by about 28%, compared with the non-isothermal model. And this result may further help to reduce the test length scale in ISL experiments. Furthermore, our theory can also be applied to other high-precision measurements, such as ultra-sensitive space accelerometers [27, 80, 86, 87], measurements of the gravitational constant G[24–26], MEMS/NEMS ac- celerometers [88, 89] and Casimir force measurements [90, 91].10 ACKNOWLEDGMENTS We thank Profs. Yun-Kau Lau and Lamberto Rondoni for their inspiring discussions and advice, as well as Drs. Zuolei Wang and Da Fan for their help. This work is sup- ported by the National Key Research and Development Program of China under Grant No. 2020YFC2200601, the National Natural Science Foundation of China under Grants No. 12175090, No. 12305045 and No. 12247101, and the 111 Project under Grant No. B20063. Appendix A: The Two Terms in the Equation (2) The first term. — Suppose the average diffusion time for one molecule to escape from the gap is τ(h), and N/τ is the number of escaping molecules per unit time, then d∆N=−N(t) τ(h)dt+N0 τ0dt, (A1) where ∆ Nis the net number of diffusing molecules, N0 andτ0are at the equilibrium point. Equation (A1) is also known as the Fick rule [38]. Considering gap spacing change ∆ h≪h0, we can have the approximation τ(h)≈ τ0. In this case, dividing Eq. (A1) by d tandV0gives 1 V0d∆N dt=−n(t) τ0+n0 τ0, (A2) where n0=N0/V0. In addition, considering ∆ V(t)≪ V0, the approximation N(t)/V0≈n(t) is made. The second term. — Assuming the TM vibrates near the equilibrium point, in the frequency domain, the dis- placement relative to the equilibrium point is ∆ h(t) = Bcos(ωt), and the instantaneous volume of the gap is V(t) =Ax(h0+Bcos(ωt)) = Axh(t), where Axis the surface area of the TM perpendicular to the x-axis. Thus, the second term on the right side of Eq. (2) can be written as −N0 V2 0d∆V dt=−n0d dtAxh V0 =−n0 h0dh dt. (A3) Appendix B: The proportion of Molecules Escaping Directly from the Test Mass Surface to the Environment Molecules undergo an average of Ncollisions inside the gap before leaving. The last time the molecules escape directly from the TM surface, the proportion is 1 /2, so the proportion of molecules escaping directly from the TM surface to the environment is approximately given by Pout≈1 21 N 2≈0.09, (B1)where we use the conclusion from Ref. [32]. For the size of sensor core for LISA mission, Nis approximately given by N ≈R2 h2lnh 1 +R h
2i≈11, (B2) where the radius Ris obtained by assuming that the area of the square is consistent with the area of the equiva- lent circle. The proportion of molecules escaping directly from the TM surface to the environment is 0.09, which is a small fraction. Appendix C: Fluctuating Force Power Spectral Density in ISL Experiment According to the theoretical model of the bi-plate finite open system, the fluctuating torque noise Srot,Mfor the middle glass block can be obtained as follows: Srot,M=s3 M,xsM,z 632mkBT π1/2 1 +π 4 p0 + s3 M,xsM,z 2+s3 M,xsM,y 6+s3 M,ysM,x 6! ×8mkBT π1/2 p0.(C1) The torque noise Srot,G,Wt for the glass blocks ( G1, G2), the TMs ( Wt1,Wt2), and gravitational compensa- tion pieces ( Wtc 1,Wtc 2) at both ends is given by: Srot,G,Wt = 8kBTDl2p0, (C2) where D=(4sG,xsG,z+πsG,xsG,z+ 2sG,ysG,z+ 2sG,xsG,y −sM,zsM,y)m 2πkBT1/2 +sWt,xsWt,zτ0,Wt hc +sWtc,x sWtc,z τ0,Wtc hc+ (sGt,y+sWt,y−sWtc,y ), (C3) where, the arm length from the suspension point to the center of the TM is l= (sG,x+sM,x)/2 = 38 .0605 mm, and the dimensions are given as sM,x×sM,y×sM,z= 61.491×8.000×12.000 mm3,sG,x×sG,y×sG,z= 14.630×19.756×27.138 mm3,sGt,x×sGt,y×sGt,z= 14.630×0.486×12.003 mm3,sWt,x×sWt,y×sWt,z = 14.630×0.200×12.003 mm3,sWtc,x×sWtc,y×sWtc,z = 14.630×0.606×15.139 mm3. The diffusion time τ0,Wtand τ0,Wtc useτcuboid in Appendix D. The parameters here may differ somewhat from those in Refs. [19, 37, 38]. The parameters in this paper are for reference only. Considering that the base damping here is almost neg- ligible, theoretically, when the diffusion time used in the11 non-isothermal model is the same as the diffusion time formula used in the bi-plate finite open model, the ratio of the power spectral density of fluctuating forces between the two models is approximately 7:6. When the damp- ing noise is equal to the intrinsic damping thermal noise, under the same pressure, the ratio of the gap spacing hc constraints between the two models is approximately 7:6. In other words, the latter is about 1/7 lower than the for- mer. As long as the base damping can be neglected, this result will not change with variations in experimental pa- rameters.Appendix D: Diffusion Time for Bi-plate Finite Open Systems Reference [39] provides theoretical result for the diffu- sion time τcuboid of square bi-plates: τx=bp 72πkBT/m1 ς√ 1 +ς2−ς
×2 arctan(1 /ς) + 2ςlnς−ςln(1 + ς2) 2ς−2√ 1 +ς2+ ln 1 +√ 1 +ς2
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2024-01-09

The residual gas damping of the test mass (TM) in the free molecular flow regime is studied in the finite open systems for high-precision gravity-related experiments. Through strict derivation, we separate the damping coefficients for two finite open systems, i.e., the bi-plate system and the sensor core system, into base damping and diffusion damping. This elucidates the relationship between the free damping in the infinite gas volume and the proximity damping in the constrained volume, unifies them into one microscopic picture, and allows us to point out three pathways of energy dissipation in the bi-plate gap. We also provide the conditions that need to be met to achieve this separation. In applications, for space gravitational wave detection, our results for the residual gas damping coefficient for the 4TM torsion balance experiment is the closest one to the experimental and simulation data compared to previous models. For the LISA mission, our estimation for residual gas acceleration noise at the sensitive axis is consistent with the simulation result, within about $5\%$ difference. In addition, in the test of the gravitational inverse-square law, our results suggest that the constraint on the distance between TM and the conducting membrane can be reduced by about $28\%$.

Damping Separation of Finite Open Systems in Gravity-Related Experiments in the Free Molecular Flow Regime

2401.04808v1

arXiv:0710.2826v2 [cond-mat.mes-hall] 17 Oct 2007Ferromagnetic resonance study of polycrystalline Fe 1−xVxalloy thin films J-M. L. Beaujour, A. D. Kent Department of Physics, New York University, 4 Washington Pl ace, New York, NY 10003, USA J. Z. Sun IBM T. J. Watson Research Center, Yorktown Heights, NY 10598 , USA (Dated: November 18, 2018) Ferromagnetic resonance has been used to study the magnetic properties and magnetization dy- namics of polycrystalline Fe 1−xVxalloy films with 0 ≤x <0.7. Films were produced by co- sputtering from separate Fe and V targets, leading to a compo sition gradient across a Si substrate. FMR studies were conducted at room temperature with a broadb and coplanar waveguide at fre- quencies up to 50 GHz using the flip-chip method. The effective demagnetization field 4 πMeffand the Gilbert damping parameter αhave been determined as a function of V concentration. The results are compared to those of epitaxial FeV films. I. INTRODUCTION A decade ago, it was predicted that a spin polar- ized current from a relatively thick ferromagnet (FM) could be used to switch the magnetization of a thin FM [1]. Since then, this effect, known as spin-transfer, has been demonstrated in spin-valves [2] and magnetic tun- nel junctions [3]. In a macrospin model with collinear layer magnetizations, there is a threshold current den- sityJcfor an instability necessary for current-induced magnetization switching of the thin FM layer [1, 4]: Jc=2eαMstf(Hk+2πMs) /planckover2pi1η, (1) whereαisthedampingconstant. tfandMsarethethick- ness and the magnetization density of the free layer, re- spectively. Hkis the in-plane uniaxial anisotropy field. η is the currentspin-polarization. In orderfor spin-transfer to be used in high density memory devices Jcmust be re- duced. From Eq. 1 it is seen that this can be achieved by employing materials with low Msandαin spin-transfer devicesor, equivalently materialswith lowGilbert damp- ing coefficients, G = αMs(gµB//planckover2pi1). Very recently, an experimental study of epitaxial FeV alloy thin films demonstrated a record low Gilbert damp- ing coefficient [5]. This material is therefore of interest for spin transfer devices. However, such devices are gen- erally composed of polycrystalline layers. Therefore it is of interest to examine polycrystalline FeV films to assess their characteristics and device potential. In this paper, we present a FMR study of thin poly- crystalline Fe 1−xVxalloy films with 0 ≤x <0.7 grown by co-sputtering. The FeV layers were embedded be- tween two Ta |Cu layers, resulting in the layer structure ||5 Ta|10 Cu|FeV|5 Cu|10 Ta||, where the numbers are layer thickness in nm. FeV polycrystalline films were prepared by dc magnetron sputtering at room tempera- ture from two separate sources, oriented at a 45oangle (Fig. 1a). The substrate, cut from a Silicon wafer with 100 nm thermal oxide, was 64 mm long and about 5 mm wide. The Fe and V deposition rates were found to vary/s49 /s50 /s51 /s52/s45/s50/s45/s49/s48/s40/s98/s41 /s32/s86/s32/s116/s97/s114/s103/s101/s116 /s52/s53/s111 /s32 /s32/s115/s117/s98/s115/s116/s114/s97/s116/s101/s70/s101/s32/s116/s97/s114/s103/s101/s116/s40/s97/s41 /s72 /s114/s101/s115/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s108/s105/s110/s101/s32/s32/s40/s97/s117/s41 /s72 /s97/s112/s112/s32/s32/s40/s107/s79/s101/s41/s49/s52/s32/s71/s72/s122 /s32/s32/s32 /s32/s120 /s61/s48/s46/s51/s55/s120 /s61/s48/s46/s53/s50 /s120 /s61/s48/s46/s49/s57/s72 /s115/s117/s98/s115/s116/s114/s97/s116/s101/s32/s104/s111/s108/s100/s101/s114 /s97/s120/s105/s115 FIG. 1: a) The co-sputtering setup. b) Typical absorp- tion curves at 14 GHz for a selection of ||Ta|Cu|7.5 nm Fe1−xVx|Cu|Ta||films, with x=0.19, 0.37 and 0.52. The res- onance field Hresand the linewidth ∆ Hare indicated. linearly across the wafer. The Fe and V rates were then adjusted to produce a film in which xvaries from 0.37 to 0.66 across the long axis of the wafer. The base pres- sure in the UHV chamber was 5 ×10−8Torr and the Ar pressure was set to 3.5 mTorr. The FeV was 7.5 nm in thickness, varying by less than 0.3 % across the sub- strate. An Fe 1−xVxfilm 3 nm thick was also fabricated. To produce films with x <0.30 the rate of the V source was decreased. Finally, pure Fe films with a thickness gradient ranging from 7 nm to 13.3 nm were deposited. TheFMRmeasurementswerecarriedoutatroomtem- perature using a coplanar wave-guide (CPW) and the flip-chip method. Details of the experimental setup and ofthe CPWstructuralcharacteristicscan be found in [6]. Adc magnetic field, up to 10 kOe, wasapplied in the film plane, perpendiculartotheacmagneticfield. Absorption lines at frequencies from 2 to 50 GHz were measured by monitoring the relative change in the transmitted signal as a function of the applied magnetic field.2 /s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s48/s53/s49/s48/s49/s53/s50/s48 /s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s50/s46/s48/s53/s50/s46/s49/s48/s50/s46/s49/s53/s49/s50/s51 /s120 /s32/s52 /s77 /s101/s102/s102/s32/s32/s40/s107/s71/s41/s55/s46/s53/s32/s110/s109/s32/s70/s101 /s49/s45 /s120/s86 /s120 /s51/s32/s110/s109/s32/s70/s101 /s48/s46/s54/s51/s86 /s48/s46/s51/s55 /s49/s50/s46/s57/s32/s110/s109/s32/s70/s101 /s120/s32 /s32/s103/s45/s102/s97/s99/s116/s111/s114/s40/s98/s41/s40/s99/s41/s32 /s32 /s32/s72 /s114/s101/s115/s32/s32/s40/s107/s79/s101/s41 /s49/s49/s32/s71/s72/s122/s40/s97/s41 FIG. 2: a) The resonance field at 11 GHz versus xand b) the effective demagnetization field versus x. The solid line is a guide to the eye. c) The Land´ e gfactor as a function of x. The dotted line shows the g-factor value of bulk Fe. II. RESULTS Typical absorption lines at 14 GHz of selected FeV al- loyfilmsareshowninFig. 1b. Thelinesarelorentzianfor most frequencies. At a fixed frequency, the FMR absorp- tion decreaseswith increasingV content. The FMR peak of a film 7.5 nm thick with x= 0.66 is about 100 times smallerthanthatofapureFeofthesamethickness. This is accompaniedby a shift of Hrestowardshigher field val- ues (Fig. 2a). The effective demagnetization field 4 πMeff and the Land´ eg-factorgwere determined by fitting the frequency dependence of the resonance field Hresto the Kittel formula [7]: f2=/parenleftBiggµB h/parenrightBig2 Hres(Hres+4πMeff),(2) where the effective demagnetization field is: 4πMeff= 4πMs−H⊥. (3) Note that in the absence of a perpendicular anisotropy fieldH⊥, the effective field would be directly related to Ms. The dependence of 4 πMeffon V concentration is shown in Fig. 2c. As xincreases the effective demagneti- zation field decreases dramatically, going from about 16 kG forx= 0 to 1.1 kG for x= 0.66. Note that the effec- tive demagnetization field of the 7.5 nm Fe film is about 25 % lower than that of bulk Fe (21.5 kG). The 12.9 nm Fe film exhibits a larger 4 πMeff, which is, however, still lower than 4 πMsof the bulk material. Similarly, the 4πMeffof an Fe 0.63V0.37film is thickness dependent: decreasing with decreasing layer thickness. The Land´ eg-factor increases monotonically with in- creasing V concentration (Fig. 2b). The minimum g- factor is measured for the Fe film: g= 2.11±0.01, which is slightly larger than the value of bulk material (g= 2.09). Note that gof a Fe film 12.9 nm thick is equal 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 /s48/s46/s48/s48/s46/s51/s48/s46/s54 /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 /s41/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s32/s40/s107/s79/s101/s41/s32 /s32/s72 /s32/s32/s40/s107/s79/s101/s41 /s72 /s48/s72 /s48/s72 /s49/s52/s32/s32 /s32/s40/s97/s41 /s120/s32 /s32/s102/s32/s32/s40/s71/s72/s122/s41 FIG. 3: a)Frequency dependence of the linewidth for 7.5 nm Fe1−xVxalloy film with x=0, 0.43 and 0.52. The solid lines are the best linear fit of the experimental data. b) ∆ H14, the linewidth at 14 GHz , and ∆ H0are shown as a function of x. c) The magnetic damping parameter versus V concentration. Fe0.63V0.37, does not appear to be thickness dependent: the 3 nm Fe 0.63V0.37layer has about the same gvalue than the 7.5 nm Fe 0.63V0.37layer. The half-power linewidth, ∆ H, was studied as a func- tion of the frequency and of the V concentration. Fig. 3b shows the dependence of the FMR linewidth on xat 14 GHz. The general trend is that ∆ Hincreases with x. However, there are two regimes. For x >0.4, the linewidth depends strongly on x, increasing by a factor 5 whenxis increased from 0.4 to 0.66. The dependence of the linewidth on xis more moderate for the films with x <0.4: it increases by about 30 %. For all samples, the linewidth scales linearly with the frequency. A least square fit of ∆ H(f) gives ∆ H0, the intercept at zero frequency, and the Gilbert damping parameter αwhich is proportional to the slope: d∆H/df= (2h/gµB)α[8]. ∆H0is typically associated with an extrinsic contribu- tion to the linewidth and related to magnetic and struc- tural inhomogeneities in the layer. For two samples with the highest Vanadium concentration, x= 0.60, 0.66, the linewith is dominated by inhomogeneous broadening and it wasnot possible to extract α. Asxincreases, ∆ H0and αincreases. The damping parameter and ∆ H0remain practically unchanged for x≤0.4 and when x >0.4, both the intercept and the slope of ∆ Hversusfincrease rapidly. III. DISCUSSION Several factors can contribute to the dependence of 4πMeffon the V concentration. First, the decrease of the effective demagnetizationfield canbe associatedwith the reduction of the alloy magnetization density Mssince the Fe content is reduced. In addition, a neutron scat- tering study showed that V acquires a magnetic moment3 antiparallel to the Fe, and that the Fe atom moment de- creases with increasing V concentration [9]. The Curie temperature of Fe 1−xVxdepends on x. In fact, Tcfor x=0.65 is near room temperature [10]. It is important to mention that a factor that can further decrease 4 πMeff is an out-of-plane uniaxial anisotropy field H⊥(Eq. 3). In thin films, the perpendicular anisotropy field is com- monly expressed as H⊥= 2K⊥/(Mst), where K⊥>0 is the anisotropy constant and tthe ferromagnetic film thickness [11]. In this simple picture, it is assumed that K⊥is nearly constant over the thickness range of our films. This anisotropy can be associated with strain due to the lattice mismatch between the FeV alloy and the adjacent Cu layers and/or with an interface contri- bution to the magnetic anisotropy. For Fe films with t= 7.5 and 12.9 nm, a linear fit of 4 πMeffversus 1/t gives 4πMs= 20.2 kG and K⊥= 2.5 erg/cm2. The value extracted for 4 πMsis in the range of the value of Febulk. AsimilaranalysisconductedonFe 0.63V0.37films of thickness t=3 and 7.5 nm gives 4 πMs= 12.2 kG and K⊥= 0.1 erg/cm2. The result suggests that the surface anisotropy constant decreases with increasing x. IV. SUMMARY The effective demagnetization field of the polycrys- talline Fe 1−xVxalloy films decreases with increasing xandalmostvanishesfor x≈0.7. AFMRstudyonepitax- ial films haveshown a similar xdependence of 4 πMeff[5]. Using the value of Mscalculated in the analysis above, we estimate the Gilbert damping constant of a 7.5 nm Fe layer and 7.5 nm Fe 0.63V0.37alloy film to be G Fe= 239 MHz and G FeV= 145 MHz respectively. The decrease of the effective demagnetization field of Fe 1−xVxwith in- creasing xis accompanied by a decrease of the Gilbert damping constant. A similar xdependence of G was observed in epitaxial films [5]. The authors explained the decrease of G by the reduced influence of spin-orbit coupling in lighter ferromagnets. Note that the Gilbert dampingofourfilmsislargerthanwhatwasfoundforthe epitaxial films (G=57 MHz for epitaxial Fe 8 nm thick). We note that the Fe 0.63V0.37alloy film, which has 4πMsapproximatly the same as that of Permalloy, has a magnetic damping constant of the same order than that of Py layer in a similar layer structure [12]. Hence, with their low Msandα, polycrystalline FeV alloy films are promising materials to be integrated in spin-tranfer de- vices. [1] J. C. Slonczewski, J. Magn. Magn. Mater. 159(1-2), L1 (1996) ; L. Berger, Phys. Rev. B 54(12), 9353 (1996). [2] see, for example, J. A. Katine et al., Phys. Rev. Lett. 84, 3149 (2000) ; B. Oezyilmaz et al., Phys. Rev. Lett. 91, 067203 (2003). [3] see, for example, G. D. Fuchs et al., J. Appl. Phys. 85 (7), 1205 (2004). [4] J. Z. Sun, Phys. Rev. B 62(1), 570 (2000). [5] C. Scheck et al., Phys. Rev. Lett. 98, 117601 (2007). [6] J-M. L. Beaujour et al., Europhys. J. B, DOI: 10.1140/epjb/e2007-00071-1 (2007). [7] C. Kittel in Introduction to Solid State Physics, Ed. 7, p.505.[8] see, for example, D. L. Mills and S. M. Rezende in Spin Dynamics in Confined Magnetic Structures II (Eds. B. Hillebrands andK.Ounadjela), pp.27-58, (Springer, Hei- delberg 2002). [9] I. Mirebeau, G. Parette, and J. W. Cable. J. Phys. F: Met. Phys. 17, 191 (1987). [10] Y. Kakehashi, Phys. Rev. B 32(5), 3035 (1985). [11] Y. K. Kim and T. J. Silva, Appl. Phys. Lett. 68, 2885 (1996). [12] S. Mizukami et al., J. Magn. Magn. Mater. 239, 42 (2002).

2007-10-15

Ferromagnetic resonance has been used to study the magnetic properties and magnetization dynamics of polycrystalline Fe$_{1-x}$V$_{x}$ alloy films with $0\leq x < 0.7$. Films were produced by co-sputtering from separate Fe and V targets, leading to a composition gradient across a Si substrate. FMR studies were conducted at room temperature with a broadband coplanar waveguide at frequencies up to 50 GHz using the flip-chip method. The effective demagnetization field $4 \pi M_{\mathrm{eff}}$ and the Gilbert damping parameter $\alpha$ have been determined as a function of V concentration. The results are compared to those of epitaxial FeV films.

Ferromagnetic resonance study of polycrystalline Fe_{1-x}V_x alloy thin films

0710.2826v2

SHARP EXPONENTIAL DECAY RATES FOR ANISOTROPICALLY DAMPED WAVES BLAKE KEELER AND PERRY KLEINHENZ Abstract. In this article, we study energy decay of the damped wave equation on com- pact Riemannian manifolds where the damping coecient is anisotropic and modeled by a pseudodierential operator of order zero. We prove that the energy of solutions decays at an exponential rate if and only if the damping coecient satises an anisotropic analogue of the classical geometric control condition, along with a unique continuation hypothesis. Fur- thermore, we compute an explicit formula for the optimal decay rate in terms of the spectral abscissa and the long-time averages of the principal symbol of the damping over geodesics, in analogy to the work of Lebeau for the isotropic case. We also construct genuinely anisotropic dampings which satisfy our hypotheses on the at torus. 1.introduction LetpM;gqbe a smooth, compact manifold without boundary and let
gbe the as- sociated Laplace-Beltrami operator (taken with the convention that
g¤0). Suppose W:L2pMqÑL2pMqis bounded and nonnegative. We consider the generalized damped wave equation given by# B2 tu

gu2WBtu0 pu;Btuq|t0pu0;u1q;(1.1) forpu0;u1qTPH:H1pMq`L2pMq, where His taken with the natural norm }pu0;u1qT}2 H}p1

gq1 2u0}2 L2pMq}u1}2 L2pMq: We study the asymptotic properties of the energy of solutions to (1.1) as tÑ8 . Here, the energy is dened by Epu;tq1 2» M|rgupt;xq|2|Btupt;xq|2dvgpxq; (1.2) wheredvgis the Riemannian volume form on M:It is straightforward to compute that d dtEpu;tq
2RexWBtu;Btuy¤0; (1.3) wherex;ydenotes the inner product on L2pM;gq:Thus, the assumption that Wis a non- negative operator guarantees that the energy of solutions to (1.1) experiences dissipation, but (1.3) does not indicate how quickly the energy decays as tÑ8 . The most straightfor- ward type of decay is uniform stabilization, i.e. when there exists a constant C¡0 and a real-valued function tÞÑrptqwithrptqÑ0 astÑ8 such that Epu;tq¤CrptqEpu;0q: Date : March 22, 2022. 1arXiv:2009.10832v2 [math.AP] 21 Mar 20222 B. KEELER AND P. KLEINHENZ In the case where Wacts via multiplication by a bounded, nonnegative function b, a great deal is known about energy decay rates. Perhaps the most well known result states that solutions to (1.1) experience uniform stabilization with an exponential rate if and only if W satises the geometric control condition (GCC) [RT75,Ral69]. The GCC is satised if there exists some T¡0 such that every geodesic with length at least Tintersects the set where bis bounded below by some positive constant. Many other works have proved weaker decay rates in the setting where the GCC is not satised (c.f. [Leb96] [Bur98] [BZ16] [Chr07] [Chr10] [BC15] [LR05] [BH07]). With more restrictive assumptions on WandM, one can obtain sharp decay rates (c.f. [AL14] [Sta17] [LL17] [Kle19a] [DK20] [Kle19b], [Sun22] [DJN19a] [Jin20]). A distinct shortcoming of the multiplicative case is that the damping force is sensitive only to positional information and not to the direction in which the solution propagates. For this reason, one can classify multiplicative damping as an isotropic force, but many physical sys- tems which experience anisotropic damping forces are studied in materials science, physics, and engineering [KKBH16,Cra08,JSC11]. However, a general analysis of the damped wave equation in the anisotropic case has not yet been done. This article aims to address this gap in the literature by studying the case where the anisotropic damping force is modeled by a pseudodierential operator. It is common in analysis of the generalized damped wave equation (1.1) to assume that W takes the form of a square, i.e. WBBfor some bounded operator B(c.f. [AL14]). This guarantees that Wis nonnegative and enables the use of certain techniques from spectral theory. We allow for a slightly more general assumption here, namely that Wtakes the form WN° j1B jBjfor some nite collection tBjuN j1 0 c`pMq, where 0 clpMqdenotes the space of classical pseudodierential operators on Mof order zero with polyhomogeneous symbols. The corresponding space of symbols is denoted S0 c`pTMq. We note that allowing Wto take the form of a sum of squares is indeed a generalization, since it is not generically possible to write°N j1B jBjasBBfor someBP 0 c`pMq, since the pseudodierential calculus only allows for the computation of square roots modulo a smoothing remainder. We denote by wPS0 c`pTMqthe principal symbol of W, taken to be positively ber-homogeneous of degree 0 outside a small neighborhood of the zero section in TM:That is,wpx;sqwpx;qfor alls¡0 and all||¥cfor somec¡0 which can be chosen to be arbitrarily small. This homogeneity allows us to treat was a function on the co-sphere bundle SM:tpx;qPTM:||g1 2u; where the choice of1 2is made for the sake of convenience in later arguments. We now state the required assumptions for the main theorem. The rst is an anisotropic analogue of the classical geometric control condition. Assumption 1 (Anisotropic Geometric Control Condition ).Let'tdenote the lift of the geodesic ow to TM. Assume that there exists a compact neighborhood Kof the zero section inTMand constants T0;c¡0such that for every px0;0qPTMzK, 1 TT» 0wp'tpx0;0qqdt¥c;forT¥T0:SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 3 That is, the long-time averages of wover geodesics are uniformly bounded below. In this case, we say Wsatises the anisotropic geometric control condition (AGCC). Note that in the case of multiplicative damping, Assumption 1 is equivalent to classical geometric control condition in [RT75]. The second key assumption requires that the kernel of Wcontain no nontrivial eigenfunc- tions of
g. Assumption 2. IfvPL2pMqsatises

gv2vwith0, thenWv0: In the case where Wbpxq, Assumption 2 is satised when bis supported on any open set, since eigenfunctions of
gcannot vanish on open sets by the unique continuation principle (c.f. [RT75]). It is for this reason that we sometimes refer to Assumption 2 as a \unique continuation hypothesis." With these assumptions stated, we then have the following equivalence. Theorem 1. All solutions uto(1.1) withWP 0 c`pMqsatisfy Epu;tq¤Ce
tEpu;0q (1.4) for someC; ¡0and for all t¥0if and only if Wsatises Assumptions 1 and 2. In other words, solutions experience uniform stabilization at an exponential rate if and only ifWsatises Assumptions 1 and 2. The existing literature on anisotropic damping coecients is quite limited. In the con- text of pseudodierential W, Sj ostrand [Sj o00] studied the asymptotic distribution of eigen- values of the stationary damped wave equation. Christianson, Schenck, Vasy, and Wun- sch [CSVW14] showed that a polynomial resolvent estimate for a related complex absorbing potential problem gives another polynomial resolvent estimate of the same order for the stationary damped wave equation. However, these results do not consider anisotropic damp- ing in a time-dependent setting and so do not provide energy decay results. Theorem 1 addresses this gap in the literature by providing conditions which guarantee exponential uniform stabilization, in analogy to the classical result of Rauch and Taylor [RT75]. Since Theorem 1 only claims the existence of some exponential decay rate , a natural question is to determine the optimal rate of decay for a given damping coecient. Given a xedWP 0 c`pMq, we dene the best exponential decay rate as in [Leb96] via :suptPR:DC¡0 such that Epu;tq¤Ce
tEpu;0q @uwhich solve (1.1) u:(1.5) Our next result shows that can be expressed in terms two fundamental quantities: the spectral abscissa, and the long-time averages of the principal symbol of Wover geodesics. The spectral abscissa is dened with respect to AW: 0 Id
g
2W ; which is the innitesimal generator of the solution semigroup for (1.1). For each R¡0, we set DpRqsuptRepq:||¡R; PSpecpAWqu: We then dene the spectral abscissa as D0lim RÑ0DpRq: (1.6)4 B. KEELER AND P. KLEINHENZ We also dene for tPRthe time-average of the damping along geodesics Lptq inf px;qPSM1 tt» 0wp'spx;qqds; and the long-time limit L8lim tÑ8Lptq: (1.7) We can then characterize as follows. Theorem 2. The best exponential decay rate for solutions to (1.1) withWP 0 c`pMqis 2 mint
D0;L8u; whereD0andL8are dened by (1.6) and(1.7) , respectively. Remark 1.1. It is of considerable note that the formula for the optimal decay rate here is an exact analogy of the multiplicative case studied by Lebeau (c.f. [Leb96, Theorem 2]). While the broad structure of our proof is similar, there are portions of the analysis which diverge greatly, particularly in Section 3 where we investigate the action of pseudodierential operators on Gaussian beams. Remark 1.2. Theorem 2 is signicantly stronger than Theorem 1, although this is not immediately obvious. The main portion of this article is dedicated to the proof of Theorem 2. We then show that Theorem 2 implies Theorem 1 in Section 6. Theorems 1 and 2 t into a broad range of existing results which attempt to reproduce the equivalence of the GCC and exponential decay under modied hypotheses on the damp- ing. It is not uncommon for such statements to be somewhat inconclusive. For example, when the damping is allowed to be time-dependent [LRLTT17] showed that for time peri- odic damping, the GCC is indeed equivalent to exponential decay, but it is not currently known if this result is true for non-periodic damping. In the setting where the damping is allowed to take negative values (commonly called \indenite damping"), the state of the art is similarly mixed. If Mis an open domain in RnwithC2boundary, [LRZ02] proves an exponential decay rate provided that the damping is positive in a neighborhood of BM (which implies the GCC) and inf xPMWpxqis not too negative. However, it is currently not known if an appropriate generalization of the GCC is equivalent to exponential stability in the indenite case. The limitations of these results illustrate that seemingly simple changes to hypotheses on the damping coecient can create substantial barriers to reproducing the classical equivalence theorem. So, the fact that Theorem 1 provides a direct analogy of the GCC for pseudodierential damping which is equivalent to exponential decay is somewhat exceptional. Note also that these other generalizations do not possess an analogy of Theo- rem 2. Although [LRZ02] and [LRLTT17] both provide a rate for the exponential decay, it is not shown to be sharp. Our nal result concerns Assumption 2, which is necessary in order to obtain Theorem 1. To see this, suppose that vsatises

gv2vwith0 andWv0. Then, the function upt;xqeitvpxq; solves (1.1), but has energy Epu;tq2}v}2 L2pMqfor allt. As previously mentioned, when W is a multiplication operator supported on any open set, unique continuation results guaranteeSHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 5 thatWdoes not annihilate any eigenfunctions of
g, making Assumption 2 unnecessary. However, in the pseudodierential setting, verifying this assumption is more dicult. A special case in which Assumption 2 is easy to check is when Wis constructed from functions of
g. Suppose WBBwithBfp

gq, wheref:RÑRsatises a \symbol-type" estimate of the form |Bk sf|¤Cp1|s|q
k for anyk:The functional calculus of Strichartz [Str72] shows that Wis pseudodierential when constructed in this way. The calculus also immediately implies that Assumption 2 holds as long as fdoes not vanish on the spectrum of

g, since for any eigenfunction v with eigenvalue , we haveWvfpq2v. However, damping coecients constructed in this fashion are somewhat uninteresting in the sense that the principal symbol is a function of||2 g, and therefore independent of direction. Thus, examples of this type are not truly anisotropic. In general, it is not obvious that one can always construct nontrivial anisotropic examples satisfying Assumption 2, although we expect that a rich class of examples do indeed exist. The following theorem demonstrates that one can always produce such examples when pM;gqis real analytic. Theorem 3. IfpM;gqis compact and real analytic, then there exists WP 0 clpMqof the formW°N j1B jBj,such that for each xPM, the principal symbol of Wvanishes on an open cone in T xMand for any van eigenfunction of
g;Wv0: The fact that the principal symbol vanishes in an open cone of directions at each point implies that theWin this theorem is not built from functions of
g, excluding the somewhat trivial case discussed previously. Using the machinery developed in the proof of Theorem 3, we are able to produce explicit examples on the at 2-torus of operators WP 0 clwhich satisfy both Assumptions 1 and 2. This construction is presented in Section 7. Remark 1.3. As mentioned previously, Assumption 2 follows directly from the geometric control condition in the multiplicative case. One might hope that this could be generalized to the scenario where Wis pseudodierential, but this problem is exceedingly dicult in general. The only result of this type known to the authors is that of [DJN19b], which uti- lized the fractal uncertainty principle to show that when Mis an Anosov surface, vis an eigenfunction of the Laplacian, and the principal symbol of Wis not identically zero, then there is a quantitative lower bound on the size of Wv. So for Anosov surfaces, Assumption 1 implies Assumption 2. But even in this specialized case, the proof involves highly sophisti- cated techniques. An analysis of the general case is an open problem, and we suspect that it would be a signicant undertaking. 1.1.Outline of the article. The majority of this article is devoted to the proof of Theo- rem 2, which spans Sections 2, 3 4, and 5. The primary tool is microlocal defect measures. We begin in Section 2 by analyzing the behavior of defect measures associated to sequences of solutions to (1.1) when propagated by the Hamiltonian ow for PB2 t

g. The formula we produce follows largely from direct computations and measure-theoretic arguments, in analogy to [Leb96,Kle17]. In Section 3, we perform a detailed study of the action of certain pseudodierential operators on coherent states, which is a critical component of constructing quasimodes for (1.1). This analysis is a key point where the pseudodierential case becomes signicantly more dicult than the multiplicative setting. In Section 4, we then combine the results of Sections 2 and 3 to produce quasimodes for the damped wave equation whose6 B. KEELER AND P. KLEINHENZ energy is strongly localized near a xed geodesic. The analysis of these quasimodes allows us to prove that ¤2 mint
D0;L8u. The proof of Theorem 2 is completed in Section 5, where we prove the lower bound ¥2 mint
D0;L8u:This section follows in close analogy to [Leb96], and so we omit some of the more technical details. In Section 6, we show that Theorem 2 implies Theorem 1. This follows directly from spectral theory analysis. Finally, in Section 7, we restrict to the case of real analytic manifolds to produce some examples. We provide a fairly generic condition on pseudodierential operators which guar- antees that they satisfy Assumption 2. We also show that one can always produce examples which fall into this category, thus proving Theorem 3. We then conclude by constructing some explicit examples on the at torus which satisfy both Assumptions 1 and 2. 1.2.Acknowledgements. The authors would like to thank J. Wunsch and A. Vasy for their frequent helpful comments throughout the course of this project. Wunsch's suggestions regarding the analytic wave front set argument in Section 7 were particularly useful. We are also grateful to M. Taylor for helping us better understand the proof of his original result with J. Rauch on exponential energy decay in [RT75]. We also wish to thank H. Christianson, Y. Canzani and J. Galkowski for their comments on an earlier version of this paper. Finally, BK was supported in part by NSF Grant DMS-1900519 through his advisor Y. Canzani. 2.Propagation of the microlocal defect measure In this section, we compute the propagation of microlocal defect measures associated to a sequence of solutions of (1.1) with WP 0 c`pMq. We begin by noting some general facts about microlocal defect measures. We will not prove these here, and we direct the reader to the seminal article of G erard [G er91] for details and proofs. Here we consider defect measures as measures on SpRMqtpt;x;;qPTpRMq:2||2 g1 2u;the co-sphere bundle ofRMtreated as a Riemannian manifold with the the product metric. For any sequence tukuHmpRMqconverging weakly to 0, there exists a sub-sequence tukjuand a positive Radon measure onSpRMqsuch that for any AP 2m c`pRMqwith compact support int, we have lim jÑ8xAukj;ukjy» SpRMqad; wherex;ydenotes the standard inner product on L2pRMq, which is the natural pairing betweenH
1pRMqandH1pRMq:Iftukuhas a defect measure without the need for passing to a subsequence, we say that tukuispure. From this point on, we specialize to the case where tukuH1pRMqis a pure sequence of solutions to the damped wave equation converging weakly to zero, with associated defect measure. A key piece of the proof of Theorem 2 is the propagation of this defect measure under the Hamiltonian ow on TpRMqgenerated by ppt;x;;q||2 g
2, the principal symbol ofPB2 t

g. We denote this ow by  s, and it can be written as spt;x;;qpt
2s;;'spx;qq; where we recall that 'sis the geodesic ow on TM. More precisely, the propagation of the defect measure refers to the behavior of under the pushforward psq. We now show that there exists a smooth function sÞÑGsPC8pSpRMqqsuch that psqG
s, and thatGscan be dened as the solution to a certain dierential equation.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 7 Lemma 2.1. For any xed pt;x;;qPSpRMq, deneGspt;x;;qas the solution to the initial value problem # G0pt;x;;q1; BsGspt;x;;qtp;Gsupt;x;;q4wpx;qGspt;x;;q;(2.1) wherepandware the principal symbols of P B2 t

gandW, respectively. Then, psqG
s. Equivalently, for any bPC8pSpRMqqwhich is compactly supported inpt;xq,» SpRMqbsd» SpRMqbG
sd: (2.2) Remark 2.2. Note that the content of Lemma 2.1 is analogous to that of [Kle17, Proposition 8], and the proof goes through in a very similar fashion for the case of pseudodierential damping. Proof. First, observe that in order to prove (2.2), it is sucient to show that for all sPR and anybPC8pSpRMqqwith compact support in pt;xq » SpRMqpbsqGsd» SpRMqbd: Furthermore, since G01, this is equivalent to showing that Bs» SpRMqpbsqGsd0: By direct computation, we see that BsrpbsqGsstp;bsuGspbsqBsGs; since sis the Hamiltonian ow generated by p:Using algebraic properties of the Poisson bracket, we have tp;bsuGstp;pbsqGsu
tp;Gsupbsq:Therefore, Bs» SpRMqpbsqGsd» SpRMqtp;pbsqGsu
tp;GsupbsqpbsqBsGsd: (2.3) To rewrite the rst term on the right-hand side above, let us extend pbsqGsto a function onTpRMqwhich is ber-homogeneous of degree 1 outside a small neighborhood of the zero section. This can be accomplished by choosing some PC8pRqwhich vanishes in a neighborhood of zero and is equal to one outside a slightly larger neighborhood. Then, for any xedsPR; p||qpbsqGsPS1 c`pTpRMqq; and hence B:Oppp||qpbsqGsqP 1 c`pMq: Thus,rB;PsP 2 c`pMq, and hence lim kÑ8xrP;Bsuk;uky » SpRMq1 itp;pbsqGsud8 B. KEELER AND P. KLEINHENZ by the denition of the defect measure. On the other hand, since each uksolves the damped wave equation, we also have xrP;Bsuk;ukyxBuk;Puky
xPuk;Buky xBuk;
2WBtukyx2WBtuk;Buky x2pBtWBBWBtquk;uky: Taking the limit of both sides as kÑ8 gives lim kÑ8xrP;Bsuk;uky» SpRMq4iwpbsqGsd: Therefore,» SpRMqtp;pbsqGsud
» SpRMq4wpbsqGsd: Combining the above with (2.3), we obtain Bs» SpRMqpbsqGsd» SpRMqpbsqpBsGs
tp;Gsu
4wGsqd; which is clearly zero if Gssatises (2.1).  Another important observation about the defect measure is its support is closely related to the characteristic set of P, dened as CharpPqtpt;x;;qPTpRMq:ppt;x;;q0u: The following is a well-known result, but we provide a short proof for the benet of the reader. Lemma 2.3. Giventukuandas above, the support of is contained in the intersection CharpPqXSpRMq. Proof. First, letPC8pRMqbe supported in a neighborhood of Char pPqand identically one on a slightly smaller neighborhood. Then, since Pis elliptic on the support of 1
, there exist a parametrix QP
2 c`pRMqsuch that pId
OppqqukQPukRukRuk for some smoothing operator Rand allk. Now x an interval Iand let PC8 0pIq. SinceR is smoothing, } ptqpI
Oppqquk}H1pRMq} ptqRuk}H1pIMq¤}uk}L2pIMq: By assumption, ukconverges weakly to zero in H1pRMq, and therefore its restriction to IMconverges weakly to zero in H1pIMq. Thus, there exists a subsequence tukjuwhich coverges strongly to zero in L2pIMq, sinceIMis compact. Therefore, }ukj}L2pIMqÑ0, which implies that ptqpId
OppqqukjÑ0 strongly in H1pIMq. Now consider the operator pB2 t
gqr ptqpI
OppqqsP 2 c`pRMq;SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 9 which is supported in IM:Since extracting a subsequence does not change the defect measure, we must have xpB2 t
gq ptqpId
Oppqqukj ;ukjqyÑ
1 2» SpRMq ptqp1
pt;x;;qqd; (2.4) recalling that 2||2 g1 2onSpRMq:However, we also have that }pB2 t
gqr ptqpId
Oppqqsukj}H
1pIMq ¤C} ptqpId
Oppqqukj}H1pIMq: Recalling that ptqpI
OppqqukjÑ0 strongly in H1pIMqand noting that tukjuis bounded in H1pIMq, we must have xpB2 t
gq ptqpId
Oppqqukj ;ukjqyÑ 0: Combining this with (2.4), we see that the support of must be disjoint from that of ptqp1
pt;x;;qq. Since was arbitrary, and the argument holds for any with the appropriate support properties, we must have that the support of is contained in Char pPqX SpRMq:  Observe that p0 exactly when  ||g, and therefore Char pPqXSpRMqis comprised of the two connected components St 1{2uXSpRMq: It is helpful to dene andG sto be the restrictions of andGstoS, respectively. The denition of Gsgiven in (2.1) then implies that BsG stp;G su 2wG s: (2.5) Now, sincewdepends only on px;qandis constant on S, we may treat G sas functions on SM. For the purposes of this article, it suces to consider only G s. It follows immediately from Lemma 2.1 that G sgives the propagation of onSunder the ow  si.e.psq G
s: As in [Kle17], we claim that G scan be realized as the solution of a much simpler dierential equation by observing that it is a cocycle map. That is, for any px;qPSMand anyr;sPR, we haveG srpx;qG rp'spx;qqG spx;q:To see this, note that by (2.5) and properties of the Poisson bracket, Bs pG r'sqG s pG r'sq tp;G su
2wG s tp;G r'suG s tp;pG r'sqG su
2wpG r'sqG s: So,pG r'sqG sandG srboth satisfy the same initial value problem, and must be equal. Using the cocycle property and the fact that G 01, we have BsG slim hÑ0G sh
G s hlim hÑ0pG h'sqG s
G s h G slim hÑ0G h's
G 0's hG sBr G r's r0:10 B. KEELER AND P. KLEINHENZ SinceG 01, we have that tp;G ru|r00. This along with the fact that 'sis independent ofrgivesBrpG r'sq r0
2w's:Thus,G scan be realized as the solution of the initial value problem# G 0px;q1 BsG spx;q
2wp'spx;qqG spx;q; which has solution G spx;qexp 
s» 02wp'rpx;qqdr : (2.6) Thus, the propagation of the defect measure exhibits exponential decay in proportion to the amount of time geodesics spend in the region where wpx;qis positive. 3.Pseudodifferential Operator Acting on Coherent States A key component of the proof of Theorem 2 is to build quasimodes for (1.1) using Guassian beams, which are strongly localized along a given geodesic. In this section, we obtain precise estimates for pseudo-dierential operators acting on slightly simpler objects, namely coherent states. A coherent state on Rnis a sequence of smooth functions thkutaking the form hkpxqkn 4eikxx
x0;0yeik 2xApx
x0q;px
x0qybpxq (3.1) for some xed px0;0q PSRn, wherebPC8 cpRnqandAPCnnhas positive denite imaginary part. Heuristically, one thinks of hkas being strongly microlicalized near px0;0q. The objective of this section is to show that if a symbol aPSm c`pR2nqvanishes to some nite order atpx0;0q, then}Oppaqhk}L2pRnqsatises a bound which depends on the symbol order mand on the order of vanishing. Remark 3.1. For the purposes of this section only, we use the more standard convention thatSRntpx;qPR2n:||1ufor the sake of convenience, but this does not alter the analysis in any way. Proposition 3.2. Fixpx0;0qPSRn,bPC8 cpRnq, and a matrix APCnn;with positive denite imaginary part. Then, for any k¥1;lethkbe given by (3.1) . LetaPSm c`pR2nqhave compact support in xand a polyhomogeneous expansion given by a¸ j¥0am
j; where each am
jPSm
j c`pR2nqsatisesam
jpx;sq sm
jam
jpx;qfor alls¡0and ||¥c¡0for some small c. Suppose there exists an `PNsuch thatam
jvanishes to order `
2jatpx0;0qfor allj¤` 2. Then, for each "¡0, there exists a C"¡0so that }Oppaqhk}L2pRnq¤C"km
` 2": (3.2) Proof. By the polyhomogeneity of a, for anyN0¥0 there exists rN0PSm
N0 c` such that aN0
1¸ j0am
jrN0: (3.3) We begin with the following lemma, which handles the remainder term in this expansion.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 11 Lemma 3.3. LetrPS
s c`pR2nqwiths¥0:Then, there exists a C¡0such that }Opprqhk}L2pRnq¤Ck
s 2: (3.4) Proof. By the quantization formula, we have }Opprqhk}2 L2pRnq» Rn» R2neixx
y;yrpx;qhkpyqdyd2 dx: Assume without loss of generality that x00. We then change variables via xÞÑk
1 2x, yÞÑk
1 2y, andÞÑk1 2:Recalling the denition of hk, we obtain }Opprqhk}2 L2pRnq» Rn» R2neixx
y;yrpk
1 2x;k1 2qeik1 2xx;0yei 2xAy;yybpk
1 2yqdyd2 dx: (3.5) For notational convenience, we dene gApyqei 2xAy;yyand lets:C8pRnqÑC8pRnqdenote dilation by s¡0. That is, sfpyq fpsyq:Then, using ^ to denote the standard Fourier transform, we dene Fkpq:» Rne
ixy;ygApyqbpk
1{2yqdykn{2rpgAk1 2pbspq: (3.6) Thus, we can rewrite (3.5) as }Opprqhk}2 L2pRnq» Rn» Rneixx;yrpk
1 2x;k1 2qFkp
k1 20qd2 dx: (3.7) We claim that for any NPNand any multi-index ;there exists a constant CN;¡0 so that B Fkpq¤CN;p1||q
Nfor allkPN: (3.8) To see this, consider the case where ||0 and note that p1||qNkn{2rpgAk1 2pbspq¤CNkn{2» Rnp1|
|qNp1||qN|pgAp
q||pbpk1 2q|d ¤CNkn{2» Rnp1||qN|pbpk1 2q|d; where the last inequality follows from the fact that pgAis Schwartz class, since Ahas positive denite imaginary part. Now, observe that kn{2» Rnp1||qN|pbpk1 2qd|¤kn{2» Rnp1k1 2||qNn1 p1k1 2||qn1|pbpk1 2q|d¤CNkn{2» Rnp1k1 2||q
n
1d; for some new CN¡0, sincepbis Schwartz-class. Changing variables via ÞÑk
1 2, we obtain that p1||qNkn{2rpgAk1 2pbspq¤CNkn{212 B. KEELER AND P. KLEINHENZ after potentially increasing CN:Dividing through by kn{2p1||qNcompletes the proof of (3.8) for||0. To obtain the estimate when ||0;simply repeat the above proof with pgAreplaced byB pgA: Now, in order to estimate (3.7) we introduce a smooth cuto function which is identically one in a neighborhood of x0. We then write }Opprqhk}2 L2pRnqIII; whereIis dened by I» Rn» Rneixx;ypxqrpk
1 2x;k1 2qFkp
k1 20qd2 dx; andIIis dened analogously with pxqreplaced by 1
pxq. To estimate I, we note that when||¤1, |rpk
1 2x;k1 2qFkp
k1 20q|¤CNp1|
k1 20|q
N¤C1 Nk
N{2; for someCN; C1 N¡0 and anyN, by (3.8) and the fact that rhas nonpositive order and is therefore uniformly bounded. Thus, I» Rn» ||¥1eixx;ypxqrpk
1 2x;k1 2qFkp
k1 20qd2 dxOpk
8q: (3.9) Now, when||¥1, |rpk
1 2x;k1 2q|¤Cp1k1 2||q
s¤Ck
s 2: Combining this with (3.9), we have I¤Ck
s}Fk}2 L1pRnq¤C1k
s; (3.10) where the nal inequality follows from (3.8). Now, consider II. Since 1
vanishes in a neighborhood of x0, we may integrate by parts arbitrarily many times in using the operatorxx;ry i|x|2, which preserves eixx;y. That is, for any¥0, we have II» Rn» Rneixx;yp1
pxqqixx;ry |x|2 
rpk
1 2x;k1 2qFkp
k1 20q d2 dx: By (3.8) and the fact that rPS
s c`, we have for any multi-index and anyN, B 
rpk
1 2x;k1 2qFkp
k1 20q ¤¸ | |¤||C ;Nk| | 2p1k1 2||q
s
| |p1|
k1 20|q
N: In the region where || ¥1, the above is bounded by CNk
s 2p1|
k1 20|q
Nfor some CN¡0. Alternatively, when || ¤ 1, we have a bound of the form CNk
N{2, since 1|
k1 20|¥Ck1 2. Combining these facts, we have » Rn» Rneixx;yp1
pxqqixx;ry |x|2 
rpk
1 2x;k1 2qFkp
k1 20q d2 dx¤CNk
s;SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 13 for someCN¡0, provided ¡n1 2so that the integral in xis convergent. Therefore, II¤CNk
s: Combining this with (3.10) and taking square roots of both sides completes the proof.  We now return to the proof of Proposition 3.2. We aim to estimate each of the terms in the sum in (3.3) separately. By denition, for each j¤N0
1, }Oppam
jqhk}2 L2pRnq» Rn» R2neixx
y;yam
jpx;qhkpyqdyd2 dx: As before, we change variables via xÞÑk
1 2x;yÞÑk
1 2y;andÞÑk1 2:This gives }Oppam
jqhk}2 L2pRnq» Rn» R2neixx
y;yam
jpk
1 2x;k1 2qFkp
k1 2q2 dx; whereFkis given by (3.6). Now, let px;qpqbe a smooth function which is identically one for||¤1 2and zero outside ||¤1. Then, for any 0 1{2 and anyk¡0, dene k;pqpk
p
k1 20qq; so thatk;is identically one on the ball of radius1 2kcentered at k1 20and zero outside the corresponding ball of radius k. Sincek;is supported on ||¡2 for suciently large k, the homogeneity of am
jimplies k;pqam
jpk
1 2x;k1 2qkm
j 2k;pqam
jpk
1 2x;q: (3.11) Recall that am
jvanishes to order `j:`
2jatpx00;0qfor allj¤` 2, so by Taylor expansion, there exists a collection of fm
j; ; gm
j; PSm
j c`pR2nqsuch that am
jpx;q¸ | |`jx fm
j; px;q ||
0 gm
j; px;q: Combining this with (3.11), we have k;pqam
jpk
1 2x;k1 2qkm
j 2¸ | |`j k
`j 2x fm
j; pk
1 2x;q ||
0 gm
j; pk
1 2x;q : Then, we dene Aj;1pxqkm
j
`j 2¸ | |`j» Rneixx;yk;pqx`jfm
j; pk
1 2x;qFkp
k1 20qd;(3.12) Aj;2pxqkm
j 2¸ | |`j» Rneixx;yk;pq ||
0 gm
j; pk
1 2x;qFkp
k1 20qd: (3.13) and Rjpxq» Rneixx;yp1
k;pqqam
jpk
1 2;k1 2qFkp
k1 2qd; (3.14) so that Oppam
jqhkAj;1Aj;2Rj:14 B. KEELER AND P. KLEINHENZ We claim that Rjis negligible for large k. SinceFkp
k1 20qis a Gaussian centered at k1 20 and 1
k;is supported at least kaway from that center, we are able to show that Rjis controlled by an arbitrarily negative power of k. Lemma 3.4. For anyN1PNthere exists CN1¡0such that }Rj}L2pRnq¤CN1k
N1: (3.15) Proof. To begin note that for any multi-index  B k;pqk
||pB qpk
p
k1 20qq: (3.16) Combining (3.16) with (3.8) shows that for any NPNand any multi-index , there exists CN;¡0 such that B rp1
k;qFkpqs¤CN;k
||1suppp1
k;qpqp1|
k1 20|q
N; where for any set ERn,1Edenotes the indicator function of E. Now, when|x| ¥1, for¥0 we may integrate by parts in (3.14) as in the proof of Lemma 3.3 to obtain Rjpxq» Rneixx;yixx;ry |x|2  p1
k;pqqam
jpk
1 2x;k1 2qFkpq d: Sinceam
jPSm
j c`pR2nq, we have that for any multi-index , |B am
jpk
1 2x;k1 2q|¤Ck|| 2p1k1 2||qm
j
||¤Ck|| 2p1k1 2||qm Thus, for any NPN, there exists a constant CNsuch that whenever |x|¥1, |Rjpxq|¤CNsup ||¤1 |x|» Rn1suppp1
k;qpqp1|
k1 20|q
Nk|| 2p1k1 2||qmd: Recall that|0|1, and so by the triangle inequality 1k1 2||¤1kk1 2|
k1 20|¤Ckp1|
k1 20|q: Thus, |Rjpxq|¤CNsup ||¤1 |x|» Rn1suppp1
k;qpqp1|
k1 20|q
Nmkm|| 2d: (3.17) Using polar coordinates 
k1 20r!withrPR; !PSn
1, we compute » Rn1suppp1
k;qpqp1|
k1 20|q
Nmd¤» Sn
18» k{2p1rq
Nmrn
1drd! ¤Ckpmn
Nq: Combining this with (3.17), we have |Rjpxq|¤CN|x|
kpmn
Nqm|| 2;if|x|¥1: SinceandNwere both arbitrary, given any N1¥0 we can choose ¥n1 2andN suciently large so that |Rjpxq|¤CN1k
N1|x|
n1 2;if|x|¥1:SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 15 By an analogous argument when |x|¤1, except without integration by parts, we have |Rjpxq|¤CN1k
N1if|x|¤1: Combining these inequalities and taking the L2norm completes the proof of (3.15).  It remains to estimate Aj;1andAj;2. It is here that we take advantage of the compatibility of the vanishing of am
jwith the particular form of the coherent state hk. We rst consider Aj;1. Lemma 3.5. For anyj¥0, there exists Cj¡0such that }Aj;1}L2pRnq¤Cjkm
j
`j 2: (3.18) Proof. Note on the support of k; k1 2
k¤||¤k1 2k: Also, recall that fm
j; PSm
j c`pR2nq, so|B fm
j; px;q|¤C||m
j
||. Therefore, sup xPRn|B fm
j; px;q|¤Ckm
j
|| 2;for allPsuppk;: (3.19) Now, when|x|¥1, we may integrate by parts as before to obtain that for any ¥0 and anyNsuciently large, |Aj;1pxq|¤km
j
`j 2¸ | |`j» Rneixx;yk;pqx fm
j; pk
1 2x;qFkp
k1 20qd ¤km
j
`j 2|x|`j» Rnixx;ry |x|2  k;pqfm
j; pk
1 2x;qFkp
k1 20qd ¤C;Nkm
j
`j 2|x|`j
» Rnp1|
k1 20|q
Nd ¤C1 ;Nkm
j
`j 2|x|`j
; where the second-to-last inequality follows from (3.8), (3.16) and (3.19). When |x|¤1, by the same argument with 0, we obtain |Aj;1pxq|¤Ckm
j
`j 2: Thus, for each ¥0, there exists a constant C¡0 so that |Aj;1pxq|¤Ckm
j
` 2p1|x|q`j
for allxPRn: Choosingso that`j
¤
n1 2and taking the L2norm gives the desired inequality.  Finally, we turn our attention to Aj;2. The estimation of this term is the most subtle of the three, and requires some very technical analysis of the relationship between the vanishing factor
 ||
0 and the structure of the support of k;. Lemma 3.6. For anyj¥0;there exists Cj¡0such that }Aj;2}L2pRnq¤Cjkm
jp
1 2q`j: (3.20)16 B. KEELER AND P. KLEINHENZ Proof. As before, we rst consider the case where |x|¥1 and the case |x|¤1 will follow from an analogous argument. When |x|¥1, we may again use integration by parts to see that for any ¥0 and anyNsuciently large, |Aj;2pxq|¤km
j 2¸ | |`j» Rneixx;yixx;ry |x|2  k;pq ||
0 gm
j; pk
1 2x;qFkp
k1 20q d: Note that on supp k;; k1 2
k¤||¤k1 2k, and so|B gm
j; |¤Ckm
j
|| 2¤Ckm
j 2. Combining this with (3.8) gives |Aj;2pxq|¤Ckm
j|x|
¸ ¤¸ | |`j» RnB  k;pq ||
0 p1|
k1 20|q
Nd;(3.21) when|x|¥1. Therefore, it is sucient to show that for any multi-indices ; with||¤ and| |`j, there exists C¡0 such that B  k;pq ||
0 ¤Ckp
1 2q`j: (3.22) To show this, it is convenient to choose coordinates on Rnso that0p1;0;:::; 0q. Writing p1;2;:::;nq, we have that on the support of k;; |
k1{20|b p1
k1{2q22 22 n¤k; by the denition of k;. Thus, |r|¤kfori1 and||¥k1{2
k: (3.23) Also, note that 1¡0 on suppk;forklarge enough, and so we can write ||
1p||1q
1p2 22 nq: Combining these facts, we have that for large k, ||
12 22 n ||1¤pn
1qk2 k1{2
k¤Ck2
1 2: We can now show (3.22) for 0. Recalling that p 1; 2;:::; nqPNnis a multi-index with| | 1 n`j;we make use of the above inequality and (3.23) to obtain  ||
0 |1
||| 1|2| 2|n| n ||`j ¤Ck2 1 k 1 2k 2k n k`j 2 Ckp
1 2q`jp
1 2q 1 ¤Ckp
1 2q`j; which proves (3.22) in the case where || 0. To handle the case where 0 we let 1;2;:::;nbe multi-indices and expand via the product rule to obtainB  ||
0 ¤¸ |12n|CjB1 1
|| || 1 B2 2 || 2 Bn n || n:(3.24)SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 17 Since1
|| ||andj ||are homogeneous of degree zero, we have that for any multi-index , B 1
|| || ¤C ||||;andB r || ¤C ||||ifr1: Furthermore, we recall that ||¥k1 2
k¥Ck1 2on suppk;;and so k;pqB 1
|| || ¤Ck
|| 2;andk;pqB r || ¤Ck
|| 2; r1: (3.25) Now consider Br 
r || r . Expanding via the product rule, we can rewrite this as a linear combination of terms of the formr || tr B1 r || Bq r || ; where each iis a multi-index with |i|¥1,1qr;andtrq r:That is, there aretrfactors ofr ||which do not have any derivatives, and the remaining r
trfactors each have at least 1 derivative applied to them. Note then that tr¥maxp0; r
|r|q:By (3.23) each factor with no derivatives is bounded by k
1 2. The homogeneous estimate (3.25) controls the factors with derivatives, givingk;pqr || tr B1 r || Bq r || ¤kp
1 2qtrk
|1| 2k
|q| 2¤Ckp
1 2qtr
|r| 2: Thus, by the triangle inequality, we have k;pqBr r || r¤Ckp
1 2qtr
|r| 2: When r
|r|¡0, we have kp
1 2qtrk
|r| 2¤kp
1 2qp r
|r|qk
|r| 2¤Ckp
1 2q r; sincetr¥ r
|r|¡0 and
1 20. On the other hand, when r¤|r|, we still have tr¥0 and so kp
1 2qtrk
|r| 2¤Ck
|r| 2¤Ck
r 2¤Ckp
1 2q r; since¡0:Thus, there exists a C¡0 so that k;pqBr r || r¤Ckp
1 2q r: (3.26) An analogous argument shows k;pqB1 1
|| || 1¤Ckp
1 2q 1; (3.27) for some potentially dierent C¡0:Combining (3.26) and (3.27) with (3.16) and (3.24) yields B  k;pq ||
0 ¤Ckp
1 2qp 1 2 nqCkp
1 2q| |Ckp
1 2q`j: We have therefore proved (3.22).18 B. KEELER AND P. KLEINHENZ Combining (3.21) and (3.22), we have that for any ¥0 and anyNlarge enough, there existsC;C1 ¡0 so that |Aj;2pxq|¤C|x|
km
j 2p
1 2q`j¸ |1|¤¸ | |`jkm
j
|1| 2» Rnp1|
k1 20|q
Nd¤C1 |x|
km
jp
1 2q`j: We then have |Aj;2pxq|¤C|x|
km
jp
1 2q`j;for all|x|¥1; (3.28) for someC¡0. To estimate Aj;2pxqwhen|x|¤1, we repeat the above argument without integrating by parts. From this, we obtain |Aj;2pxq|¤Ckm
jp
1 2q`j;for all|x|¤1: (3.29) Choosing¡n
1 2, we can combine (3.28) with (3.29), then take L2norms to obtain (3.20) as desired.  Recalling the constructions of Rj;Aj;1;andAj;2, we combine Lemmas 3.4, 3.5, and 3.6 to obtain that for each 0 ¤j¤N0
1, }Oppam
jqhk}L2pRnq¤}Aj;1}L2pRnq}Aj;2}L2pRnq}Rj}L2pRnq¤Cjkm
jp
1 2q`j;(3.30) for someCj¡0. Since`j`
2j, (3.4) and (3.30) imply that for any N0¥m, there exists a collection of constants tCjuN0 j0so that ||Oppaqhk||L2¤N0
1¸ j0||Oppam
jqhk||L2||OpprN0qhk||L2 ¤N0
1¸ j0Cjkm
jp
1 2q`jCN0km
N0 2 N0
1¸ j0Cjkm
` 2p`
2jqCN0km
N0 2 ¤Ckm
` 2`CN0km
N0 2; for someC¡0:ChoosingN0large enough and small enough completes the proof of Proposition 3.2.  4.The upper bound for  In this section, we show that ¤2 mint
D0;L8u, whereD0andL8are dened as in Section 1. That
2D0is an upper bound is straightforward to show. To do so, let jPSpecpAWqzt0u. Thus there exists u pu0;u1q 0 such that AWuju, where we recall AW 0 Id
g
2W : It is then immediate that upxq etju0pxqsolves the damped wave equation with initial datapu0;u1q, and Epu;tqe2tRepjqEpu;0q:SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 19 SinceEpu;0q0, we have that ¤
2Repjqfor allj. Furthermore, by the denition of D0, there must either exist some j0with Repj0qD0, or a sequence of jwith RepjqÑD0. In either case, we must have ¤
2D0. Showing that 2 L8is also an upper bound is more complicated. Our technique for this is inspired by the method of Gaussian beams introduced by Ralston [Ral69, Ral82]. Using Gaussian beams, one can produce quasimodes for the wave equation with energy strongly localized near a single geodesic. Intuitively, solutions to (1.1) should decay only when they interact with the damping coecient. Motivated by this, we modify the Gaussian beam construction using the propagation of the defect measure derived in Section 2, in analogy to [Kle17]. From this, we obtain solutions whose energy decays at a rate proportional to the integral of the symbol of Walong the chosen geodesic. To begin, we recall Ralston's original Gaussian beam construction on Rnwith a Riemann- ian metricg. LetAptqbe annnsymmetric matrix-valued function with positive denite imaginary part. Let tÞÑpxt;tqdenote a geodesic trajectory and set px;tqxt;x
xty1 2xAptqpx
xtq;x
xty: LetbPC8pRRnq. Then, we dene ukpx;tqk
1n{4bpt;xqeik px;tq: (4.1) The work of [Ral82] guarantees that there exist appropriate choices of bandAptqso thatuk is a quasimode of the undamped wave equation with positive energy, which is concentrated along the geodesic pxt;tq. We summarize some notable facts from [Ral82] in the following Lemma. Lemma 4.1 [Ral82] .FixT¡0andpx0;0qPSM. For'tpx0;0qpxt;tq;there exists a bPC8pRRnqand annnsymmetric matrix-valued function tÞÑAptqso that for any k¥1;for theukdened in (4.1) sup tPr0;Ts}B2 tukp;tq

gukp;tq}L2pRnq¤Ck
1 2: (4.2) Furthermore, for all tPr0;Ts, lim kÑ8Epuk;tq¡0; (4.3) and the limit is always nite and independent of t. Remark 4.2. By (4.3), we may assume without loss of generality that lim kÑ8Epuk;tq1 for alltPr0;Ts: Remark 4.3. Using coordinate charts and a partition of unity, we can extend this con- struction to the case of manifolds, which results in a sequence tuku C8pRMqsuch that lim kÑ8Epuk;tq1 and the appropriate analogue of (4.2) holds. Next, we modify tukuusing the propagation of the defect measure from Section 2 to produce a sequence of quasimodes for the damped wave equation. Recall that G t:SMÑC is given by G tpx0;0qexp

³t 02wpxs;sqds wherepxs;sq'spx0;0q. Recall also the time averaging function tÞÑLptq Lptq1 tinf px0;0qPSMt» 0wpxs;sqds:20 B. KEELER AND P. KLEINHENZ Note thatLptqcan be rewritten in terms of G t Lptq
1 tsup px;qPSMln G tpx;q : Motivated by the form of G t, we xpx0;0qPSMand set vkpt;xqG tpx0;0qukpt;xq: That is, we modify the quasimode for the free wave equation so that it decays exponentially at a rate proportional to integral of wpxt;tqalong the geodesic it is concentrated on. We now show that for any "¡0; vkis anOpk
1 2"qquasimode of (1.1). Proposition 4.4. Givenpx0;0qPSM, letukpt;xqbe as specied in Remark 4.3 and set vkpt;xqG tpx0;0qukpt;xq. For anyT¡0and"¡0, there exists a constant C";T¡0so that sup tPr0;Ts}pB2 t

g2WBtqvkpt;q}L2pMq¤C";Tk
1 2": (4.4) Proof. By direct computation pB2 t

g2WBtqvkG tpB2 t

gquk2BtG tBtukpB2 tG tquk2WBtpG tukq G tpB2 t

gquk
2wpxt;tqG tBtuk
Btpwpxt;tqG tquk 2WG tBtuk
2wpxt;tqWG tuk G tpB2 t

gquk2pW
wpxt;tqqG tBtuk wpxt;tq2
2wpxt;tqW
Btwpxt;tq G tuk: By the construction of ukand the boundedness of G t, we have sup tPr0;Ts}G tpB2 t

gqupt;q}L2pMq¤Opk
1 2q: SinceWis order zero, and therefore bounded on L2pMq, we obtain sup tPr0;Tspwpxt;tq2
2wpxt;tqW
Btwpxt;tqqG tuk L2pMq¤Csup tPr0;Ts||ukpt;q||L2Opk
1q; where the nal equality follows from the fact that³ Rnkn 4e
k|y|2dyis uniformly bounded in k. To estimate W
wpxt;tqqG tBtukpt;qwe will apply Proposition 3.2 with m0 and`1. To do so, note W
wpxt;tqis a pseudo with appropriate vanishing properties. Furthermore Btukpx;tqk
1n 4Btbpt;xqeik px;tqikn 4bpt;xqBt px;tqeik px;tq; and for xed t, both of these terms take the form of a coherent state hkas dened in Proposition 3.2 (the fact that the rst has an extra factor of k
1is irrelevant, as it only improves the estimate). Since all quantities depend on tin aC8fashion, for any "¡0 sup tPr0;Ts2pW
wpxt;tqqG tBtukpt;q L2pMq¤Cpk
1{2"q: (4.5) By the triangle inequality, we obtain (4.4), which completes the proof.  The next step in the proof of the upper bound for is given apx0;0qproduce a sequence ofexact solutions to (1.1) whose energy approaches |G tpx0;0q|2.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 21 Proposition 4.5. Given any T¡0, any"¡0, and anypx0;0q PSM, there exists an exact solution uof the generalized damped wave equation (1.1) with |Epu;0q
1|" andEpu;Tq
|G Tpx0;0q|2": (4.6) Proof. Letukandvkbe as dened previously. Then, dene !kas the unique solution of the damped wave equation with initial conditions !kpx;0qvkpx;0qandBt!kpx;0qBtvkpx;0q. It is immediate that Ep!k;0qEpvk;0qEpuk;0qÑ1;askÑ8: To see (4.6), rst note by the triangle inequality |Ep!k;tq1 2
Epvk;tq1 2|¤Ep!k
vk;tq1 2: (4.7) Thus, it suces to prove that lim kÑ8Epvk;tq|G tpx0;0q|2and that lim kÑ8Ep!k
vk;tq0. To see that lim kÑ8Epvk;tq|G tpx0;0q|2, note that by the denition of vkand properties ofG t Epvk;tq1 2» M|G tpx0;0qBtukpx;tq
wpxt;tqG tpx0;0qukpx;tq|2 |G tpx0;0qrgukpx;tq|2dvgpxq: Now sincewpxt;tqandG tare bounded » Mwpxt;tqG tpx0;0qukpx;tq2dvgpxq¤C||ukpt;q||2 L2¤C1k
2; for someC; C1¡0. Thus, lim kÑ8Epvk;tqlim kÑ81 2» MG tpx0;0qBtukpx;tq2G tpx0;0qrgukpx;tq2dvgpxq G tpx0;0q2lim kÑ8Epuk;tq G tpx0;0q2; (4.8) where in the nal equality we used that lim kÑ8Epuk;tq1. To controlEp!k
vk;tq, letfkpB2 t

2WBtqvk:Then pB2 t

2WBtqpvk
!kqfk: By Proposition 4.4, for any ";T¡0 there exists a C";T¡0 such that sup tPr0;Ts}fkpt;q}L2pMq¤C";Tk
1 2": (4.9)22 B. KEELER AND P. KLEINHENZ By direct computation BtEp!k
vk;tq» MpB2 t

gqp!k
vkqBtp!k
vkqpB2 t

gqp!k
vkqBtp!k
vkqdvgpxq 2Re» Mrfk
2WBtp!k
vkqsBtp!k
vkqdvgpxq 2Re» MfkBtp!k
vkqdvgpxq
4RexWBtp!k
vkq;Btp!k
vkqyL2pMq: Note that the second term on the right-hand side above is nonpositive, since Wis a non- negative operator. Now, using (4.9) and that ||Btpvk
!kqpt;q||L2is uniformly bounded for kPNandtPr0;Ts, there exists C1 ";T¡0 such that sup tPr0;Ts2Re» MfkBtp!k
vkqdvgpxq¤2}fkpt;q}L2}Btp!k
vkqpt;q}L2¤C1 ";Tk
1 2": Thus, for any "¡0 sup tPr0;Ts|BtEp!k
vk;tq|¤C1 ";Tk
1 2": Integrating in tgives sup tPr0;TsEpvk
!k;tq¤C1 ";TTk
1 2": Combining this with (4.7) and (4.8) yields (4.6).  For the penultimate step in the proof of the upper bound for , we will show that tÞÑtLptq is superadditive. That is, for r;t¥0,ptrqLptrq¥tLptqrLprq. To see this observe ptrqLptrq inf px0;0qPSM»rt 0wpxs;sqds  inf px0;0qPSM»t 0wpxs;sqds»tr twpxs;sqds ¥ inf px0;0qPSM»t 0wpxs;sqds inf px0;0qPSM»tr twpxs;sqds  inf px0;0qPSM»t 0wpxs;sqds inf px0;0qPSM»r 0wpxs;sqds tLptqrLprq: Now by Fekete's lemma, L8:lim tÑ8Lptq sup tPr0;8qLptq, and thusLptq¤L8for allt. That the supremum is not innite follows from the fact that wpx;qis uniformly bounded on TM: We are now ready to show that ¤2L8. Assume for the sake of contradiction that 2L83for some¡0. Then since 2 pL8q, there exists a C¡0 such that for allt¥0 and all solutions uof (1.1), Epu;tq¤CEpu;0qe
2tpL8q: (4.10)SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 23 For the next step, it is convenient to remove the factor of C:To accomplish this, choose T¡0 large enough so that max pC;1qeT. Then Ce
2TpL8qe
Tp2L8q: SinceLptq¤L8for allt, we obtain Ce
2TpL8qe
2TL8
T¤e
2TLpTq
T: (4.11) Now, we recall that
TLpTq sup px;qPSMlnG Tpx;q: Thus, there exists a point px0;0qPSMsuch that ln G Tpx0;0q¡
TLpTq
1 2T:Therefore, e
2TLpTq
T|G Tpx0;0q|2: So by (4.11) there exists a ¡0 such that Ce
2TpL8q|G Tpx0;0q|2
: Now, by Proposition 4.5, there exists an exact solution uof (1.1) such that 1¡Epu;0q
 2andEpu;Tq¡|G Tpx0;0q|2
 2: Thus, Epu;Tq¡Epu;Tq Epu;0q
 2 ¡Epu;0q |G Tpx0;0q|2
 2
 2Epu;Tq ¡Epu;0q |G Tpx0;0q|2
 2
 2Epu;0q Epu;0q |G Tpx0;0q|2
 : Therefore, Epu;Tq¡Epu;0qp|G Tpx0;0q|2
q¡CEpu;0qe
2TpL8q; but this contradicts (4.10). Thus, we must have ¤2L8:Combining this with the discussion at the beginning of this section, we have proved the upper bound ¤2 mint
D0;L8u: We complete the proof of Theorem 2 in the next section by proving the corresponding lower bound for: 5.The lower bound for  In this section, we prove that the best exponential decay rate satises ¥2 mint
D0;L8u; (5.1) which is the nal component of the proof of Theorem 2. In contrast to the proof of the upper bound, this section proceeds in direct analogy to the work of Lebeau, and so we omit many of the details which can be found in [Leb96, Kle17]. While the proofs presented here are not new, we include them to introduce notation that is used later in Section 6, where we use Theorem 2 to prove Theorem 1.24 B. KEELER AND P. KLEINHENZ We begin with the following energy inequality, which for the multiplicative case is presented as Lemma 1 in [Leb96]. Lemma 5.1. For everyT¡0and every"¡0, there exists a constant cp";Tq¡0so that for every solution uof(1.1) , Epu;Tq¤p 1"qe
2TLpTqEpu;0qcp";Tq}pu0;u1q}2 L2ÀH
1; (5.2) This inequality is proved using straightforward properties of the propagation of the defect measure, so the proof from [Leb96] goes through with no modication. To obtain the desired lower bound on we must further control the }pu0;u1q}2 L2ÀH
1on the right hand side. Given Lemma 5.1, we proceed by introducing the adjoint A W 0
Id

g
2W of the semigroup generator AW. Note that the spectrum of A Wis the conjugate of the spectrum of AW. Thus, we denote by E jthe generalized eigenspace of A Wwith associated eigenvalue j. Recall that HH1pMq`L2pMq, equipped with the natural norm. It is also useful to introduce the 9Hseminorm dened for elements of Hby }pu0;u1qT}2 9H}ru0}2 L2}u1}2 L2: For eachN¥1, dene the subspace HN# 'PH:x'; yH0;@ Pà |j|¤NE j+ : Our rst observation is that HNis invariant under the action of the semigroup etAW. To demonstrate this, let t kube a basis of the nite dimensional spaceÀ |j|¤NE jDpA Wq:Now, sinceE jis invariant under A W, we can express each A W las a nite linear combination of thet ku. Thus, for each `and any'PHN, we have BtxetAW'; `yH t0xetAW';A W `yH t0¸ c`;kx'; kyH0; by the denition of HN. Repeating this argument, we see that Bj txetAW'; `yH t00 for allj:Observing that xetAW'; `yHis an analytic function of t, we havexetAW'; `yH0 for alltPR:ThereforeetAW'PHN. Now, dene H1L2`H
1and letNdenote the norm of the embedding of HNin H1, which is well-dened since Mis compact. Since Wis bounded on L2, it is compact as an operator from L2ÑH
1. Therefore A W:HÑH1is a compact perturbation of the skew-adjoint operator 0
Id
P 0 . Thus, the family tE ju8 j0is total in H, and so limNÑ8N0 (c.f. [GK69, Ch. 5, Theorem 10.1]). We can now proceed with the proof of (5.1). Assume that 2 min t
D0;L8u¡0, otherwise the statement is trivial. Choose ¡0 small enough so that 2 mint
D0;L8u
¡0 and takeTlarge enough so that 4 |L8
LpTq|andeT 2¡3. Then, by Lemma 5.1 with "1, there exists a constant cp1;Tqsuch that for every solution uof (1.1) Epu;tq¤2e
2TLpTqEpu;0qcp1;Tq}pu0;u1q}2 H1: (5.3)SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 25 Next, choose Nlarge enough so that cp1;Tq2 N¤e
2TLpTq. Then, for solutions uof (1.1) with initial data pu0;u1qTPHN Epu;Tq¤3e
2TLpTqEpu;0q: SinceHNis invariant under evolution by etAW Epu;kTq¤3ke
2kTLpTqEpu;0q;@kPN: Then, we can use the fact that 4 |L8
LpTq|andT 2¡ln 3 to obtain that Epu;kTq¤3ke
2kTpL8
{4qEpu;0q ¤
eln 3
T 2 k e
2kTL8Epu;0q ¤e
kTEpu;0q; where the nal inequality follows from the fact that ¤2L8
2L8by denition. Since the energy is nondecreasing, it follows that Epu;tq¤Ce
tEpu;0q @t¥0; (5.4) for some constant C¡0: To extend (5.4) to all solutions of (1.1), let  denote the orthogonal projection from H ontoÀ |j|¤NEj:Then for any v pu0;u1qTPH, there is an orthogonal decomposition of the formvvpId
qv. SinceEjandE kare orthogonal for jk, we have that pId
qvPHN;and henceHK NÀ |j|¤NEj. SinceEjis invariant under etAWandHK Nis nite dimensional, we have that there exists a C¡0 so that for all solutions uof (1.1) with initial data in HK N, Epu;tq¤Ce2D0Epu;0q¤Ce
tEpu;0q;@t¥0: (5.5) Finally, since  and Id
 are continuous with respect to the 9Hseminorm, for some C¡0 Epu;0qEppId
qu;0q¤CEpu;0q: Therefore, using the decomposition  pId
qon the initial data of any solution uwe can apply (5.4) and (5.5) to obtain Epu;tq¤Ce
tEpu;0q;@t¥0; (5.6) for some possibly larger C¡0:By denition of the best possible decay rate, ¥ 2 mint
D0;L8u
. Sincecan be taken arbitrarily small, this proves (5.1). Combining this with the upper bound obtained in Section 4 completes the proof of Theorem 2. 6.Proof of Theorem 1 In this section we show that Theorem 2 implies Theorem 1. First, we will assume both Assumptions 1 and 2 are satised. We will show this implies ¡0, which is equivalent to exponential energy decay. Note that Assumption 1 immediately implies that L8¥c¡0. Thus, we only need to show that D00. For this, we introduce the quantity D8:lim RÑ8suptRepq:||¡R; PSpecAWu: We claim that D8¤
L8. To show this, rst recall EjandHNfrom Section 5. Let ube a solution to (1.1) with initial data pu0;u1qTPEjwith|j|¡N. ThenuetAWpu0;u1qT26 B. KEELER AND P. KLEINHENZ etjpu0;u1qT. Note that EjHNwhenever|j| ¡N. Combining this with the proof of (5.4) e2RepjqtEpu;0qEpu;tq¤Ce
tEpu;0q; for every 02L8. Hence, 2Re pjq¤
whenever|j|¥N, and so Repjq¤
L8 for suchj. It immediately follows that D8¤
L80. By the abstract spectral theory arguments in the proof of [AL14, Lemma 4.2], the spectrum ofAWconsists only of isolated eigenvalues and Re pq¤0 for allPSpecpAWq. Thus, in order to have D00, eitherD80 or there exists a nonzero eigenvalue of AWon the imaginary axis. Since we have already shown D80, we need only rule out nonzero imaginary eigenvalues. Suppose iPSpecpAWqwithPRand corresponding eigenvector pv0;v1qT:Thenv1v0, and
gv02v0
2iWv 00: (6.1) Taking the L2inner product of both sides with v0and then taking the imaginary part gives
2xWv 0;v0y0: If0, the equation is trivially satised. However, if 0, thenxWv 0;v0y0. Recalling thatW°B jBjfor some collection of operators Bj, we must have Wv 00. Then by (6.1)v0is an eigenfunction of
gwith eigenvalue
2andv0PkerW. But by Assumption 2, this is impossible. Thus, the only possible eigenvalue of AWon the imaginary axis is zero and we cannot have D00:Combining this with the fact that L8¡0, we have shown that Assumptions 1 and 2 imply ¡0, which in turn demonstrates that solutions to (1.1) experience exponential energy decay. We now prove the reverse implication in Theorem 1. For this, we assume that (1.4) holds with some ¡0 for all solutions uand we want to see that Assumptions 1 and 2 hold. By denition, ¥¡0, and hence both
D0andL8are strictly positive. Because L8¥{2¡0 Assumption 1 holds. Similarly, since D00, there cannot be any eigenvalues ofAWon the imaginary axis except possibly at zero. Now suppose that vPL2satises

gv2vwith0 andWv0:Thenpv;ivqTis an eigenvector of AWwith eigenvalue i0, which is a contradiction. Thus, Assumption 2 must also hold, which completes the proof of Theorem 1. 7.A Class of Examples on Analytic Manifolds One of the key hypotheses of Theorem 1 was that the damping coecient Wmust not annihilate any eigenfunctions of
gassociated with nonzero eigenvalues. In the case where W is a multiplication operator whose support satises the classical geometric control condition, this is always satised by the unique continuation properties of elliptic operators [RT75]. However, when the damping is pseudodierential it is much more dicult to check this hypothesis. In this section, we produce a collection of operators on real analytic manifolds which satisfy Assumption 2 and are not multiplication operators. We also give an example of an explicit pseudodierential damping coecient on T2which satises Assumptions 1 and 2. The primary tool in this discussion is the analytic wavefront set, and so we begin by providing some background denitions for the reader's convenience. More details can be found in [H or83, §8.4-8.6].SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 27 Given a set XRnand a distribution uPD1pXq, ifuis real analytic on an open neighborhood of x0we write that uPCanearx0PX:In analogy with the relationship between the standard wavefront set and C8singularities, one can resolve Casingularities by dening the analytic wavefront set, written WFApuqand dened as follows. Denition 7.1. We say that a point px0;0qPTXz0is not inWFApuq, if there exists an open neighborhood Uofx0, a conic neighborhood of0and a bounded sequence uNPE1pXq, which are equal to uonU, and each satisfy |puNpq|¤CN1 || N ; (7.1) for allP: By [H or83, Prop. 8.4.2], we have that uPCanearx0if and only if WFApuqcontains no points of the form px0;qwith0: We also introduce a set, which we can be thought of as the analytically invertible directions ofudenoted by Apuq. Its complement is commonly called the (analytic) characteristic set ofu[H or83]. Denition 7.2. We say that 0PRnz0is in Apuqif there exists a complex conic neighbor- hoodVof0and a function , which is holomorphic in tPV:||¡cufor somec¡0; satisfying pu1inVXRnand there exists C;N¡0such that |pq|¤C||N; forPV: The nal preliminary we require is the notion of the normal set of a closed region F contained within a manifold M. For the purposes of this denition, we only require that M beC2. Denition 7.3. LetFbe a closed region in a C2manifoldM:The exterior normal set, NepFq, is dened as the set of all px0;0q PTMz0such thatx0PFand such that there exists a real valued function fPC2pMqwithdfpx0q00and fpxq¤fpx0q; xPF: The interior normal set of Fis then dened by NipFqtpx;q:px;
qPNepFquand the full normal set is dened as NpFqNepFqNipFq. We write NpFqto denote the closure of the normal set of F. Note that the projection of NepFqontoMis dense inBFbut might not be equal to BF[H or83, Prop. 8.5.8]. With these denitions in hand, we are able to describe a class of pseudodierential oper- ators which do not annihilate any eigenfunctions of
g. Lemma 7.4. LetpM;gqbe a compact, real analytic manifold of dimension n. Suppose ;rPC8 cpMqare cuto functions supported entirely within a single coordinate patch, with r1on an open neighborhood of the support of . Let PC8pRnqbe homogeneous of degree 0 outside a compact neighborhood of the origin, and dene BP 0 clpMqin local coordinates by BurOpp pqqu. Letq denote the inverse Fourier transform of and 2:RnRnÑRndenote the natural projection onto the ber variables , if 2pNpsuppqqXApq qH;28 B. KEELER AND P. KLEINHENZ then for any eigenfunction uof
g, we haveBu0: Proof. We proceed by contradiction, so assume Bu0 for some eigenfunction uof
g. ThusWFApBuq  H and we aim to show there exists some px0;0q PWFApBuq. First, by [H or83, Thm 8.5.6'], we have NpsuppuqWFApuq: Sinceuis an eigenfunction, it cannot vanish identically on any open set. We claim that this implies BpsuppqBp suppuq: (7.2) To see this, suppose xPBpsuppqand letVbe any open neighborhood of x. Sincepxq0, we have that pxqupxq0, so it is enough to show that uis not identically zero on all of V. Without loss of generality, we may assume that Vlies entirely within the same coordinate patch containing supp . Sincexis a boundary point of the support, does not vanish identically on V. By continuity, this implies the existence of a smaller open neighborhood rVV(not containing x) whereis never zero. Since uis an eigenfunction, it cannot vanish identically on rV, and hence uis not identically zero on rVV, which proves (7.2). Now we want to show Npsuppq Npsuppuq. Takepx0;0q PNpsuppq;notex0 maximizes a function fon suppwithdfpx0q0, sox0PBpsuppqBp suppuq. That isx0is not an interior point. Furthermore, since supp suppuandfis maximized at x0in suppit must also be maximized at x0when restricted to the smaller set supp u: ThereforeNpsuppqNpsuppuqandNpsuppqNpsuppuq. Hence, NpsuppqWFApuq: (7.3) Since the cuto function is supported in a single coordinate patch, we can treat u and Opp quas functions on Rn:Now, observe that q uOpp qu, wheredenotes standard convolution. This, along with [H or83, Thm 8.6.15] gives WFApuqWFApOpp quqYpRnApq qcq: (7.4) Applying (7.3), we obtain NpsuppqWFApOpp quqYpRnApq qcq; and therefore, NpsuppqXpRnApq qqWFApOpp quq: By hypothesis, there exists a point px0;0qPNpsuppqXpRnApq qqWFApOpp quq: In particular, x0Psupp, and since r1 on a neighborhood of supp , we see thatpx0;0q must also lie inside WFAprOpp quqWFApBuq. This contradicts the assumption that Bu0, and thus the proposition is proved.  Remark 7.5. It is worth noting that the argument of this lemma works for when
gis replaced by P, an elliptic second order pseudodierential operator, as long as P's eigenfunc- tions do not vanish identically on open sets. Given Proposition 7.4, the proof of Theorem 3 is straightforward.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 29 Proof of Theorem 3. Given a real analytic manifold pM;gq;take;ras in the statement of Proposition 7.4. Let px0;0qPNepsuppqbe an arbitrary exterior normal. Then, take any PC8pRnqwhich is identically one in a conic neighborhood of 0, zero on the complement of a slightly larger conic neighborhood, and homogeneous of degree 0 away from the origin. Then Apq qcontains0because 1 on a conic neighborhood of 0, and so one may take 1 in the denition of A:Proposition 7.4 then guarantees that BrOpp qdoes not annihilate any eigenfunctions of
g;and thus neither does WBB:One can repeat this process in any nite number of coordinate patches to show that there exists W°N j1B jBj with the same property.  We now construct a pseudodierential damping coecient on T2which satises Assump- tions 1 and 2. Example 7.6. LetT2R2{Z2denote the two-dimensional torus equipped with the at metric, and let
be the associated Laplace-Beltrami operator. Let ¡0 and let1P C8 cpT2qbe supported in the vertical strip tpxp1q;xp2qq PT2:1 2
¤xp1q¤1 2uand equal to one on a smaller vertical strip. Dene r1in a similar way, but with r11 on the support of 1. Analogously, let 2PC8 cpT2qbe supported in the horizontal strip tpxp1q;xp2qqPT2:1 2
¤xp2q¤1 2uand equal to one on a smaller horizontal strip, and dener2similarly with r21 on the support of 2. Now, let"¡0 and let 1PC8pS1qbe supported in the set 2"

 4
2"; 42" Y3 4
2";5 42" and equal to one on the smaller set  ". Similarly, let 2PC8pS1qbe nonzero on  2" 2 and equal to one on  " 2. ChoosePC8 cpRqto be supported in r1 4;8qand equal to one onr1 2;8q. Then dene symbols bjPS0 clpTT2qby bjpq jpqprq; j1;2; where pr;qin standard polar coordinates on T xT2:Figure 1 illustrates the cone of directions in T x0T2in whichb1is supported at some arbitrary x0Psupp1:Now dene BjrjOppbjqj, and set the damping coecient Wto be WB 1B1B 2B2: To see kerWcontains no nontrivial eigenfunctions of
we apply Proposition 7.4. Note Npsupp1qcontains all points of the form px;qwithxPBpsupp1qandpr;q, where 0 or. Sinceb1is constant in a conic neighborhood of both of these cotangent directions, the hypotheses of Proposition 7.4 are satised. Thus, ker B1contains no eigenfunctions of the Laplacian. An analogous argument holds for B2, and since B 1B1andB 2B2are nonnegative operators,Wcannot annihilate any eigenfunctions of the Laplacian. To show exponential energy decay with Was the damping coecient, we must also demon- strate that Wsatises the AGCC. For this, it is convenient to observe that the AGCC is equivalent to the existence of some T0¡0 andc¡0 such that every trajectory tÞÑ'tpx0;0q encounters the set Wctpx;qPTT2:wpx;q¥c¡0u30 B. KEELER AND P. KLEINHENZ Figure 1. The cone of directions containing the support of b1px0;qinT x0T2. in timeT¤T0:Recall that the geodesics on T2are the projections of straight lines in R2 under the quotient map. Thus, the geodesic ow on ST2is given by px;qÞÑppxtqmodZ2;q: Given an arbitrary point px0;0qPST2, we will show that p ptq; 1ptqqppx0t0qmodZ2;0q must intersect Wcin some xed time T0¡0. Let us write 0PS1aspcos0;sin0q, and consider the case where 0lies in  ". Suppose rst that 0P

 4
"; 4" ; which implies b1p0q0. Then, if x0pxp1q 0;xp2q 0q, the horizontal coordinate of ptqis given by pxp1q 0tcos0qmodZ; which must reach1 2in some time less than1 cos0¤1 cosp{4"q. Therefore,p ptq; 1ptqqintersects the region where b1is strictly positive in time less than1 cosp{4"q. The same argument holds if instead0Pp3 4
";5 4"q, and so whenever 0P", we have that there exists a c¡0 such thatp ptq; 1ptqqintersectstb1px;q¥?cuin nite time. Analogously, if 0P" 2, then the vertical component of ptq, given bypxp2q 0tsin0qmodZ, must equal1 2in some time less than1 sinp{4
"q. Therefore, p ptq; 1ptqqintersectstb2px;q ¥?cuin nite time. Since T2
"Yp" 2q ST2; and sincewpx;q b2 1px;qb2 2px;q, we have that for every px0;0q PST2, the curve 'tpx0;0qintersects Wcin some xed time T0¡0. We have therefore shown that Was dened here satises both Assumptions 1 and 2. Thus by Theorem 1, all solutions to the damped wave equation on T2with damping coecient Wexperience exponential energy decay. Remark 7.7. In the previous example, one may notice that on the intersection of the vertical and horizontal strips, the principal symbol of the damping coecient is supportedSHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 31 in all directions PTT2z0:So in this region, Wbehaves very much like a multiplication operator for frequencies away from zero. A natural question is whether or not there must always be a point of \full microsupport" if the hypotheses of Theorem 1 are to be satised. In fact, there need not be such a point. To see this, we can modify our example above as follows. Dene1;r1; 2;r2andb1in a similar fashion to the previous example, but now dene b2to be supported only in the directions with angle Pp 4
2";3 42"qand identically one onp 4
";3 4"q. Next, we introduce another horizontal strip, disjoint from the rst, with a corresponding pair of cuto functions 3;r3. Then, dene 3PC8pS1qto be supported inp5 4
2";7 42"qand equal to one on p5 4
";7 4"q, and letb3pq 3pqprq, where pr;qas before. This is illustrated in Figure 2. Then, if we dene B3r3Oppb3q3and setW°3 j1B jBj;we can apply arguments similar to those above to see that Assumptions 1 and 2 are still satised, but there does not exist any point xPT2wherewpx;qis supported in all directions. Figure 2. The cones containing the supports of b2px0;qandb3px1;q. References [AL14] N. Anantharaman and M. L eautaud. Sharp polynomial decay rates for the damped wave equation on the torus. Anal. PDE , 7(1):159{214, 2014. doi:10.2140/apde.2014.7.159. With an appendix by St ephane Nonnenmacher. [BC15] N. Burq and H. Christianson. Imperfect geometric control and overdamping for the damped wave equation. Communications in Mathematical Physics , 336(1):101{130, 2015. [BH07] N. Burq and M. Hitrik. Energy decay for damped wave equations on partially rectangular domains. Mathematical Research Letters , 14(1):35{47, 2007. [Bur98] N. Burq. Contr^ ole de l' equation des ondes dans des ouverts comportant des coins. Bull. Soc. Math. France , 126(4):601{637, 1998. URL http://www.numdam.org/item?id=BSMF_1998_ _126_4_601_0 . Appendix B written in collaboration with Jean-Marc Schlenker. [BZ16] N. Burq and C. Zuily. Concentration of laplace eigenfunctions and stabilization of weakly damped wave equation. Communications in Mathematical Physics , 345(3):1055{1076, 2016. [Chr07] H. Christianson. Semiclassical non-concentration near hyperbolic orbits. Journal of Functional Analysis , 246(2):145{195, 2007. [Chr10] H. Christianson. Corrigendum to \semiclassical non-concentration near hyperbolic orbits" [j. funct. anal. 246(2) (2007) 145{195]. Journal of Functional Analysis , 258(3):1060{1065, 2010.32 B. KEELER AND P. KLEINHENZ [Cra08] I. J. D. Craig. Anisotropic viscous dissipation in compressible magnetic x-points. Astrony & Astrophysics , 487(3):1155{1161, 2008. doi:10.1051/0004-6361:200809960. [CSVW14] H. Christianson, E. Schenck, A. Vasy, and J. Wunsch. From resolvent estimates to damped waves. J. Anal. Math. , 121(1):143{162, 2014. [DJN19a] S. Dyatlov, L. Jin, and S. Nonnenmacher. Control of eigenfunctions on surfaces of variable curvature. arXiv:1906.08923 , 2019. [DJN19b] S. Dyatlov, L. Jin, and S. Nonnenmacher. Control of eigenfunctions on surfaces of variable curvature. arXiv preprint arXiv:1906.08923 , 2019. [DK20] K. Datchev and P. Kleinhenz. Sharp polynomial decay rates for the damped wave equation with h older-like damping. Proc. Amer. Math. Soc. , 2020. doi:10.1090/proc/15018. [G er91] P. G erard. Microlocal defect measures. Communications in Partial Dierential Equations , 16(11):1761{1794, 1991, https://doi.org/10.1080/03605309108820822. doi:10.1080/03605309108820822. [GK69] I. C. Gohberg and M. G. Kre n. Introduction to the theory of linear nonselfadjoint operators . Translations of Mathematical Monographs, Vol. 18. American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. [H or83] L. H ormander. The Analysis of Linear Partial Dierential Operators I . Berlin: spring-verlag, 1983. doi:10.1007/978-3-642-61497-2. [Jin20] L. Jin. Damped wave equations on compact hyperbolic surfaces. Communications in Mathemat- ical Physics , 373(3):771{794, 2020. [JSC11] S. V. Joubert, M. Y. Shatalov, and C. E. Coetzee. Analysing manufacturing imperfections in a spherical vibratory gyroscope. In 2011 4th IEEE International Workshop on Advances in Sensors and Interfaces (IWASI) , pages 165{170. IEEE, 2011. [KKBH16] D. Krattiger, R. Khajehtourian, C. L. Bacquet, and M. I. Hussein. Anisotropic dissipation in lattice metamaterials. AIP Advances , 6(12):121802, 2016. [Kle17] G. Klein. Best exponential decay rate of energy for the vectorial damped wave equation. SIAM Journal on Control and Optimization , 56, 07 2017. doi:10.1137/17M1142636. [Kle19a] P. Kleinhenz. Stabilization Rates for the Damped Wave Equation with H older-Regular Damping. Commun. Math. Phys. , 369(3):1187{1205, 2019. [Kle19b] P. Kleinhenz. Decay rates for the damped wave equation with nite regularity damping. arXiv preprint arXiv:1910.06372 , 2019. [Leb96] G. Lebeau. Equation des ondes amorties. In Algebraic and Geometric Methods in Mathematical Physics: Proceedings of the Kaciveli Summer School, Crimea, Ukraine, 1993 , pages 73{109. Springer Netherlands, Dordrecht, 1996. [LL17] M. L eautaud and N. Lerner. Energy decay for a locally undamped wave equation. Annales de la facult e des sciences de Toulouse S er.6 , 26(1):157{205, 2017. [LR05] Z. Liu and B. Rao. Characterization of polynomial decay rate for the solution of linear evolution equation. Zeitschrift f ur angewandte Mathematik und Physik ZAMP , 56(4):630{644, 2005. [LRLTT17] J. Le Rousseau, G. Lebeau, P. Terpolilli, and E. Tr elat. Geometric control condition for the wave equation with a time-dependent observation domain. Analysis & PDE , 10(4):983{1015, 2017. [LRZ02] K. Liu, B. Rao, and X. Zhang. Stabilization of the wave equations with potential and indenite damping. Journal of mathematical analysis and applications , 269(2):747{769, 2002. [Ral69] J. Ralston. Solutions of the wave equation with localized energy. Communications on Pure and Applied Mathematics , 22(6):807{823, 1969. [Ral82] J. Ralston. Gaussian beams and the propagation of singularities. Studies in Partial Dierential Equations, MAA Studies in Mathematics , 23:206{248, 1982. [RT75] J. Rauch and M. Taylor. Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. , 24(1):79{86, 1975. [Sj o00] J. Sj ostrand. Asymptotic distribution of eigenfrequencies for damped wave equations. Publica- tions of the Research Institute for Mathematical Sciences , 36(5):573{611, 2000. [Sta17] R. Stahn. Optimal decay rate for the wave equation on a square with constant damping on a strip. Zeitschrift f ur angewandte Mathematik und Physik , 68(2):36, 2017. [Str72] R. Strichartz. A functional calculus for elliptic pseudo-dierential operators. American Journal of Mathematics , 94(3):711{722, 1972.SHARP DECAY FOR ANISOTROPICALLY DAMPED WAVES 33 [Sun22] C. Sun. Sharp decay rate for the damped wave equation with convex-shaped damping. Interna- tional Mathematics Research Notices , 2022. Email address :bkeeler@live.unc.edu Department of Mathematics and Statistics, McGill University, 805 Rue Sherbrooke Ouest, Montr eal, QC H3A 0B9 Email address :pkleinhe@gmail.com Department of Mathematics, Michigan State University, 619 Cedar River Rd, East Lans- ing, MI 48823

2020-09-22

In this article, we study energy decay of the damped wave equation on compact Riemannian manifolds where the damping coefficient is anisotropic and modeled by a pseudodifferential operator of order zero. We prove that the energy of solutions decays at an exponential rate if and only if the damping coefficient satisfies an anisotropic analogue of the classical geometric control condition, along with a unique continuation hypothesis. Furthermore, we compute an explicit formula for the optimal decay rate in terms of the spectral abscissa and the long-time averages of the principal symbol of the damping over geodesics, in analogy to the work of Lebeau for the isotropic case. We also construct genuinely anisotropic dampings which satisfy our hypotheses on the flat torus.

Sharp exponential decay rates for anisotropically damped waves

2009.10832v2

arXiv:1208.1462v1 [cond-mat.str-el] 7 Aug 2012ObservationofCoherent HelimagnonsandGilbertdamping in anItinerant Magnet J. D. Koralek1,∗,†, D. Meier2,∗,†, J. P. Hinton1,2, A. Bauer3, S. A. Parameswaran2, A. Vishwanath2, R. Ramesh1,2R. W. Schoenlein1, C. Pfleiderer3, J. Orenstein1,2 1Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, California 94720, USA 2Department of Physics, University of California, Berkeley , California 94720, USA and 3Physik Department E21, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany (Dated: DatedAugust 30, 2018) We study the magnetic excitations of itinerant helimagnets by applying time-resolved optical spectroscopy to Fe0.8Co0.2Si. Optically excited oscillations of the magnetization in the helical state are found to disperse to lower frequency as the applied magnetic field is increased ; the fingerprint of collective modes unique to helimagnets,knownashelimagnons. Theuseoftime-resolve dspectroscopyallowsustoaddressthefundamen- tal magnetic relaxation processes by directly measuring th e Gilbert damping, revealing the versatility of spin dynamics inchiralmagnets. Theconceptofchiralitypervadesallofscience,havingpro - found implications in physics, chemistry and biology alike . In solids, relativistic spin-orbit coupling can give rise t o the Dzyaloshinskii-Moriya (DM) interaction,2,3imparting a ten- dency for the electron spins to form helical textures with a well-definedhandednessincrystalslackinginversionsymm e- try. Helical spinorderisespeciallyinterestingwhenthem ag- netismarisesfromthesameelectronsresponsibleforcondu c- tion as is the case in doped FeSi which displays unconven- tional magnetoresitence,4,5helimagnetism,6and the recently discovered Skyrmion lattice.7,8The excitations of helimag- nets have been studied over the past 30 years, culminating recently in a comprehensive theory of spin excitations call ed helimagnons.9,10Signatures of helimagnons have been ob- served in neutron scattering11and microwave absorption,12 yet little is known about their magnetodynamics and relax- ation phenomenaon the sub-picosecondtimescales on which magnetic interactions occur. Understanding the dynamics, however,isofgreatimportanceregardingspintransfertor que effects in chiral magnets, and related proposed spintronic s applications.13–15 In this work we study the dynamics of collective spin ex- citationsin the itineranthelimagnetFe 0.8Co0.2Si. Ouroptical pump-probemeasurementsidentifyanomalousmodesatzero wavevector ( q=0) which we identify unmistakably as heli- magnons. These helimagnons manifest as strongly damped magnetization oscillations that follow a characteristic s caling relation with respect to temperature and magnetic field. The sub-picosecond time resolution of our technique enables de - terminationof the intrinsic Gilbert dampingparameterwhi ch is foundto be oneorderof magnitudelargerthan in localized systems, revealing the versatility of the spin-lattice int erac- tions available in the emergent class of DM-driven helimag- nets. Despite being a non-magnetic insulator, FeSi is trans- formed into an itinerant magnet upon doping with cobalt.4,16 We have chosen Fe 0.8Co0.2Si for our study because it can easily be prepared in high quality single crystals17with a reasonably high magnetic ordering temperature TN, and its exotic equilibrium properties are well characterized, ope n- ing the door for non-equilibrium dynamical studies. Small- angle neutron scattering8was used to determine the phase diagram and has revealed helimagnetic spin textures belowTN=30 K that emerge from the interplay between the fer- romagnetic exchange and DM interactions. In zero magnetic field the spins form a proper helix with a spatial period of ≈350˚A,18whereasfinitefieldscantthespinsalongthehelix wavevector, kh, (see Fig. 1(c))inducinga conicalstate witha net magnetization. Sufficiently high fields, H≥Hc, suppress the conical order in favor of field alignment of all spins. In theexperimentsreportedhere,femtosecondpulsesoflinea rly polarized 1.5 eV photons from a Ti:Sapphire oscillator were used to excite a (100) oriented single crystal of Fe 0.8Co0.2Si at near normal incidence. The changes induced in the sam- ple by the pump pulse were probed by monitoringthe reflec- tion and Kerr rotation of time-delayed probe pulses from the same laser. In order to minimize laser heating of the sam- ple the laser repetition rate was reduced to 20 MHz with an electro-optic pulse picker, ensuring that thermal equilib rium wasreachedbetweensuccessivepumppulses. Signaltonoise was improved by modulation of the pump beam at 100 KHz andsynchronouslock-indetectionofthereflectedprobe. Ke rr rotation was measured using a Wollaston prism and balanced photodiode. All temperature and field scans presented in thi s work were performed from low to high TandH||(100)after zero-fieldcooling. Fig. 1 shows the transient reflectivity, ΔR/R, as a function of temperature and magnetic field. At high temperature we observe a typical bolometric response from transient heati ng of the sample by the pump pulse (Fig 1 (a)).19This is char- acterized by a rapid increase in reflectivity, followed by tw o- component decay on the fs and ps timescales, corresponding to the thermalizationtimes between differentdegreesof fr ee- dom (electron, spin, lattice, etc.).20As the sample is cooled belowTN, the small thermal signal is beset by a much larger negative reflectivity transient (Fig. 1 (b)) with a decay tim e of roughly τR≈175 ps at low temperature (Fig. 3 (b)). A natural explanation for this is that the pump pulse weakens the magnetic order below TN, which in turn causes a change in reflectivity via the resulting shift of spectral weight to low energy.21Thetemperaturedependenceof the peak ΔR/Rval- ues is plotted in Fig. 1 (c) for several applied fields, showin g onlyweakfielddependence. Toaccessthemagnetizationdynamicsmoredirectlywean- alyze the polarizationstate of the probe pulses, which rota tes by an angle θKupon reflection from the sample surface, in2 FIG.1: Timedependence ofthepump-inducedtransientreflec tivityΔR/Rinthe(a)paramagneticand(b)helimagneticstates. Thetem perature dependence of the maximum ΔR/Ris plottedin(c)for several applied magnetic fields. proportion to the component of the magnetization along the light trajectory. The change in Kerr rotation induced by the optical pump, ΔθK, is shown in Fig. 2 as a function of tem- perature and field. The upper panels show temperature scans at fixedmagneticfield,while afieldscan atfixedtemperature isshowninpanel(d). Weobservethat ΔθKchangessignas H isreversed(notshown),andgoestozeroas Hgoesto zeroor as temperatureis raised above TN. Oscillationsof the magne- tization are clearly visible in the raw data below 25 K in the helimagneticphase. In order to analyze the magnetization dynamics, we use a simple phenomenological function that separates the oscil la- tory and non-oscillatorycomponentsseen in the data. It con - sists ofadecayingsinusoidaloscillation, ΔθK=e−t τK[A+Bsin(ωt)] (1) witha timedependentfrequency, ω(t)=2πf0/bracketleftBig 1+0.8/parenleftBig e−t τK/parenrightBig/bracketrightBig (2) which decays to a final value ω0. We emphasize that there is only a single decay time τKdescribing the magneto dynam- ics, and it is directly related to the Gilbert damping parame - terα=(2πf0τK)−1. This function produces excellent fits to the data as illustrated in Fig. 3 (a), allowing accurate extr ac- tion of the oscillation frequencies and decay times shown in Figs. 3 (b)-(d). The oscillation frequency is reduced as ei- ther field or temperature is increased, while the decay time τKis roughly constant and equal to τRbelow 20 K. As the temperature is raised towards the phase transition, the rel ax- ation time τKdiverges, which can be understood in terms of a diverging magnetic correlation length due to the presence of a critical point. The similarity between the decay times τR andτKwithin the ordered phase reflects strongly correlated charge and spin degrees of freedom, and supports the notion thatΔR/Risdeterminedbythemagneticorder. Themagneticoscillationfrequencyreaches f0≈4.8GHzat lowtemperature,whichcorrespondstoaLarmorprecessiono f spinssubjectedtoafieldof170mT,whichisroughlythecrit- ical field Hcrequiredto destroy the spin helix. This, togetherwith the fact that the oscillation frequencyis nonzero only in thehelicalstate,suggeststhattheoscillationsarecomin gfrom excitations unique to the helical structure. It is well know n that magnetization oscillations can be optically induced b y ultrafast generation of coherent magnons,24–26however, or- dinary magnons cannot explain our data as their frequency wouldincreasewith H,oppositetowhatisseenin Fig. 3(c). Based on these observations, we propose the following in- terpretation of our results: In the helical magnetic phase, the pump photons weaken the magnetic order through the ultra- fast demagnetization process.27As described above, this re- duction in magnetic order gives rise to a decrease in the re- flectivity at 1.5 eV which is nearly field independent. As a magnetic field is applied the spins become canted along the helix wavevector,giving rise to a macroscopic magnetizati on whichweobserveinKerrrotationviaitscomponentalongthe probelight trajectory. The demagnetizationfromthe pumpi s responsible for the initial peak seen in the ΔθKtime traces, and is captured by the exponential component of our fitting function (green curve in Fig 3 (a)). The pump photons also launch a coherentspin wave, giving rise to the oscillations in ΔθK(red curve in Fig. 3 (a)). The form of the oscillatory component goes like sin (ωt)rather than [1−cos(ωt)], sug- gesting impulsive stimulated Raman scattering as the mecha - nism of excitation.25The anomalousfield dependenceshown in Fig. 3 (c) leads to the unambiguous conclusion that the optically excited spin waves are the fundamental modes of helimagnetstermedhelimagnons.10Specifically,theoptically accessible helimagnon mode consists of the constituents of the spin helix precessing in-phase about their local effect ive field. Since this local effective field is reduced during the u l- trafast demagnetizationprocess, the oscillation frequen cyde- creases as a function of time delay as the field recovers, ne- cessitatingthetimedependentfrequencyinEq. 1. Theabili ty to resolve helimagnons with femtosecond time resolution at q=0isuniquetoouropticalprobe,andcomplimentsneutron scatteringwhichisrestrictedtomappinghelimagnonbands at higherq. This regionof reciprocalspace is particularlyinter- estinginthecaseofhelimagnetsastheperiodicityintrodu ced bythehelicalspintexturegeneratesbandsthatarecentere dat q=±khand therefore have finite frequency modes at q=03 FIG. 2: (a),(b),(c) Time dependence of the pump-induced cha nge in Kerr rotation, ΔθK, as a function of temperature for several applied magneticfields. (d) ΔθKasafunctionofmagneticfieldforseveraltemperatures. Cur vesareoffsetforclarity. Alsoshownisaschematicphase diagram, adapted from Reference 8,withredarrows illustra tingthe temperature and fieldscans usedin(a)-(d). even in the absence of a gap. This is in contrast to ordinary magnonsinwhichthebandsaregenerallycenteredat q=0so thattheassociatedmodehaszerofrequency. Wenotethatour observationsare in agreementwith previouswork on the col- lectivemodesofskyrmions28whichcoexistwithhelimagnons in theA-Phase(see Fig. 3).12Theappearanceofthese modes is not expected in our data as their corresponding oscillati on periodsexceedtheobserveddampingtimein Fe 0.8Co0.2Si. Inordertoquantitativelytestthehelimagnoninterpretat ion wetaketheexpressionforthe q=0helimagnonfrequencyin anexternalmagneticfield, f0=gµBHc/radicalbig 1+cos2θ (3) wheregistheeffectiveelectron g-factor,µBistheBohrmag- neton,andπ 2−θistheconicalanglei.e. theamountthespins are canted away from khby the applied field H. Ignoringde- magnetization effects of the spin waves themselves, we can write sinθ=H Hc, whereHcis the critical field at which the spinsall alignwiththe field andthe helimagnonceasesto ex- ist asa well-definedmode. Thenweobtain,9 f0=gµBHc/radicalBigg 1−1 2/parenleftbiggH Hc/parenrightbigg2 (4) which expresses the magnon frequency as a function of ap- plied field. This expression fits the data remarkably well as shown in Fig. 3 (c), capturingthe decrease in frequencywith increasing Hwhich is unique to helimagnons. However, due tothefactthattheoscillationperiodexceedsthedampingt imeforfieldsabove75mT,itisnotpossibletoextractthevalueo f thecritical field Hcinthis system. The solidline in Fig. 3(d) isafittotheform f0∝/radicalBig 1−T TNwhichgives TNasafunction ofHin reasonableagreementwithpublisheddata.8 The Gilbert damping parameter can be directly obtained from the measured decay times through the relation α= (2πf0τK)−1, which gives a value of α≈0.4 for the heli- magnetic phase of Fe 0.8Co0.2Si. This is an order of magni- tude larger than what was seen in insulating Cu 2OSeO3,12 where helimagnetism arises from localized rather than itin - erant spins. The contrast in dynamics between these systems is critical in the context of potential spintronic applicat ions basedonhelimagnetismwherethereisatradeoffbetweenfas t switching which requires large damping, and stability whic h reliesonlowdamping. In summary, this work demonstrates ultrafast coherent op- tical excitation of spin waves in an itinerant DM-driven spi n system and reveals the underlying spin dynamics. We iden- tifytheseexcitationsashelimagnonsthroughtheiranomal ous field dependence and explain our observations with a com- prehensive model. Our experiments directly yield the intri n- sic Gilbertdampingparameter,revealingastrikingdiffer ence in spin relaxationphenomenabetweenitinerant andlocaliz ed helimagnets. The results elucidate the dynamicsof collect ive modes common to the actively studied B20 transition metal compounds that codetermine their performance in potential spinbasedapplications. Acknowledgments: The work in Berkeley was supported by the Director, Office of Science, Office of Basic Energy4 FIG.3: (a) Exemplary ΔθKoscillation data (blue circles) and fit (black line) using th e model described inthe text. The fitis decomposed into an exponential term (green curve) and an oscillatory term (r ed curve). The fitting function uses a single time constant τKfor all terms which is 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 all fields. The solid lines are guides to the eye. Panels (c) an d (d) show the reduced magnetization oscillation frequency for field scans and temperature scans respectively, andsolidlines are fitstot he data as described inthe maintext. Sciences,MaterialsSciencesandEngineeringDivision,of the U.S. Department of Energy under Contract No. DE-AC02- 05CH11231. C.P. and A.B. acknowledge support through DFGTRR80(FromElectronicCorrelationstoFunctionality) , DFG FOR960 (Quantum Phase Transitions), and ERC AdG (291079, TOPFIT). A.B. acknowledges financial support through the TUM graduate school. D.M. acknowledges sup-portfromtheAlexandervonHumboldtfoundationandS.A.P. acknowledgessupportfrom the SimonsFoundation. C.P. and A.B. also thank S. Mayr, W. M¨ unzer, and A. Neubauer for assistance. ∗Theseauthorscontributedequallytothiswork. †Email address: jdkoralek@lbl.gov and meier@berkeley.edu 2I. 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Lett. 102, 037204 (2009). 19R.W.Schoenlein,W.Z.Lin,G.FujimotoandG.L.Besley,Phys . Rev. Lett. 581680 (1987). 20S. I. Anisimov, B. L. Kapeliovich and T. L. Perelman, Sov. Phy s. JETP39, 375 (1975). 21F.P.Mena etal.,Phys. Rev. B 73, 085205 (2006). 22S.A.Brazovskii,S.G.Dmitriev,Sov.Phys.JETP 42,497(1976). 23M. Janoschek, M. Garst, A. Bauer, P. Krautscheid, R. Georgii , P.B¨ oni, andC.Pfleiderer,arXiv:1205.4780v1(2012). 24M. van Kampen etal.,Phys. Rev. Lett. 88, 227201 (2002). 25A. M.Kalashnikova et al.,Phys.Rev. B 78, 104301 (2008). 26D. Talbayev et al.,Phys. Rev. Lett. 101, 097603 (2008). 27A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010). 28M. Mochizuk, Phys. Rev. Lett. 108, 017601 (2012).

2012-08-07

We study the magnetic excitations of itinerant helimagnets by applying time-resolved optical spectroscopy to Fe0.8Co0.2Si. Optically excited oscillations of the magnetization in the helical state are found to disperse to lower frequency as the applied magnetic field is increased; the fingerprint of collective modes unique to helimagnets, known as helimagnons. The use of time-resolved spectroscopy allows us to address the fundamental magnetic relaxation processes by directly measuring the Gilbert damping, revealing the versatility of spin dynamics in chiral magnets. (*These authors contributed equally to this work)

Observation of Coherent Helimagnons and Gilbert damping in an Itinerant Magnet

1208.1462v1

arXiv:0708.3323v1 [cond-mat.mes-hall] 24 Aug 2007Enhancement of the Gilbert damping constant due to spin pump ing in non-collinear ferromagnet / non-magnet / ferromagnet trilayer systems Tomohiro Taniguchi1,2, Hiroshi Imamura2 1Institute for Materials Research, Tohoku University, Send ai 980-8577, 2Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan (Dated: October 29, 2018) We analyzed the enhancement of the Gilbert damping constant due to spin pumping in non- collinear ferromagnet / non-magnet / ferromagnet trilayer systems. We show that the Gilbert damping constant depends both on the precession angle of the magnetization of the free layer and on the direction of the magntization of the fixed layer. We find the condition to be satisfied to realize strong enhancement of the Gilbert damping constant . PACS numbers: 72.25.Mk, 75.70.Cn, 76.50.+g, 76.60.Es There is currently great interest in the dynamics of magnetic multilayers because of their potential applica- tions in non-volatile magnetic random access memory (MRAM) and microwave devices. In the field of MRAM, much effort has been devoted to decreasing power con- sumption through the use of current-induced magnetiza- tion reversal (CIMR) [1, 2, 3, 4, 5, 6, 7]. Experimentally, CIMR is observed as the current perpendicular to plane- type giant magnetoresistivity (CPP-GMR) of a nano pil- lar, in which the spin-polarized current injected from the fixed layer exerts a torque on the magnetization of the free layer. The torque induced by the spin current is utilized to generate microwaves. The dynamics of the magnetization Min a ferromag- net under an effective magnetic field Beffis described by the Landau-Lifshitz-Gilbert (LLG) equation dM dt=−γM×Beff+α0M |M|×dM dt,(1) whereγandα0are the gyromagnetic ratio and the Gilbert damping constant intrinsic to the ferromagnet, respectively. The Gilbert damping constant is an im- portant parameter for spin electronics since the critical current density of CIMR is proportional to the Gilbert damping constant [8, 9] and fast-switching time magne- tization reversal is achieved for a large Gilbert damp- ing constant [10]. Several mechanisms intrinsic to ferro- magnetic materials, such as phonon drag [11] and spin- orbit coupling [12], have been proposed to account for the origin of the Gilbert damping constant. In addition to these intrinsic mechanisms, Mizukami et al.[13, 14] and Tserkovnyak et al.[15, 16] showed that the Gilbert dampingconstantinanon-magnet(N) /ferromagnet(F) / non-magnet(N) trilayersystem is enhanced due to spin pumping. Tserkovnyak et al.[17] also studied spin pump- ing in a collinear F/N/F trilayer system and showed that enhancement of the Gilbert damping constant depends on the precession angle of the magnetization of the free layer. On the other hand, several groups who studied CIMR in a non-collinear F/N/F trilayer system in which theFIG. 1: (Color online) The F/N/F trilayer system is schemat- ically shown. The magnetization of the F 1layer (m1) pre- cesses around the z-axis with angle θand angular velocity ω. The magnetization of the F 2layer (m2) is fixed with tilted angleρ. The precession of the magnetization in the F 1layer pumpsspin current Ipump sintotheNandF 2layer, andcreates the spin accumulation µNin the N layer. The spin accumu- lation induces the backflow spin current Iback(i) s(i= 1,2). magnetization of the free layer is aligned to be perpen- dicular to that of the fixed layer have reported the reduc- tion of the critical current density [5, 6, 7]. Therefore, it is intriguing to ask how the Gilbert damping constant is affected by spin pumping in non-collinear F/N/F trilayer systems. In this paper, we analyze the enhancement of the Gilbert damping constant due to spin pumping in non- collinear F/N/F trilayer systems such as that shown in Fig. 1. Following Refs. [15, 16, 17, 18], we calculate the spin current induced by the precession of the magnetiza- tion of the free layer and the enhancement of the Gilbert damping constant. We show that the Gilbert damping constant depends not only on the precession angle θof the magnetization of a free layer but also on the angle ρ betweenthemagnetizationsofthefixedlayerandthepre- cession axis. The Gilbert damping constant is strongly enhanced if angles θandρsatisfy the condition θ=ρor θ=π−ρ. The system we consider is schematically shown in Fig. 1. A non-magnetic layer is sandwiched between two fer- romagnetic layers, F 1and F 2. We introduce the unit2 vectormito represent the direction of the magnetiza- tion of the i-th ferromagnetic layer. The equilibrium direction of the magnetization m1of the left free fer- romagnetic layer F 1is taken to exist along the z-axis. When an oscillatingmagnetic field is applied, the magne- tization of the F 1layer precesses around the z-axis with angleθ. The precession of the vector m1is expressed asm1= (sinθcosωt,sinθsinωt,cosθ), whereωis the angular velocity of the magnetization. The direction of the magnetization of the F 2layer,m2, is assumed to be fixed and the angle between m2and thez-axis is repre- sented byρ. The collinear alignment discussed in Ref. [17] corresponds to the case of ρ= 0,π. Before studying spin pumping in non-collinear sys- tems, we shall give a brief review of the theory of spin pumping in a collinear F/N/F trilayer system [17]. Spin pumping is the inverse process of CIMR where the spin current induces the precession of the magnetization. Contrary to CIMR, spin pumping is the generation of the spin current induced by the precession of the mag- netization. The spin current due to the precession of the magnetization in the F 1layer is given by Ipump s=/planckover2pi1 4πg↑↓m1×dm1 dt, (2) whereg↑↓is a mixing conductance [18, 19] and /planckover2pi1is the Dirac constant. Spins are pumped from the F 1layer into the N layer and the spin accumulation µNis cre- ated in the N layer. Spins also accumulate in the F 1 and F 2layers. In the ferromagnetic layers the trans- verse component of the spin accumulation is assumed to be absorbed within the spin coherence length defined as λtra=π/|k↑ Fi−k↓ Fi|, wherek↑,↓ Fiis the spin-dependent Fermi wave number of the i-th ferromagnet. For fer- romagnetic metals such as Fe, Co and Ni, the spin co- herence length is a few angstroms [20]. Hence, the spin accumulation in the i-th ferromagnetic layer is aligned to be parallel to the magnetization, i.e., µFi=µFimi. The longitudinal component of the spin accumulation decays on the scale of spin diffusion length, λFi sd, which is of the order of 10 nm for typical ferromagnetic metals [21]. The difference in the spin accumulation of ferromag- netic and non-magnetic layers, ∆ µi=µN−µFimi(i= 1,2), induces a backflow spin current, Iback(i) s, flowing into both the F 1and F 2layers. The backflow spin cur- rentIback(i) sis obtained using circuit theory [18] as Iback(i) s=1 4π/braceleftbigg2g↑↑g↓↓ g↑↑+g↓↓(mi·∆µi)mi +g↑↓mi×(∆µi×mi)/bracerightbig ,(3) whereg↑↑andg↓↓are the spin-up and spin-down con- ductances, respectively. The total spin current flowing out of the F 1layer is given by Iexch s=Ipump s−Iback(1) s [17]. The spin accumulation µFiin the F ilayer is ob- tained by solving the diffusion equation. We assume that spin-flip scattering in the N layer is so weak thatwe can neglect the spatial variation of the spin current within the N layer, Iexch s=Iback(2) s. The torque τ1 acting on the magnetization of the F 1layer is given by τ1=Iexch s−(m1·Iexch s)m1=m1×(Iexch s×m1). For the collinear system, we have τ1=g↑↓ 8π/parenleftbigg 1−νsin2θ 1−ν2cos2θ/parenrightbigg m1×dm1 dt,(4) whereν= (g↑↓−g∗)/(g↑↓+g∗) is the dimensionless parameter introduced in Ref. [17]. The Gilbert damping constant in the LLG equation is enhanced due to the torqueτ1asα0→α0+α′with α′=gLµBg↑↓ 8πM1dF1S/parenleftbigg 1−νsin2θ 1−ν2cos2θ/parenrightbigg ,(5) wheregLis the Land´ e g-factor,µBis the Bohr magneton, dF1is the thickness of the F 1layer andSis the cross- section of the F 1layer. Next, we move on to the non-collinear F/N/F trilayer system with ρ=π/2, in which the magnetization of the F2layer is aligned to be perpendicular to the z-axis. Fol- lowing a similar procedure, the LLG equation for the magnetization M1in the F 1layer is expressed as dM1 dt=−γeffM1×Beff+γeff γ(α0+α′)M1 |M1|×dM1 dt,(6) whereγeffandα′are the effective gyromagnetic ratio and the enhancement of the Gilbert damping constant, respectively. The effective gyromagnetic ratio is given by γeff=γ/parenleftbigg 1−gLµBg↑↓νcotθcosψsinωt 8πMdF1Sǫ/parenrightbigg−1 ,(7) where cosψ= sinθcosωt=m1·m2and ǫ= 1−ν2cos2ψ−ν(cot2θcos2ψ−sin2ψ+sin2ωt).(8) The enhancement of the Gilbert damping constant is ex- pressed as α′=gLµBg↑↓ 8πMdF1S/parenleftbigg 1−νcot2θcos2ψ ǫ/parenrightbigg .(9) Itshouldbenotedthat, fornon-collinearsystems, both thegyromagneticratioandtheGilbert dampingconstant are modified by spin pumping, contrary to what occurs in collinear systems. The modification of the gyromag- netic ratio and the Gilbert damping constant due to spin pumping can be explained by considering the pumping spincurrentandthe backflowspincurrent[SeeFigs. 2(a) and 2(b)]. The direction of the magnetic moment car- ried by the pumping spin current Ipump sis parallel to the torque of the Gilbert damping for both collinear and non-collinear systems. The Gilbert damping constant is enhanced by the pumping spin current Ipump s. On the otherhand, the directionofthe magneticmoment carried3 FIG. 2: (Color online) (a) Top view of Fig. 1. The dotted circle in F 1represents the precession of magnetization M1 and the arrow pointing to the center of this circle represent s the torque of the Gilbert damping. The arrows in Ipump sand Iback(1) srepresent the magnetic moment of spin currents. (b) The back flow Iback(1) shas components aligned with the di- rection of the precession and the Gilbert damping. by the backflow spin current Iback(1) sdepends on the di- rection of the magnetization of the F 2layer. As shown in Eq. (3), the backflowspin current in the F 2layerIback(2) s has a projection on m2. Since we assume that the spin current is constant within the N layer, the backflow spin current in the F 1layerIback(1) salso has a projection on m2. For the collinear system, both Ipump sandIback(1) s are perpendicular to the precession torque because m2 is parallel to the precession axis. However, for the non- collinear system, the vector Iback(1) shas a projection on the precession torque, as shown in Fig. 2(b). Therefore, the angular momentum injected by Iback(1) smodifies the gyromagnetic ratio as well as the Gilbert damping in the non-collinear system. Let us estimate the effective gyromagnetic ratio using realistic parameters. According to Ref. [17], the con- ductancesg↑↓andg∗for a Py/Cu interface are given byg↑↓/S= 15[nm−2] andν≃0.33, respectively. The Land´ eg-factor is taken to be gL= 2.1, magnetization is 4πM= 8000[Oe] and thickness dF1= 5[nm]. Substitut- ing these parameters into Eqs. (7) and (8), one can see that|γeff/γ−1| ≃0.001. Therefore, the LLG equation can be rewritten as dM1 dt≃ −γM1×Beff+(α0+α′)M1 |M1|×dM1 dt.(10) The estimated value of α′is of the order of 0.001. How- ever, we cannot neglect α′since it is of the same order as the intrinsic Gilbert damping constant α0[22, 23]. Experimentally, the Gilbert damping constant is mea- suredasthe width ofthe ferromagneticresonance(FMR) absorptionspectrum. LetusassumethattheF 1layerhas no anisotropy and that an external field Bext=B0ˆzisapplied along the z-axis. We also assume that the small- angle precession of the magnetization around the z-axis is excited by the oscillating magnetic field B1applied in thexy-plane. The FMR absorption spectrum is obtained as follows [24]: P=1 T/integraldisplayT 0dtαγMΩ2B2 1 (γB0−Ω)2+(αγB0)2,(11) where Ω is the angular velocity of the oscillating mag- netic field, T= 2π/Ω andα=α0+α′. Sinceαis very small, the absorption spectrum can be approximately ex- pressedasP∝α0+∝an}bracketle{tα′∝an}bracketri}htandthehighestpointofthepeak proportional to ∝an}bracketle{t1/(α0+α′)∝an}bracketri}ht, where∝an}bracketle{tα′∝an}bracketri}htrepresents the time-averaged value of the enhancement of the Gilbert damping constant. In Fig. 3(a), the time-averaged value ∝an}bracketle{tα′∝an}bracketri}htfor a non-collinear system in which ρ=π/2 is plot- ted by the solid line as a function of the precession an- gleθ. The dotted line represents the enhancement of the Gilbert damping constant α′for the collinear system given by Eq. (5). The time-averaged value of the en- hancement of the Gilbert damping constant ∝an}bracketle{tα′∝an}bracketri}httakes its maximum value at θ= 0,πfor the collinear system (ρ= 0,π). Contrary to the collinear system, ∝an}bracketle{tα′∝an}bracketri}htof the non-collinear system in which ρ=π/2 takes its maxi- mum value at θ=π/2. As shown in Fig. 2(b), the backflow spin current gives a negative contribution to the enhancement of the Gilbert damping constant. This contribution is given by the projection of the vector Iback(1) sonto the direction of the torque of the Gilbert damping, which is repre- sented by the vector m1×˙m1. Therefore, the condition to realize the maximum value of the enhancement of the Gilbert damping is satisfied if the projection of Iback(1) s ontom1×˙m1takes the minimum value; i.e., θ=ρor θ=π−ρ. We can extend the above analysis to the non-collinear systemwith anarbitraryvalue of ρ. After performingthe appropriate algebra, one can easily show that the LLG equation for the magnetization of the F 1layer is given by Eq. (6) with γeff=γ/bracketleftBigg 1−gLµBg↑↓νsinρsinωt(cotθcos˜ψ−cscθcosρ) 8πMdS˜ǫ/bracketrightBigg−1 (12) α′=gLµBg↑↓ 8πMdS/braceleftBigg 1−ν(cotθcos˜ψ−cscθcosρ)2 ˜ǫ/bracerightBigg , (13) where cos ˜ψ= sinθsinρcosωt+ cosθcosρ=m1·m2 and ˜ǫ=1−ν2cos2˜ψ −ν{(cotθcos˜ψ−cscθcosρ)2−sin2˜ψ+sin2ρsin2ωt}. (14) Substituting the realistic parameters into Eqs. (12) and (14), we can show that the effective gyromagnetic ratio4   D E  /c50/c0f/c12 /c50 /c51/c1c/c41 /c1e  /c50/c0f/c12 /c50 /c52/c50/c0f/c12/c50 /c51/c1c/c41 /c1e FIG. 3: (Color online) (a) The time-averaged value of the en- hancement of the Gilbert damping constant α′is plotted as a function of the precession angle θ. The solid line corresponds to the collinear system derived from Eq. (9). The dashed line corresponds to the non-collinear system derived from E q. (5). (b) The time-averaged value of the enhancement of the Gilbert damping constant α′of the non-collinear system is plotted as a function of the precession angle θand the an- gleρbetween the magnetizations of the fixed layer and the precession axis.γeffcan be replaced by γin Eq. (6) and that the LLG equation reduces to Eq. (10). Figure 3(b) shows the time-averaged value of the enhancement of the Gilbert damping constant ∝an}bracketle{tα′∝an}bracketri}htof Eq. (13). Again, the Gilbert damping constant is strongly enhanced if angles θandρ satisfy the condition that θ=ρorθ=π−ρ. In summary, we have examined the effect of spin pumping on the dynamics of the magnetization of mag- netic multilayers and calculated the enhancement of the Gilbertdampingconstantofnon-collinearF/N/Ftrilayer systems due to spin pumping. The enhancement of the Gilbert damping constant depends not only on the pre- cession angle θof the magnetization of a free layer but also on the angle ρbetween the magnetizations of the fixed layerand the precession axis, as shown in Fig. 3(b). We have shown that the θ- andρ-dependence of the en- hancement of the Gilbert damping constant can be ex- plained by analyzing the backflow spin current. The con- dition to be satisfied to realizestrongenhancement of the Gilbert damping constant is θ=ρorθ=π−ρ. The authors would like to acknowledge the valuable discussions we had with Y. Tserkovnyak, S. Yakata, Y. Ando, S. Maekawa, S. Takahashi and J. Ieda. This work was supported by CREST and by a NEDO Grant. [1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [2] L. Berger, Phys. Rev. B 54, 9353 (1996). [3] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em- ley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature425, 380 (2003). [4] A. Deac, K. J. Lee, Y. Liu, O. Redon, M. Li, P. Wang, J. P. Nozi´ eres, and B. Dieny, J. Magn. Magn. Mater. 290-291 , 42 (2005). [5] A. D. Kent, B. Ozyilmaz, and E. del Barco, Appl. Phys. Lett.84, 3897 (2004). [6] K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86, 022505 (2005). [7] T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl. Phys. Lett. 89, 172504 (2005). [8] J. Z. Sun, Phys. Rev. B 62, 570 (2000). [9] J. Grollier, V. Cros, H. Jaffres, A. Hamzic, J. M. George, G. Faini, J. B. Youssef, H. L. LeGall, and A. Fert, Phys. Rev. B67, 174402 (2003). [10] R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett. 92, 088302 (2004). [11] H. Suhl, IEEE Trans. Magn. 34, 1834 (1998). [12] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970). [13] S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater.239, 42 (2002).[14] S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002). [15] Y. Tserkovnyak and A. Brataas and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [16] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B66, 224403 (2002). [17] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B67, 140404(R) (2003). [18] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur. Phys. J. B 22, 99 (2001). [19] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. 84, 2481 (2000). [20] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 (2002). [21] J. Bass and W. P. Jr., J. Phys.: Condens. Matter 19, 183201 (2007). [22] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000). [23] F. Schreiber, J. Pflaum, Th. M¨ uhge, and J. Pelzl, Solid State Commun. 93, 965 (1995). [24] S. V. Vonsovskii, ed., FERROMAGNETIC RESO- NANCE (Israel Program for Scientific Translations Ltd., Jersalem, 1964).

2007-08-24

We analyzed the enhancement of the Gilbert damping constant due to spin pumping in non-collinear ferromagnet / non-magnet / ferromagnet trilayer systems. We show that the Gilbert damping constant depends both on the precession angle of the magnetization of the free layer and on the direction of the magntization of the fixed layer. We find the condition to be satisfied to realize strong enhancement of the Gilbert damping constant.

Enhancement of the Gilbert damping constant due to spin pumping in noncollinear ferromagnet/nonmagnet/ferromagnet trilayer systems

0708.3323v1

Magnetization Dynamics and Damping due to Electron-Phonon Scattering in a Ferrimagnetic Exchange Model Alexander Baral,Svenja Vollmar, and Hans Christian Schneidery Physics Department and Research Center OPTIMAS, Kaiserslautern University, P. O. Box 3049, 67663 Kaiserslautern, Germany (Dated: June 4, 2018) We present a microscopic calculation of magnetization damping for a magnetic \toy model." The magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless band of localized spins, and the magnetization damping is due to coupling of the itinerant carriers to a phonon bath in the presence of spin-orbit coupling. Using a mean-eld approximation for the kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we derive Boltzmann scattering integrals for the distributions and spin coherences in the case of an antiferromagnetic exchange splitting, including a careful analysis of the connection between lifetime broadening and the magnetic gap. For the Elliott-Yafet type itinerant spin dynamics we extract dephasing and magnetization times T1andT2from initial conditions corresponding to a tilt of the magnetization vector, and draw a comparison to phenomenological equations such as the Landau- Lifshitz (LL) or the Gilbert damping. We also analyze magnetization precession and damping for this system including an anisotropy eld and nd a carrier mediated dephasing of the localized spin via the mean-eld coupling. PACS numbers: 75.78.-n, 72.25.Rb, 76.20.+q I. INTRODUCTION There are two widely-known phenomenological ap- proaches to describe the damping of a precessing mag- netization in an excited ferromagnet: one introduced originally by Landau and Lifshitz1and one introduced by Gilbert,2which are applied to a variety of prob- lems3involving the damping of precessing magnetic mo- ments. Magnetization damping contributions and its in- verse processes, i.e., spin torques, in particular in thin lms and nanostructures, are an extremely active eld, where currently the focus is on the determination of novel physical processes/mechanisms. Apart from these ques- tions there is still a debate whether the Landau-Lifshitz or the Gilbert damping is the correct one for \intrin- sic" damping, i.e., neglecting interlayer coupling, inter- face contributions, domain structures and/or eddy cur- rents. This intrinsic damping is believed to be caused by a combination of spin-orbit coupling and scattering mechanisms such as exchange scattering between s and d electrons and/or electron-phonon scattering.4{6Without reference to the microscopic mechanism, dierent macro- scopic analyses, based, for example, on irreversible ther- modynamics or near equilibrium Langevin theory, prefer one or the other description.7,8However, material param- eters of typical ferromagnetic heterostructures are such that one is usually rmly in the small damping regime so that several ferromagnetic resonance (FMR) experiments were not able to detect a noticeable dierence between Landau Lifshitz and Gilbert magnetization damping. A recent analysis that related the Gilbert term directly to the spin-orbit interaction arising from the Dirac equa- tion does not seem to have conclusively solved this dis- cussion.9 The dephasing term in the Landau-Lifshitz form isalso used in models based on classical spins coupled to a bath, which have been successfully applied to out-of-equilibrium magnetization dynamics and magnetic switching scenarios.10The most fundamental of these are the stochastic Landau-Lifshitz equations,10{13from which the Landau-Lifshitz Bloch equations,14,15can be derived via a Fokker-Planck equation. Quantum-mechanical treatments of the equilibrium magnetization in bulk ferromagnets at nite temper- atures are extremely involved. The calculation of non-equilibrium magnetization phenomena and damp- ing for quantum spin systems in more than one dimen- sion, which include both magnetism and carrier-phonon and/or carrier-impurity interactions, at present have to employ simplied models. For instance, there have been microscopic calculations of Gilbert damping parameters based on Kohn-Sham wave functions for metallic ferro- magnets16,17and Kohn-Luttinger p-dHamiltonians for magnetic semiconductors.18While the former approach uses spin density-functional theory, the latter approach treats the anti-ferromagnetic kinetic-exchange coupling between itinerant p-like holes and localized magnetic moments originating from impurity d-electrons within a mean-eld theory. In both cases, a constant spin and band-independent lifetime for the itinerant carriers is used as an input, and a Gilbert damping constant is ex- tracted by comparing the quantum mechanical result for !!0 with the classical formulation. There have also been investigations, which extract the Gilbert damping for magnetic semiconductors from a microscopic calcula- tion of carrier dynamics including Boltzmann-type scat- tering integrals.19,20Such a kinetic approach, which is of a similar type as the one we present in this paper, avoids the introduction of electronic lifetimes because the scat- tering is calculated dynamically.arXiv:1405.2347v1 [cond-mat.mtrl-sci] 9 May 20142 The present paper takes up the question how the spin dynamics in the framework of the macroscopic Gilbert or Landau-Lifshitz damping compare to a microscopic model of relaxation processes in the framework of a rel- atively simple model. We analyze a mean-eld kinetic exchange model including spin-orbit coupling for the itin- erant carriers. Thus the magnetic mean-eld dynamics is combined with a microscopic description of damping pro- vided by the electron-phonon coupling. This interaction transfers energy and angular momentum from the itin- erant carriers to the lattice. The electron-phonon scat- tering is responsible both for the lifetimes of the itiner- ant carriers and the magnetization dephasing. The lat- ter occurs because of spin-orbit coupling in the states that are connected by electron-phonon scattering. To be more specic, we choose an anti-ferromagnetic coupling at the mean-eld level between itinerant electrons and a dispersion-less band of localized spins for the magnetic system. To keep the analysis simple we use as a model for the spin-orbit coupled itinerant carrier states a two-band Rashba model. As such it is a single-band version of the multi-band Hamiltonians used for III-Mn-V ferromag- netic semiconductors.18,21{24The model analyzed here also captures some properties of two-sublattice ferrimag- nets, which are nowadays investigated because of their magnetic switching dynamics.25,26The present paper is set apart from studies of spin dynamics in similar mod- els with more complicated itinerant band structures19,20 by a detailed comparison of the phenomenological damp- ing expressions with a microscopic calculation as well as a careful analysis of the restrictions placed by the size of the magnetic gap on the single-particle broadening in Boltzmann scattering. This paper is organized as follows. As an extended introduction, we review in Sec. II some basic facts con- cerning the Landau-Lifshitz and Gilbert damping terms on the one hand and the Bloch equations on the other. In Sec. III we point out how these dierent descriptions are related in special cases. We then introduce a micro- scopic model for the dephasing due to electron-phonon interaction in Sec. IV, and present numerical solutions for two dierent scenarios in Secs. V and VI. The rst scenario is the dephasing between two spin subsystems (Sec. V), and the second scenario is a relaxation process of the magnetization toward an easy-axis (Sec. VI). A brief conclusion is given at the end. II. PHENOMENOLOGIC DESCRIPTIONS OF DEPHASING AND RELAXATION We summarize here some results pertaining to a single- domain ferromagnet, and set up our notation. In equilib- rium we assume the magnetization to be oriented along its easy axis or a magnetic eld ~H, which we take to be thezaxis in the following. If the magnetization is tilted out of equilibrium, it starts to precess. As illustrated in Fig. 1 one distinguishes the longitudinal FIG. 1. Illustration of non-equilibrium spin-dynamics in pres- ence of a magnetic eld without relaxation (a) and within relaxation (b). component Mk, inzdirection, and the transverse part M?q M2M2 k, precessing in the x-yplane with the Larmor frequency !L. In connection with the interaction processes that re- turn the system to equilibrium, the decay of the trans- verse component is called dephasing. There are three phenomenological equations used to describe spin de- phasing processes: 1. The Bloch(-Bloembergen) equations27,28 @ @tMk(t) =Mk(t)Meq T1(1) @ @tM?(t) =M?(t) T2(2) describe an exponential decay towards the equilib- rium magnetization Meqinzdirection. The trans- verse component decays with a time constant T2, whereas the longitudinal component approaches its equilibrium amplitude with T1. These time con- stants may be t independently to experimental results or microscopic calculations. 2. Landau-Lifshitz damping1with parameter  @ @t~M(t) = ~M~H~M M~M~H
(3) where is the gyromagnetic ratio. The rst term models the precession with a frequency !L= j~Hj, whereas the second term is solely responsible for damping. 3. Gilbert damping2with the dimensionless Gilbert damping parameter  @ @t~M(t) = G~M~H+~M M@t~M (4) It is generally accepted that is independent of the static magnetic elds ~Hsuch as anisotropy elds,18,29and thus depends only on the material and the microscopic interaction processes.3 The Landau-Lifshitz and Gilbert forms of damping are mathematically equivalent2,7,30with = (5) G= (1 +2) (6) but there are important dierences. In particular, an in- crease oflowers the precession frequency in the dynam- ics with Gilbert damping, while the damping parameter in the Landau-Lifshitz equation has no impact on the precession. In contrast to the Bloch equations, Landau- Lifshitz and Gilbert spin-dynamics always conserve the lengthj~Mjof the magnetization vector. An argument by Pines and Slichter,31shows that there are two dierent regimes for Bloch-type spin dynamics depending on the relation between the Larmor period and the correlation time. As long as the correlation time is much longer than the Larmor period, the system \knows" the direction of the eld during the scattering process. Stated dierently, the scattering process \sees" the mag- netic gap in the bandstructure. Thus, transverse and longitudinal spin components are distinguishable and the Bloch decay times T1andT2can dier. If the correlation time is considerably shorter than the Larmor period, this distinction is not possible, with the consequence that T1 must be equal to T2. Within the microscopic approach, presented in Sec. IV D, this consideration shows up again, albeit for the energy conserving functions resulting from a Markov approximation. The regime of short correlation times has already been investigated in the framework of a microscopic calcula- tion by Wu and coworkers.32They analyze the case of a moderate external magnetic eld applied to a non- magnetic n-type GaAs quantum well and include dier- ent scattering mechanisms (electron-electron Coulomb, electron-phonon, electron-impurity). They argue that the momentum relaxation rate is the crucial time scale in this scenario, which turns out to be much larger than the Larmor frequency. Their numerical results conrm the identity T1=T2expected from the Pines-Slichter argument. III. RELATION BETWEEN LANDAU-LIFSHITZ, GILBERT AND BLOCH We highlight here a connection between the Bloch equations (1, 2) and the Landau-Lifshitz equation (3). To this end we assume a small initial tilt of the mag- netization and describe the subsequent dynamics of the magnetization in the form ~M(t) =0 @M?(t) cos(!Lt) M?(t) sin(!Lt) MeqMk(t)1 A (7) whereM?andMjjdescribe deviations from equilib- rium. Putting this into eq. (3) one gets a coupled set ofequations. @ @tM?(t) =HMeqMk(t) j~M(t)jM?(t) (8) @ @tMk(t) =H1 j~M(t)jM2 ?(t) (9) Eq. (8) is simplied for a small deviation from equilib- rium, i.e.,M(t)Meqandj~M(t)jMeq: M?(t) =Cexp(Ht) (10) Mk(t) =C2 2Meqexp(2Ht) (11) whereCis an integration constant. For small excitations the deviations decay exponentially and Bloch decay times T1andT2result, which are related by 2T1=T2=1 H: (12) Only this ratio of the Bloch times is compatible with a constant length of the magnetization vector at low exci- tations. By combining Eqs. (12) and (5) one can connect the Gilbert parameter and the dephasing time T2 =1 T2!L: (13) If the conditions for the above approximations apply, the Gilbert damping parameter can be determined by t- ting the dephasing time T2and the Larmor frequency !L to computed or measured spin dynamics. This dimen- sionless quantity is well suited to compare the dephasing that results from dierent relaxation processes. Figure 2 shows the typical magnetization dynamics that results from (3), i.e., Landau-Lifshitz damping. As an illustration of a small excitation we choose in Fig. 2(a) an angle of 10for the initial tilt of the magnetization, which results in an exponential decay with 2 T1=T2. From the form of Eq. (3) it is clear that this behavior persists even for large !Land. Obviously the Landau- Lifshitz and Gilbert damping terms describe a scenario with relatively long correlation times (i.e., small scat- tering rates), because only in this regime both decay times can dier. The microscopic formalism in Sec. IV works in the same regime and will be compared with the phenomenological results. For an excitation angle of 90, the Landau-Lifshitz dynamics shown in Fig. 2(b) become non-exponential, so that no well-dened Bloch decay times T1,T2exist. IV. MICROSCOPIC MODEL In this section we describe a microscopic model that in- cludes magnetism at the mean-eld level, spin-orbit cou- pling as well as the microscopic coupling to a phonon bath treated at the level of Boltzmann scattering inte- grals. We then compare the microscopic dynamics to4 0 5000.51δM⊥/Meq time (ps)0 5000.51 time (ps)δM/bardbl/Meq0 5000.010.02δM/bardbl/Meq time (ps)0 5000.10.2 time (ps)δM⊥/Meq T1= 5.02 ps(a) (b)T2= 10.04 ps FIG. 2. Dynamics of M?andMkcomputed using to Landau-Lifshitz damping ( !L= 1 ps1,H= 106A m 1:26
104Oe,= 107m A ps). (a) An angle of 10leads to exponential an exponential decay with well dened T1andT2 times. (b). For an angle of 90, the decay (solid line) is not exponential as comparison with the exponential t (dashed line) clearly shows. the Bloch equations (1), (2), as well as the Landau- Lifshitz (3) and Gilbert damping terms (4). The mag- netic properties of the model are dened by an anti- ferromagnetic coupling between localized magnetic im- purities and itinerant carriers. As a prototypical spin- orbit coupling we consider an eectively two-dimensional model with a Rashba spin-orbit coupling. The reason for the choice of a model with a two-dimensional wave vector space is not an investigation of magnetization dy- namics with reduced dimensionality, but rather a reduc- tion in the dimension of the integrals that have to be solved numerically in the Boltzmann scattering terms. Since we treat the exchange between the localized and itinerant states in a mean-eld approximation, our two- dimensional model still has a \magnetic ground state" and presents a framework, for which qualitatively dif- ferent approaches can be compared. We do not aim at quantitative predictions for, say, magnetic semiconduc- tors or ferrimagnets with two sublattices. Finally, we include a standard interaction hamiltonian between the itinerant carriers and acoustic phonons. The correspond- ing hamiltonian reads ^H=^Hmf+^Hso+^Heph+^Haniso: (14) Only in Sec. VI an additional eld ^Haniso is included, which is intended to model a small anisotropy.A. Exchange interaction between itinerant carriers and localized spins The \magnetic part" of the model is described by the Hamiltonian ^Hmf=X ~k~2k2 2m^cy ~k^c~k+J^~ s
^~S: (15) which we consider in the mean-eld limit. The rst term represents itinerant carriers with a k-dependent disper- sion relation. In the following we assume s-like wave functions and parabolic energy dispersions. The eective mass is chosen to be m= 0:5me, wheremeis the free electron mass, and the ^ c(y) ~koperators create and annihi- late carriers in the state j~k;iwherelabels the itinerant bands, as shown in Fig. 3(a). The second term describes the coupling between itiner- ant spins~ sand localized spins ~Svia an antiferromagnetic exchange interaction ^~ s=1 2X ~kX 0h~k;0j^~ j~k;i^cy ~k^c~k0 (16) ^~S=1 2X 0h0j^~ jiX ~K^Cy ~K^C~K0 (17) Here, we have assumed that the wave functions of the lo- calized spins form dispersionless bands, i.e., we have im- plicitly introduced a virtual-crystal approximation. Due to the assumption of strong localization there is no or- bital overlap between these electrons, which are therefore considered to have momentum independent eigenstates jiand a at dispersion, as illustrated in Fig. 3(a). The components of the vector ^~ are the Pauli matrices ^ iwith i=x;y;z , and ^C(y) ~Kare the creation and annihilation op- erators for a localized spin state. We do notinclude interactions among localized or itin- erant spins, such as exchange scattering. For simplicity, we assume both itinerant and localized electrons to have a spin 1=2 and therefore andto run over two spin- projection quantum numbers 1=2. In the following we chosse an antiferromagnetic ( J > 0) exchange constant J= 500 meV, which leads to the schematic band struc- ture shown in Fig. 3(b). In the mean eld approximation used here, the itiner- ant carriers feel an eective magnetic eld ^Hloc ~Hloc=JB g~S (18) caused by localized moments and vice versa. Here B is the Bohr magneton and g= 2 is the g-factor of the electron. The permeability is assumed to be the vac- uum permeability 0. This time-dependent magnetic eld~Hloc(t) denes the preferred direction in the itiner- ant sub-system and therefore determines the longitudinal and transverse component of the itinerant spin at each time.5 r#k (a) (b) E(k) k  k ,k ,kE(k) EF FIG. 3. Sketch of the band-structure with localized ( at dispersions) and itinerant (parabolic dispersions) electrons. Above the Curie-Temperature TCthe spin-eigenstates are de- generate (a), whereas below TCa gap between the spin states exists. B. Rashba spin-orbit interaction The Rashba spin-orbit coupling is given by the Hamil- tonian ^Hso=R(^xky^ykx) (19) A Rashba coecient of R= 10 meV nm typical for semi- conductors is chosen in the following calculations. This value, which is close to the experimental one for the InSb/InAlSb material system,33is small compared to the exchange interactions, but it allows the exchange of an- gular momentum with the lattice. C. Coherent dynamics From the above contributions (15) and (19) to the Hamiltonian we derive the equations of motion contain- ing the coherent dynamics due to the exchange interac- tion and Rashba spin-orbit coupling as well as the inco- herent electron-phonon scattering. We rst focus on the coherent contributions. In principle, one has the choice to work in a basis with a xed spin-quantization axis or to use single-particle states that diagonalize the mean- eld (plus Rashba) Hamiltonian. Since we intend to use a Boltzmann scattering integral in Sec. IV D we need to apply a Markov approximation, which only works if one deals with diagonalized eigenenergies. In our case this is the single-particle basis that diagonalizes the entire one- particle contribution of the Hamiltonian ^Hmf+^Hso. In matrix representation this one-particle contribution for the itinerant carriers reads: ^Hmf+^Hso= ~2k2 2m+
loc z(
loc ++R~k)
loc ++R~k~2k2 2m
loc z! (20) where we have dened
loc i=J1 2h^SiiandR~k= iRkexp(i'k) with'k= arctan(ky=kx). The eigenen- ergies are  ~k=~2k2 2mq j
loczj2+jR~k+
loc +j2: (21)and the eigenstates j~k;+i= 1 ~k ;j~k;i= ~k 1 (22) where ~k=
loc ++R~k
locz+q j~
locj2+jR~kj2(23) In this basis the coherent part of the equation of mo- tion for the itinerant density matrix 0 ~k h^cy ~k^c~k0i reads @ @t0 ~k coh=i ~  ~k0 ~k
0 ~k: (24) No mean-eld or Rashba terms appear explicitly in these equations of motion since their contributions are now hid- den in the time-dependent eigenstates and eigenenergies. Since we are interested in dephasing and precessional dynamics, we assume a comparatively small spin-orbit coupling, that can dissipate angular momentum into the lattice, but does not have a decisive eect on the band- structure. Therefore we use the spin-mixing only in the transition matrix elements of the electron-phonon scat- teringM~k00 ~k(31). For all other purposes we set R~k= 0. In particular, the energy-dispersion  ~kis assumed to be unaected by the spin-orbit interaction and therefore it is spherically symmetric. With this approximation the itinerant eigenstates are always exactly aligned with the eective eld of the local- ized moments ~Hloc(t). Since this eective eld changes with time, the diagonalization and a transformation of the spin-density matrix in \spin space" has to be re- peated at each time-step. This eort makes it easier to identify the longitudinal and transverse spin compo- nents with the elements of the single-particle density matrix: The o-diagonal entries of the density matrix  ~k, which precess with the k-independent Larmor fre- quency!L= 2
loc=~, always describe the dynamics of the transverse spin-component. The longitudinal compo- nent, which does not precess, is represented by the diag- onal entries  ~k. Since both components change their spatial orientation continuously, we call this the rotating frame. The components of the spin vector in the rotating frame are h^ski=1 2X ~k ++ ~k ~k
(25) h^s?i=X ~k+ ~k (26) The components in the xed frame are obtained from Eq. (16) h^~ si=1 2X ~kX 0h~k;0j^~ j~k;i0 ~k(27)6 In this form, the time-dependent states carry the infor- mation how the spatial components are described by the density matrix at each time step. No time-independent \longitudinal" and \transverse" directions can be identi- ed in the xed frame. In a similar fashion, the diagonalized single-particle states of the localized spin system are obtained. The eigenenergies are E=~
itin (28) where
itin i=J1 2h^siiis the localized energy shift caused by the itinerant spin component si. The eigenstates are again always aligned with the itinerant magnetic mo- ment. In this basis the equation of motion of the localized spin-density matrix 0 locP ~Kh^Cy ~K^C~K0iis simply @ @t0 loc=i ~(EE0)0 loc (29) and does not contain explicit exchange contributions. Eqs. (25), (26), and (27) apply in turn to the components hSkiandhS?iof the localized spin and its spin-density matrix0 loc. D. Electron-phonon Boltzmann scattering with spin splitting Relaxation is introduced into the model by the interac- tion of the itinerant carriers with a phonon bath, which plays the role of an energy and angular momentum sink for these carriers. Our goal here is to present a derivation of the Boltzmann scattering contributions using stan- dard methods, see, e.g., Refs. 34 and 36. However, we emphasize that describing interaction as a Boltzmann- like instantaneous, energy conserving scattering process is limited by the existence of the magnetic gap. Since we keep the spin mixing due to Rashba spin-orbit coupling only in the Boltzmann scattering integrals, the resulting dynamical equations describe an Elliott-Yafet type spin relaxation. The electron-phonon interaction Hamiltonian reads34 ^Heph=X ~ q~!ph q^by ~ q^b~ q +X ~k~k0X 0 M~k00 ~k^cy ~k^b~k~k0^c~k00+ h.c.
(30) where ^b(y) ~ qare the bosonic operators, that create or an- nihilate acoustic phonons with momentum ~ qand linear dispersion!ph(q) =cphj~ qj. The sound velocity is taken to becph= 40 nm/ps and we use an eectively two- dimensional transition matrix element35 M~k00 ~k=Dq j~k~k0jh~k;j~k0;0i (31) where the deformation potential is chosen to be D= 60 meVnm1=2. The scalar-product between the initialstatej~k0;0iand the nal state j~k;iof an electronic transition takes the spin-mixing due to Rashba spin-orbit coupling into account. The derivation of Boltzmann scattering integrals for the itinerant spin-density matrix (24) leads to a memory integral of the following shape @ @tj(t) inc=1 ~X j0Zt 1ei(
Ejj0+i )(tt0)Fjj0[(t0)]dt0; (32) regardless whether one uses Green's function36or equation-of-motion techniques.34Since we go through a standard derivation here, we highlight only the impor- tant parts for the present case and do not write the equa- tions out completely. In particular, for scattering process j0=j0;~k0i!j=j;~ki, we useFjj0[(t0)] as an abbre- viation for a product of dynamical electronic spin-density matrix elements , evaluated at time t0<t, and equilib- rium phononic distributions. The corresponding energy dierence is denoted by
Ejj0=EjEj0~!ph(j~k~k0j), and describes the decay of the exponential function due to dissipation and/or higher order correlation functions. In general, the integral has to be evaluated numerically and contains memory eects. To apply the Markov ap- proximation one needs to compare two time scales: the \memory depth" 1 = , i.e., the time scale on which the exp[ (tt0)] factor essentially cuts o the integral, and the typical time scale on which the Fterm changes. In this paper we deal with relaxation processes not too far away from equilibrium, so that the typical time scale of the components of the spin-density matrix contained in Fis set by the Bloch times T1andT2. We can thus approximate Fjj0[(t0)] byFjj0[(t)], for all transitions labeled byjandj0, if the memory depth is shorter than the Bloch time(s), or 1 T1: (33) Provided condition (33) holds, the integral (32) can be done using the Markov approximation Fjj0[(t0)]' Fjj0[(t)] @ @tj(t) incoh=i ~X j0Fjj0[(t)]1
Ejj0+i~ :(34) As it is customary, we neglect in the following the real part of the complex energy denominator, which results in shifts of the single-particle energies. While these shifts may play an important role in non-Markovian problems with discrete energy levels37, the imaginary parts yield the relaxation contributions that are important for the present paper @ @tj(t) incoh=X j0Fjj0[(t)]~ (
Ejj0)2+ (~ )2(35) All transitions are thus weighted by a Lorentzian peaked at resonant transitions (
Ejj0= 0) with a broadening7 of~ that may be interpreted as an energy uncertainty. For relaxation processes in a system with a spin-splitting (due to internal elds and/or spin-orbit coupling), this broadening must not be so large as to blur the distinc- tion between the split bands. Consequently, only if the broadening is smaller than the magnetic splitting, i.e,, if !L, it is possible to distinguish between longi- tudinal and transverse components of the spin-density matrix. With Eq. (33) the inequality !Lyields the condition !L1 T1(36) for the Larmor frequencies and Bloch times, for which it is permissible to replace the Lorentzian by an energy conserving function ~ (
Ejj0)2+ (~ )2 !0!(
Ejj0): (37)This reduces the numerical eort very considerably, be- cause it allows one to eliminate an integration from the scattering term, and the energy conserving function is therefore often used without explicitly checking its valid- ity. The considerations leading to the connection between Eqs. (36) and (37) are a microscopic version of an argu- ment due to Pines and Slichter,31according to which T1 andT2can dier only for correlation times that long in comparison to a Larmor period. The microscopic Boltz- mann scattering terms, which contain the energy con- servingfunctions and will be used in the following, do not apply in a regime outside of condition (36). If !L'1=T1, a nite broadening has to be taken into account. Together the full equation of motion in the regime (36) for the itinerant-carrier spin density matrix thus reads @ @t0 ~k=i ~  k0 k
0 ~k + ~X k0X 123M~k ~k01M~k02 ~k3
E~k02~k3
h 1 +Nph j~k0~kj
12 ~k0 3030 ~k
Nph j~k0~kj30 ~k 1212 ~k0
i  ~X ~k0X 123M~k ~k01M~k02 ~k3
E~k3~k02
h 1 +Nph j~k~k0j
30 ~k 1212 ~k0
Nph j~k~k0j12 ~k0 3030 ~k
i + ~X ~k0X 123M~k01 ~k0M~k3 ~k02
E~k02~k3
h 1 +Nph j~k0~kj
21 ~k0 33 ~k
Nph j~k0~kj3 ~k 2121 ~k0
i  ~X ~k0X 123M~k01 ~k0Mk3 ~k02
E~k3~k02
h 1 +Nph j~k~k0j
3 ~k 2121 ~k0
Nph j~k~k0j21 ~k0 33 ~k
i (38) Here,
E~k~k00= k0 k0~!ph j~k~k0j, andNph qis the occupation function of a thermalized phonon bath, given by a Bose-Einstein distribution Nph q=1 e~!ph1(39) where= 1=(kBTph). The numerical results for the mi- croscopic dynamics in the following sections are obtained by numerically solving the equations of motion (29) and (38). In the numerical calculations the spin-density ma- trix is transformed to the single-particle basis of the in- stantaneous, diagonalized eigenstates (22). V. DEPHASING BETWEEN LOCALIZED AND ITINERANT SPINS: NUMERICAL RESULTS Since we are interested in this paper in a comparison of the model described above with Landau-Lifshitz and Gilbert damping, we investigate magnetization dynam- ics with an initial spin-density matrix that correspondsto a tilting of the spins out of their equilibrium posi- tion without changing the kinetic energy of the carriers, because we need initial conditions that lead to generic magnetization dephasing without carrier heating and the corresponding demagnetization dynamics. A. Initial state and spin dynamics Thus we take as the equilibrium initial state the steady-state that is reached for the coupled spins inter- acting with the phonon bath at low temperature ( Tph= 1 K), as shown in Fig. 4(a). In this equilibrium state the spin density matrix is characterized by shifted Fermi functions for the distributions  ~k=f( kEF;Tph) and vanishing coherences + ~k= 0. In particular, the steady-state calculation determines the equilibrium mag- netic gap. We then change the itinerant density matrix to that corresponding to an itinerant spin tilted by = 10 out of equilibrium, see Fig. 4(b). This initial condition8 FIG. 4. (a) Localized spin h~Siand itinerant spin h~ siin ther- mal equilibrium. (b) Itinerant spins tilted out of equilibrium by an angle . 0 1 2 300.51 k(nm−1)Occupations 0 1 2 3−0.500.5 k(nm−1)Coherences ρ|k|− − ρ|k|+ + ℜ(ρ|k|− +) ℑ(ρ|k|− +) FIG. 5. Initial spin density-matrix (occupations and coher- ences) for itinerant electrons corresponding to a tilt = 50. The deviation from equilibrium occurs only between the Fermi wave-vectors of the \+" and \ " bands. achieves a tilting of the spins without heating and avoids generic de- and remagnetization dynamics. Microscop- ically the tilted spin corresponds to the spin density- matrix shown in Fig. 5. The perturbation for distribu- tions and coherences exists only between the two Fermi wave vectors for the = + and=bands. For smaller tilt angles the deviation is much less pronounced. From this initial condition, both spins start to precess around the instantaneous direction, along which the ex- change interaction tries to align them. This direction is determined for the itinerant carriers by the localized spins, and vice versa. A return back into equilibrium re- quires the scattering of itinerant electrons with phonons. If we switch o spin mixing, the dynamics shown in Fig. 6(a) result: No angular momentum is exchanged with the phonon bath, the excited system cannot relax into equilibrium and the precession goes on indenitely. Fig. 6(b) shows the same result including spin-mixed itin- erant states. Now angular momentum can be transferred from the itinerant sub-system into the lattice and the to- tal spinh~Si+h~ sichanges. In the presence of spin-orbit coupling, electron-phonon scattering, which is by itself spin-diagonal, can return the spin system into equilib- rium, characterized by aligned spins and vanishing trans- verse components. Since we consider here a small Rashba (a) (b) 5&:P; 5&:P; O&:P; O&:P; FIG. 6. Non-equilibrium dynamics of localized (red) and itin- erant (blue) spins including electron-phonon scattering with- out spin-mixing (a) and within spin-mixing (b). 0 1 2 30.120.130.14 time (ps)|s|0 1 2 3−0.0400.04 time (ps)sx 0 1 2 3−0.0400.04 time (ps)sy 0 1 2 3−0.14−0.13−0.12 time (ps)sz FIG. 7. Relaxation dynamics of itinerant spins in the xed frame. coupling, the interaction with the phonon bath removes energy much faster than angular momentum. The car- rier temperature therefore stays practically equal to the phonon temperature Tphduring the entire relaxation pro- cess, and no heat-induced demagnetization processes oc- cur. The nal magnetization is, however, not necessarily oriented in the zdirection of the xed frame, because in the results discussed in the this section there is no external eld or anisotropy to induce such an alignment. We plot the resulting dynamics of the itinerant spins during the dephasing process in Fig. 7, which shows that9 00.511.522.53−0.133−0.132−0.131 time(ps)s/bardbl 00.511.522.5300.010.020.03 time(ps)s⊥ FIG. 8. Relaxation dynamics of itinerant spins in the rotating frame. An exponential t determines the Bloch decay times toT1= 0:5 ps andT2= 1:0 ps. all itinerant spatial components precess, and no spatially xed component can be considered to be longitudinal. First, the localized spin turns away from the zdirec- tion due to the tilted itinerant spin and subsequently both spin-systems precess around each other because the quantization axis of each system changes continuously due to the mutual interaction. During the entire relax- ation process the absolute value of the itinerant spin is conserved to better than 1%. Fig. 8 shows the same dynamics in the rotating frame, where the longitudinal and transverse dynamics can be seen clearly. Both itin- erant components show exponential dynamics, which are therefore well described by decay times T1andT2. If well- dened decay times exist, one expects a ratio T2=T1= 2 as long as the length of the spin is conserved. The t for our numerical results indeed gives 2 T1'T2. Figure 9 plots the T1andT2values extracted from the dynamics as a function of the strength of the electron- phonon coupling, or deformation potential, D. The de- pendence of the decay times on Dcan be t extremely well by a 1=D2relation, which demonstrates the propor- tionality of the Bloch decay times T1;2/1=D2. Further, the ratioT2=T1stays equal to 2 for all coupling strengths. As discussed in Sec. IV D about the Markov approxima- tion, our microscopic description cannot reach regimes whereT1andT2are indistinguishable and therefore equal. However, these results show that for small tilting angles, not even a pronounced electron-phonon coupling leads to a noticeable deviation from the T2= 2T1be- havior. Because this relation between T1andT2holds, the dynamics in Fig. 8 can be equally well described by an Landau-Lifshitz or Gilbert damping term. By tting the dephasing time T21 ps and the Larmor-frequency !L281 ps1, equation (13) yields the corresponding Gilbert damping parameter iso3:6
103. 204060801001201401.822.2 D(meV√nm)T2/T1 2040608010012014010−2100102 D(meV√nm)T1,2(ps) FIG. 9. Top: Longitudinal (black squares) and the transverse (blue circles) Bloch decay times vs. electron-phonon coupling strengthD. Two t curves /1=D2show thatT1;2/1=D2. Bottom:The ratio of both decay times remains almost con- stantT2=T12 over the entire range of coupling. 369 0 0.5 1279279.5280280.5281 1/T2(ps−1)ω(ps−1) LL Gilbert Microscopic FIG. 10. Precession frequency with respect to the damping strength in terms of the decay rate 1 =T2. B. Precession-frequency shift due to dephasing In the Landau-Lifshitz equation the contributions de- scribing, respectively, the precession and the damping are completely independent, so that the damping constant  has no impact on the precession. By contrast, an increase ofin the Gilbert equation does not only increase the dephasing rate, it lowers the precession frequency as well. In this section we investigate the change of the preces- sion frequency in the microscopic calculation and com- pare it with the macroscopic descriptions. To this end, we use the o-diagonal components of the itinerant den- sity matrix +(t) =P ~k+ ~k(t), which describe the dy- namics of the transverse spin components in the rotating frame. The modulus of its Fourier transform j+(!)j shows a distinct peak, which is exactly at the precession frequency. To compare the precession frequency for dierent damping parameters, we use the dependence on T2, be- cause all dephasing parameters can be related to T2for10 small excitations. Fig. 10 plots the Larmor frequency vs. the transverse relaxation rate 1 =T2. The precession pa- rameters of the Landau-Lifshitz and the Gilbert equation are chosen such that the Larmor frequency in the un- damped limiting case is equal to that of the microscopic simulation. In order to stay within the bounds set by con- dition (36), we do not extend the plot in Fig. 10 to higher dephasing rates. Fig. 10 shows that the microscopic cal- culation yields a reduction of the precession frequency with the damping rate. Although the Gilbert dynamics also show such a reduction, it occurs only at shorter T2. As mentioned above, for the Landau-Lifshitz damping, the frequency is independent of the damping parame- ter. Even though the change of precession frequency in the microscopic calculation is small, the Landau-Lifshitz damping completely fails to include this eect. While Gilbert damping does show a reduction of precession fre- quency, it is not at all close to the microscopic calcula- tion on the frequency scale considered here. Both phe- nomenological damping expressions thus do not repro- duce the dependence of the precession frequency on T2. Even though the numerical dierences are small, these dierences already occur in the small-excitation regime, and may perhaps be detectable. C. Dephasing at larger excitation angles The phenomenological Landau-Lifshitz and Gilbert damping contributions describe an exponential decay only for small excitation angles, as studied in the pre- vious section. In this section we investigate the eect of larger excitation angles ( >10) on the spin dynamics in the microscopic calculation. Apart from this the initial condition of the dynamics is the same as before, in par- ticular, the itinerant spin is tilted such that the absolute value of the spin is unchanged. Figure 11 shows the time development of the skand s?components of the itinerant spin in the rotating frame for an initial tilt angle = 140. While the transverse component s?in the rotating frame can be well described by an exponential decay, the longitudinal component sk shows a dierent behavior. It initially decreases with a time constant of less than 1 ps, but does not reach its equilibrium value. Instead, the eventual return to equi- librium takes place on a much longer timescale, during which the s?component is already vanishingly small. The long-time dynamics are therefore purely collinear. For the short-time dynamics, the transverse component can be t well by an exponential decay, even for large ex- citation angles. This behavior is dierent from Landau- Lifshitz and Gilbert dynamics, cf. Fig. 2, which both ex- hibit non-exponential decay of the transverse spin com- ponent. In Fig. 12 the dependence of T2on the excitation an- gle is shown. From small up to almost 180, the decay time decreases by more than 50%. This dependence is exclusively due to the \excitation condition," which in- 0 1 2 3 4−0.100.1 time (ps)s/bardbl 0 1 2 3 400.050.1 time (ps)s⊥FIG. 11. Dynamics of the longitudinal and transverse itiner- ant spin components in the rotating frame (solid lines) for a tilt angle of = 140, together with exponential ts toward equilibrium (dashed lines). The longitudinal equilibrium po- larization is shown as a dotted line. volves only spin degrees of freedom (\tilt angle"), but no change of temperature. Although one can t such a T2 time to the transverse decay, the overall behavior with its two stages is, in our view, qualitatively dierent from the typical Bloch relaxation/dephasing picture. To highlight the similarities and dierences from the Bloch relaxation/dephasing we plot in Fig. 13 the mod- ulus of the itinerant spin vector j~ sjin the rotating frame, whose transverse and longitudinal components were shown in Fig. 11. Over the 2 ps, during which the transverse spin in the rotating frame essentially decays, the modulus of the spin vector undergoes a fast initial decrease and a partial recovery. The initial length of ~ s is recovered only over a much larger time scale of several hundred picoseconds (not shown). Thus the dynamics can be seen to dier from a Landau-Lifshitz or Gilbert- like scenario because the spin does not precess toward equilibrium with a constant length. Additionally they dier from Bloch-like dynamics because there is a com- bination of the fast and slow dynamics that cannot be described by a single set of T1andT2times. We stress that the microscopic dynamics at larger excitation angles show a precessional motion of the magnetization with- out heating and a slow remagnetization. This scenario is somewhat in between typical small angle-relaxation, for which the modulus of the magnetization is constant and which is well described by Gilbert and Landau-Lifshitz damping, and collinear de/remagnetization dynamics. VI. EFFECT OF ANISOTROPY So far we have been concerned with the question how phenomenological equations describe dephasing pro- cesses between itinerant and localized spins, where the11 0 50 100 1500.40.60.81 β(◦)T2(ps) FIG. 12.T2time extracted from exponential t to s?dynam- ics in rotating frame for dierent initial tilting angles . 0 0.5 1 1.5 20.040.060.080.10.120.14 time (ps)|s| 10° 50° 90° 140° FIG. 13. Dynamics of the modulus j~ sjof the itinerant spin for dierent initial tilt angles . Note the slightly dierent time scale compared to Fig. 11. magnetic properties of the system were determined by a mean-eld exchange interaction only. Oftentimes, phe- nomenological models of spin dynamics are used to de- scribe dephasing processes toward an \easy axis" deter- mined by anisotropy elds.29 In order to capture in a simple fashion the eects of anisotropy on the spin dynamics in our model, we sim- ply assume the existence of an eective anisotropy eld ~Haniso, which enters the Hamiltonian via ^Haniso =gB^~ s
~Haniso (40) and only acts on the itinerant carriers. Its strength is assumed to be small in comparison to the eld of the localized moments ~Hloc. This additional eld ~Haniso has to be taken into account in the diagonalization of the coherent dynamics as well, see section IV C. For the investigation of the dynamics with anisotropy, we choose a slightly dierent initial condition, which is shown in Fig. 14. In thermal equilibrium, both spins are now aligned, with opposite directions, along the anisotropy eld ~Haniso, which is assumed to point in the zdirection. At t= 0 they are both rigidly tilted by an 5&:P; O&:P; U T V E *_lgqm FIG. 14. Dynamics of the localized spin ~Sand itinerant spin ~ s. Att= 0, the equilibrium conguration of both spins is tilted (= 10) with respect to an anisotropy eld ~Haniso. The anisotropy eld is only experienced by the itinerant sub- system. 01002003004005006000.490.4950.5 time(ps)Sz(t) 010020030040050060000.050.1 time(ps)/radicalBig S2x(t)+S2y(t) FIG. 15. Relaxation dynamics of the localized spin toward the anisotropy direction for longitudinal component Szand the transverse componentp S2x+S2y. An exponential t yields Bloch decay times of Taniso 1 = 67:8 ps andTaniso 2 = 134:0 ps. angle= 10with respect to the anisotropy eld. Figure 14 shows the time evolution of both spins in the xed frame, with zaxis in the direction of the anisotropy eld for the same material parameters as in the previous sections and an anisotropy eld ~Haniso =108A m
~ ez. The dynamics of the entire spin-system are somewhat dierent now, as the itinerant spin precesses around the combined eld of the anisotropy and the localized mo- ments. The localized spin precesses around the itinerant spin, whose direction keeps changing as well. Figure 15 contains the dynamics of the components of the localized spin in the rotating frame. Both com- ponents show an exponential behavior that allows us to extract well dened Bloch-times Taniso 1 andTaniso 2. Again we nd the ratio of 2 Taniso 1Taniso 2, because the abso- lute value of the localized spin does not change, as it is not coupled to the phonon bath. In Fig. 16 the Larmor-frequency !aniso L, which is the precession frequency due to the anisotropy eld, and the Bloch decay times Taniso 2 are plotted vs. the strength of the anisotropy eld ~Haniso. The Gilbert damping pa-12 0 5 10 150510 Haniso(107A/m)ωaniso L (ps−1) 0 5 10 1505001000 Haniso(107A/m)Taniso 2 (ps) 0 5 10 1501020 Haniso(107A/m)αaniso (10−4) FIG. 16. Larmor frequency !aniso L and Bloch decay time Taniso 2 extracted from the spin dynamics vs. anisotropy eld Haniso, as well as the corresponding damping parameter aniso. rameteraniso for the dephasing dynamics computed via Eq. (13) is also presented in this gure. The plot reveals a decrease of the dephasing time Taniso 2 and a almost linear increase of the Larmor frequency !aniso L with the strength of the anisotropy eld Haniso. The Gilbert damping parameter aniso shows only a neg- ligible dependence on the anisotropy eld Haniso. This conrms the statement that, in contrast to the dephas- ing rates, the Gilbert damping parameter is independent of the applied magnetic eld. In the investigated range we nd an almost constant value of aniso'9104. The Gilbert damping parameter aniso for the de- phasing toward the anisotropy eld is about 4 times smaller than iso, which describes the dephasing between both spins. This disparity in the damping eciency (aniso<  iso) is obviously due to a fundamental dier- ence in the dephasing mechanism. In the anisotropy case the localized spin dephases toward the zdirection with- out being involved in scattering processes with itinerant carriers or phonons. The dynamics of the localized spins is purely precessional due to the time-dependent mag- netic moment of the itinerant carriers ~Hitin(t). Thus, only this varying magnetic eld, that turns out to be slightly tilted against the localized spins during the en- tire relaxation causes the dephasing, in presence of the coupling between itinerant carriers and a phonon bath, which acts as a sink for energy and angular momentum. The relaxation of the localized moments thus occurs only indirectly as a carrier-meditated relaxation via their cou- pling to the time dependent mean-eld of the itinerant spin. Next, we investigate the dependence of the Gilbert pa- rameteraniso on the bath coupling. Fig. 17 shows that 0 50 100 150 20000.0040.0080.012 D(meV√nm)αaniso FIG. 17. Damping parameter aniso vs. coupling constant D (black diamonds). The red line is a quadratic t, indicative ofaniso/D2. aniso increases quadratically with the electron-phonon coupling strength D. Since Fig. 9 establishes that the spin-dephasing rate 1=T2for the fast dynamics discussed in the previous sec- tions, is proportional to D2, we ndaniso/1=T2. We brie y compare these trends to two earlier calculations of Gilbert damping that employ p-dmodels and assume phenomenological Bloch-type rates 1 =T2for the dephas- ing of the itinerant hole spins toward the eld of the localized moments. In contrast to the present paper, the localized spins experience the anisotropy elds. Chovan and Perakis38derive a Gilbert equation for the dephasing of the localized spins toward the anisotropy axis, assum- ing that the hole spin follows the eld ~Hlocof the localized spins almost adiabatically. Tserkovnyak et al.39extract a Gilbert parameter from spin susceptibilities. The re- sulting dependence of the Gilbert parameter aniso on 1=T2in both approaches is in qualitative accordance and exhibits two dierent regimes. In the the low spin- ip regime, where 1 =T2is small in comparison to the p-dex- change interaction a linear increase of aniso with 1=T2 is found, as is the case in our calculations with micro- scopic dephasing terms. If the relaxation rate is larger than thep-ddynamics,aniso decreases again. Due to the restriction (36) of the Boltzmann scattering integral to low spin- ip rates, the present Markovian calculations cannot be pushed into this regime. Even though the anisotropy eld ~Haniso is not cou- pled to the localized spin ~Sdirectly, both spins precess around the zdirection with frequency !aniso L. In analogy to Sec. V B we study now the in uence of the damping process on the precession of the localized spin around the anisotropy axis and compare it to the behavior of Landau-Lifshitz and Gilbert dynamics. Fig. 18 reveals a similar behavior of the precession frequency as a function of the damping rate 1 =Taniso 2 as in the isotropic case. The microscopic calculation predicts a distinct drop of the Larmor frequency !aniso L for a range of dephasing rates where the precession frequency is unchanged according to the Gilbert and Landau-Lifshitz damping models. Al- though Gilbert damping eventually leads to a change in precession frequency for larger damping, this result shows a qualitative dierence between the microscopic and the13 0 0.02 0.04 0.06 0.087.127.167.27.24 1/Taniso 2(ps−1)ωaniso(ps−1) Gilbert LL Microscopic FIG. 18. Precession frequency of the localized spin around the anisotropy eld vs. Bloch decay time 1 =Taniso 2. phenomenological calculations. VII. CONCLUSION AND OUTLOOK In this paper, we investigated a microscopic descrip- tion of dephasing processes due to spin-orbit coupling and electron-phonon scattering in a mean-eld kinetic exchange model. We rst analyzed how spin-dependent carrier dynamics can be described by Boltzmann scat- tering integrals, which leads to Elliott-Yafet type relax- ation processes. This is only possible for dephasing rates small compared to the Larmor frequency, see Eq. (36). The microscopic calculation always yielded Bloch times 2T1=T2for low excitation angles as it should be due to the conservation of the absolute value of the mag- netization. A small decrease of the eective precession frequency occurs with increasing damping rate, which is a fundamental dierence to the Landau-Lifshitz descrip- tion and exceeds the change predicted by the Gilbert equation in this regime.We modeled two dephasing scenarios. First, a relax- ation process between both spin sub systems was studied. Here, the dierent spins precess around the mean-eld of the other system. In particular, for large excitation an- gles we found a decrease of the magnetization during the precessional motion without heating and a slow remag- netization. This scenario is somewhat in between typi- cal small angle-relaxation, for which the modulus of the magnetization is constant and which is well described by Gilbert and Landau-Lifshitz damping, and collinear de/remagnetization dynamics. Also, we nd important deviations from a pure Bloch-like behavior. The second scenario deals with the relaxation of the magnetization toward a magnetic anisotropy eld expe- rienced by the itinerant carrier spins for small excitation angles. The resulting Gilbert parameter aniso is inde- pendent of the static anisotropy eld. The relaxation of the localized moments occurs only indirectly as a carrier- meditated relaxation via their coupling to the time de- pendent mean-eld of the itinerant spin. To draw a meaningful comparison with Landau- Lifshitz and Gilbert dynamics we restricted ourselves throughout the entire paper to a regime where the elec- tronic temperature is equal to the lattice temperature Tph at all times. In general our microscopic theory is also ca- pable of modeling heat induced de- and remagnetization processes. We intend to compare microscopic simulations of hot electron dynamics in this model, including scat- tering processes between both types of spin, with phe- nomenological approaches such as the Landau-Lifshitz- Bloch (LLB) equation or the self-consistent Bloch equa- tion (SCB)40. We nally mention that we derived relation (13) con- necting the Bloch dephasing time T2and the Gilbert damping parameter . Despite its simplicity and obvious usefulness, we were not able to nd a published account of this relation. baral@rhrk.uni-kl.de yhcsch@physik.uni-kl.de 1L. Landau and E. Lifshitz, Phys. Z. Sowj. 8, 153 (1935). 2T. L. Gilbert, IEEE Trans. Magnetics, 40, 6 (2004). 3D. L. Mills and R. Arias The damping of spin motions in ultrathin lms: Is the Landau-Lifschitz-Gilbert phe- nomenology applicable? Invited paper presented at the VII Latin American Workshop on Magnetism, Magnetic Materials and their Applications, Renaca, Chile, 2005. 4D. A. Garanin, Physica A 172, 470 (1991). 5V. V. 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2014-05-09

We present a microscopic calculation of magnetization damping for a magnetic "toy model." The magnetic system consists of itinerant carriers coupled antiferromagnetically to a dispersionless band of localized spins, and the magnetization damping is due to coupling of the itinerant carriers to a phonon bath in the presence of spin-orbit coupling. Using a mean-field approximation for the kinetic exchange model and assuming the spin-orbit coupling to be of the Rashba form, we derive Boltzmann scattering integrals for the distributions and spin coherences in the case of an antiferromagnetic exchange splitting, including a careful analysis of the connection between lifetime broadening and the magnetic gap. For the Elliott-Yafet type itinerant spin dynamics we extract dephasing and magnetization times T_1 and T_2 from initial conditions corresponding to a tilt of the magnetization vector, and draw a comparison to phenomenological equations such as the Landau-Lifshitz or the Gilbert damping. We also analyze magnetization precession and damping for this system including an anisotropy field and find a carrier mediated dephasing of the localized spin via the mean-field coupling.

Magnetization dynamics and damping due to electron-phonon scattering in a ferrimagnetic exchange model

1405.2347v1

Current-driven motion of magnetic topological defects in ferromagnetic superconductors Se Kwon Kim1,∗and Suk Bum Chung2, 3,† 1Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea 2Department of Physics and Natural Science Research Institute, University of Seoul, Seoul 02504, Republic of Korea 3School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea (Dated: May 24, 2023) Recent years have seen a number of instances where magnetism and superconductivity intrinsically coexist. Our focus is on the case where spin-triplet superconductivity arises out of ferromagnetism, and we make a hydrodynamic analysis of the effect of a charge supercurrent on magnetic topological defects like domain walls and merons. We find that the emergent electromagnetic field that arises out of the superconducting order parameter provides a description for not only the physical quantities such as the local energy flux density and the interaction between current and defects but also the energy dissipation through magnetic dynamics of the Gilbert damping, which becomes more prominent compared to the normal state as superconductivity attenuates the energy dissipation through the charge sector. In particular, we reveal that the current-induced dynamics of domain walls and merons in the presence of the Gilbert damping give rise to the nonsingular 4 πand 2 π phase slips, respectively, revealing the intertwined dynamics of spin and charge degrees of freedom in ferromagnetic superconductors. I. INTRODUCTION While magnetism has traditionally been regarded as inimical to superconductivity, recent years have seen ob- servation of ferromagnetism and superconductivity co- existing or cooperating in varieties of materials which includes uranium heavy-fermion compounds [1–3] and two-dimensional moir´ e materials such as twisted bilayer graphene [4–6]. It has been known that such coexistence can be naturally accommodated by the Cooper pairing of spin-polarized electrons [7]. In such cases, it is natural to question what effect, if any, ferromagnetism may have on superconductivity and vice versa. It is well established in magnetism and spintronics that the current-induced motions of spin textures such as domain walls in magnetic metals give rise to the spin and energy dissipation into the baths of quasiparticles or phonons, commonly known as the Gilbert damping [8, 9]. The conservation of energy dictates that the dissipated energy should be externally supplied by the input power. In the case of normal metals, however, resistivity-induced energy dissipation is present regardless of the presence or the absence of any spin textures. Hence the Gilbert damping gives rise to only an additional term in the en- ergy dissipation and, in this sense, its presence can be difficult to confirm solely through charge transport. Charge transport detection of the Gilbert damping in ferromagnetic superconductors may be more straightfor- ward despite involving a feature unconventional for su- perconductors. To maintain a steady-state motion of spin textures in the presence of the Gilbert damping, a fer- romagnetic superconductor needs the finite input power ∗sekwonkim@kaist.ac.kr †sbchung0@uos.ac.kr (a)(b) FIG. 1. (a) The illustration of the mutually orthogonal unit vectors ˆs,ˆu, and ˆvthat describe the directional degrees of freedom of the order parameter of a ferromagnetic supercon- ductor. (b) The configuration of the triad {ˆs,ˆu,ˆv}for a do- main wall in a ferromagnetic superconductor with easy-axis spin anisotropy along the zdirection. that goes out of the superconductor solely in the form of the Gilbert damping. This indicates voltage arising in- side the superconductor in the direction of the current by the dynamics of spin textures. The mechanisms by which a superconductor acquires a finite voltage difference be- tween two points is referred to as phase slips [10, 11]. In conventional superconductors, these phase slips gen- erally accompany the singularities, i.e.the vanishing of the order parameter at a certain time during the phase slips. In this paper, we show that this is not necessar- ily the case for ferromagnetic superconductors by us- ing the concrete example of the current-induced mo- tions of two types of magnetic defects, domain walls and merons, which are schematically illustrated in Fig. 1(b) and Fig. 3, respectively. To this end, we begin by ex- amining the order parameter of the spin-polarized super- conductor and show how the Cooper pair spin rotation around the spin polarization direction is actually equiv- alent to the twisting of the overall phase. This gives rise to a channel for the interaction between ferromagnetism and superconductivity, namely the coupling of CooperarXiv:2305.13564v1 [cond-mat.supr-con] 23 May 20232 pairs to the effective gauge field arising from spin tex- ture [7, 12, 13]. We then proceed to show how such formalism can be used to obtain the current-induced motion of topologi- cal spin defects such as a domain wall and a meron in presence of a background superflow. First, for a domain wall, we show that the current-induced motion of do- main walls in the presence of the Gilbert damping ac- companies the precessional dynamics of the local spin polarization and this in turn gives rise to the nonsingu- lar 4πphase slips through the generation of an emergent electric field. The induced phase slip opens a channel through which the ferromagnetic superconductor can ac- quire input power, which is shown to be dissipated by the spin dynamics entirely via the Gilbert damping. Also, a current-induced motion of a meron is shown to give rise to the nonsingular 2 πphase slips perpendicular to its motion, engendering a channel for the input power that is dissipated via the Gilbert damping. The gener- ation of the 2 πphase slips can be understood from the emergent electromagnetic field associated with the meron dynamics. For ferromagnetic metals, the emergent elec- tromagnetic fields associated with spin textures and their dynamics have been discussed theoretically [14–17] and confirmed experimentally [18–21]. However, their man- ifestations in the dynamics of magnetic defects in fer- romagnetic superconductors and the resultant nonsingu- lar phase slips have not been discussed yet. Our work reveals that the current-induced dynamics of magnetic defects exemplify the intertwined dynamics of spin and charge degrees of freedom in ferromagnetic superconduc- tors, where the emergent electromagnetic fields play cru- cial roles. The paper is organized as follows. The general formal- ism for the order parameter and its dynamics of ferro- magnetic superconductors is developed phenomenologi- cally in Sec. II. The current-induced dynamics of a do- main wall and its relation to the nonsingular 4 πphase slips are discussed in Sec. III. Section IV concerns the current-induced dynamics of a meron and its relation to the nonsingular 2 πphase slips. We conclude the paper in Sec. V with discussions. II. GENERAL FORMALISM A. Order parameter The order parameter of a fully spin-polarized triplet su- perconductor provides a starting point for understanding how superconductivity and magnetism are intertwined through the emergent gauge field. In the d-vector for- malism defined by i(d·σσy)s,s′≡∆s,s′, it is given by [7, 12, 13] d=√ρ 2eiϕ(ˆu+iˆv) =rρ 2eiϕ(ˆu+iˆv)√ 2≡rρ 2ˆd,(1)where ρ= 2d∗·dis the number density of the Cooper pairs, ˆuandˆvare perpendicular unit vectors, and ˆd∗·ˆd= 1; the simplest example would be ˆu=ˆx,ˆv=ˆywhich gives ∆ s,s′= 0 except for ∆ ↑↑(see Appendix A for the details). There is an ambiguity here in defining ϕas the above order parameter remains invariant under the following simultaneous change of ϕandˆuandˆv: eiϕ(ˆu+iˆv) =ei(ϕ+δϕ) e−iδϕ(ˆu+iˆv) ≡ei(ϕ+δϕ)(ˆu′+iˆv′), (2) where ˆu′,ˆv′are obtained by rotating ˆu,ˆvby +δϕaround ˆu׈v. As the spin density in units of ℏcan be written as s= 2id×d∗=ρˆu׈v≡ρˆs, (3) Eq. (2) denotes the U(1) ϕ+sorder parameter redundancy [7, 12], i.e.the invariance of the order parameter when the angle of the spin rotation around ˆsequals the change inϕ. Such redundancy implies the existence of an effec- tive gauge field arising from the spin degrees of freedom. See Fig. 1(a) for the illustration of the three mutually orthogonal unit vectors ˆs,ˆu, and ˆv, which are depicted by red, green, and blue arrows, respectively. For deriving the vector potential and magnetostatics of this effective gauge field, the above order parameter suffices. From the spin rotation angle around ˆsdefined asα, the effective vector gauge can be written as [14] ai≡ℏ q∂iα=ℏ qˆs·(ˆu×∂iˆu) ; (4) hence the emergent gauge is a direct consequence of the U(1) ϕ+sorder parameter redundancy of Eq. (2). Indeed, the emergent gauge field of spatial curvature in the chi- ral superconductor has been attributed to the analogous order parameter redundancy there [22–25]. Here, while we have kept the charge, q, generic, q=−2e <0 holds in superconductors. From this emergent vector potential, it is straightforward to obtain the emergent magnetic field, bi=ϵijk∂jak=−ℏϵijk 2qˆs·(∂jˆs×∂kˆs) ; (5) note that this is in the same form as the well-known Mermin-Ho relation between the orbital angular momen- tum texture and superfluid velocity in the 3He-A super- fluid [26, 27]. Yet this discussion does not include any dynamics, for which we shall adopt a two-step approach of first formulating the simplest free energy for the order parameter [Eq. (2)] and then use its Lagrangian to obtain the equations of motion. B. Free energy Given that we seek results relevant to wide-ranging su- perconductors whose common attributes may not extend beyond the spin-polarized Cooper pairing [1–6] we will consider for our free energy the simplest minimal model3 that includes the spin anisotropy and the Zeeman cou- pling: F[d] =Z dVF′ 0[d] +Z dVU 2 2|d|2−ρ0
2,(6) where F′ 0=A′ d 2|(∇−iq ℏA)d|2+ρA′ s 2|∇ˆs|2−D 2(ˆs·ˆz)2−Hsz+qV , where A′ srepresents the excess spin stiffness; note that, in contrast to previous analysis [12], our treatment will encompass both the easy-axis anisotropy D > 0 and the easy-plane anisotropy D < 0. As we will focus on the cases where the fluctuation of the condensate density ρ≡ 2|d|2is strongly suppressed, it is convenient to separatelygroup together terms dependent on ρfluctuations [12, 13] F=Z dV ρF0+Z dVA′ d 16ρ(∇ρ)2+U 2(ρ−ρ0)2 , where F0=A′ d 2|(∇−iq ℏA)ˆd|2+A′ s 2|∇ˆs|2−D 2(ˆs·ˆz)2−Hsz+qV is the free energy density per unit density. The gauge transformation is implemented as A7→A+∇Λ,d7→eiqΛ/ℏd. The free energy can be recast into the following form: F=Z dV ρAc 2h ∂iϕ−q ℏAi−ˆs·(ˆu×∂iˆu)i2 +As 2(∂iˆs)2−D 2(ˆs·ˆz)2−Hsz+qV +Z dVAc 8ρ(∇ρ)2+U 2(ρ−ρ0)2 , (7) where Ac=A′ dandAs=A′ s+A′ d/2. Here, Acand Asrepresent the charge stiffness and the spin stiffness, respectively. The similar expression without the sec- ond and the third terms can be found in Eq. (2.3) and Eq. (2.5) of Ref. [28]. Accordingly, the charge supercurrent density is modi- fied to Ji=−δF δAi=q ℏρAc(∂iϕ−q ℏAi−q ℏai), (8) with the velocity field is given by vi=Ji qρ=Ac ℏ(∂iϕ−q ℏAi−q ℏai). (9) It satisfies the following equation (by assuming a nonsin- gular θ): ∇×v=−qAc ℏ2(B+b). (10) The free energy formalism provides a convenient springboard for extending our analysis to dynamics as well. In particular, such analysis helps us understand how emergent electric field would arise, in analogy with the standard electrodynamics. This is accomplished by considering the Langrangian for this minimal model. C. Equations of motion For the dynamics analysis, we now obtain the classical equation of motion for both the charge and the spin com-ponent of the order parameter through considering the Lagrangian of the spin-polarized superconductor. This can be written as L=Z dV2iℏd∗·∂td−F =−Z dVℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]−F . (11) The first term of the above Lagrangian arises from 2 iℏd∗ being the conjugate variable to d; the detailed derivation of its relation to ρ, ϕ,ˆscan be found in Appendix B. The low-energy dynamics of the order parameter can be described by the three Euler-Lagrange equations for ϕ, ρ, andˆs. The equations for ρandϕare basically analogous to those of the conventional superconductors. The equation of motion for the density ρ, ˙ρ=−1 ℏ∂in ρAch ∂iϕ−q ℏAi−ˆs·(ˆu×∂iˆu)io =−1 q∂iJi, =−∇·(ρv) (12) is obtained from δL/δϕ = 0 and is none other than the continuity equation for the Cooper pair density. Simi- larly, the equation of motion for the phase ϕ −ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =F0+U(ρ−ρ0)−Ac 4ρ∇2ρ(13) obtained from δL/δρ = 0 (where only terms constant or linear in ρare retained) comes out to be the Josephson relation. We, however, want to obtain a hydrodynamic equation of motion for Cooper pairs, for which purpose we take the spatial derivative of the Josephson relation:4 −ℏh ∂t∂iϕ−q ℏ(Ei+ei)−q ℏ∂t(Ai+ai)i =ℏ2 Acv·∂iv+∂i ρ′ e+U(ρ−ρ0)−Ac 4ρ∇2ρ , where ei=−ℏ qˆs·(∂iˆs×∂tˆs) (14) is the emergent electric field and ρ′ e=As(∂iˆs)2 2−Ds2 z 2−Hsz (15) is the magnetic energy density (per unit density). By using v·∂iv=1 2∂i(v2) = (v·∇)vi+ϵijkvj(∇×v)k= (v·∇)vi−qAc ℏ2ϵijkvj(Bk+bk) and defining the material derivative Dt≡∂t+v·∇and the effective mass of a Cooper pair, m≡ℏ2/Ac, we obtain mDtv=q(E+e) +qv×(B+b)−∂i ρ′ e+U(ρ−ρ0)−Ac 4ρ∇2ρ . (16) The novelty in the ferromagnetic superconductor is the equation of motion for the spin direction ˆsthat is derived from δL/δˆs= 0 (see Appendix C for details): ℏρ∂tˆs=−ℏJi q∂iˆs+∂i[ρAs(ˆs×∂iˆs)]+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z, (17) which is identical to the Landau-Lifshitz equation [29] augmented by the adiabatic spin-transfer torque [9, 30, 31]. This can be also written as ℏρDtˆs=∂i[ρAs(ˆs×∂iˆs)] +ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z. By using ˙ ρ=−∂iJi/q, we can obtain the spin continuity equation: ∂t(ℏρˆs) =−∂iJs i+ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z,(18) where Js i=ℏJi qˆs−ρAs(ˆs×∂iˆs), is the spin current density. The first term and the second term on the right-hand side are longitudinal spin currents proportional to the charge current and the transverse spin current that is carried by a spin texture, respec- tively. A complete set of equations describing the hydrody- namics of a ferromagnetic superconductor in the absence of external fields ( E= 0 and B= 0) can now be given; the analogous equations have been written down for a spinor BEC [32]. It is convenient to measure energy in the unit of the anisotropy energy absolute value |D|andlength in the unit derived from the combination of |D| with the spin stiffness As,i.e. l=s As |D|, ϵ=|D|. Also, we will use ˜ ρ≡ρ/ρ0, Uρ/D ≡η. This gives us the dimensionless equations −Dt˜ρ= ˜ρ(∇·v), ˜m(∇×v) =−b, ˜mDtv=e+v×b −∂i[ρe+η(˜ρ−1)− ∇2˜ρ/4], ˜ρDtˆs=∂i[˜ρ(ˆs×∂iˆs)] + ˜ρν(ˆs·ˆz)ˆs׈z+ ˜ρhˆs׈z, (19) where ν≡sgn(D), ˜m≡As/Acthe dimensionless mass which is on the order of unity, h≡H/D the dimension- less external field, and ρe=1 2(∂iˆs)2−ν1 2(ˆs·ˆz)2−hsz the dimensionless magnetic energy density; for zero ex- cess spin stiffness ˜ m= 1/2. The emergent electromag- netic fields are now re-defined as ei=−ˆs·(∂iˆs×∂tˆs), b i=−ϵijk 2ˆs·(∂jˆs×∂kˆs), where the charge qis absorbed into the fields.5 D. Gilbert damping Due to the inevitable nonconservation of spin angu- lar momentum in solids, it is reasonable to expect the damping of spin dynamics and the associated energy dis- sipation, which are not included in the hydrodynamics equations [Eq. (19)], to play an important role in spin dynamics of the ferromagnetic superconductors as in any other solid-state systems. The spin sinks can be quasi- particles, phonons, and any other excitations that can possess angular momentum [33–36]. The spin dissipation can be treated phenomenologically with the addition of the Gilbert damping term α˜ρˆs×∂tˆs[8] to the spin equa- tion of motion in Eq. (19), ˜ρDtˆs+α˜ρˆs×∂tˆs=∂i[˜ρ(ˆs×∂iˆs)]+ ˜ρν(ˆs·ˆz)ˆs׈z+ ˜ρhˆs׈z. (20) In the incompressible limit η→ ∞ where ˜ ρis uniform and constant, this gives us the energy continuity equation, ∂tρe+∇·je=−v·e−α(∂tˆs)2, (21) where je=−∂tˆs·∇ˆs (22) is the magnetic energy flux density (per unit density). This continuity equation, which has been previously noted in literature [37, 38], can be derived by taking the product of both sides of Eq. (20) with ˆs×∂tˆs. One can note that the first term on the right-hand side of Eq. (21) (−v·e) is the power dissipated (supplied) by the su- perflow vflowing parallel (antiparallel) to the direction of the emergent electric field e, while the second term (−α(∂tˆs)2) is the energy dissipation through the Gilbert damping. Equation (21) implies that, in the incompressible limit, the energy dissipated by the Gilbert damping is equal to the work done by the emergent electric field when spin texture is transported without any distortion. This is because the total magnetic energy should be unchanged in this process and hence the left-hand side of Eq. (21) integrated over the whole system should be zero. III. DOMAIN WALL For the supercurrent-driven motion of topological de- fects, we first consider an easy-axis ferromagnetic super- conductor with spin-anisotropy sign ν= sgn( D) = +1. A domain wall is a generically stable topological defect between two different ground states for easy-axis spin systems and intrinsically has no skyrmion density, i.e. no emergent magnetic field. Therefore for the rest of this section, we will take the Cooper pair velocity to be ir- rotational, i.e.∇×v= 0 [Eq. (19)]. In addition, we also set the applied magnetic field to be zero and hence h= 0.A. Deriving dynamics from a static solution The solution for a domain-wall motion in the back- ground of the constant and uniform background super- flow can be straightforwardly constructed from the static domain wall solution in absence of any background su- perflow for the incompressible limit η→ ∞ . For this case, the absence of the emergent magnetic field allows us to consider only the spin equation of motion [40] (∂t+v·∇+αˆs×∂t)ˆs=∂i[(ˆs×∂iˆs)] + (ˆs·ˆz)ˆs׈z; (23) from the hydrodynamic equations Eq. (19); others are either irrelevant due to b= 0 or merely provides the constraint ˜ ρ= 1 in the incompressible limit. Given the intrinsically quasi-one-dimensional nature of the domain wall, we can set the boundary condition ˆs(x→ ±∞ ) =±ˆz, for any domain-wall configuration. For the static domain wall at x= 0 in absence of background superflow, the solution is given by the following Walker ansatz [41]: ˆs0= (ˆxcosφ0+ˆysinφ0)sech x+Qˆztanhx, (24) where Q=±1 represents the domain wall type, satisfies the static domain-wall equation 0 =∂x[(ˆs0×∂xˆs0)] + ( ˆs0·ˆz)ˆs0׈z, (25) derived from Eq. (23), for an arbitrary domain-wall angle φ0; it is important to note here that Eq. (25) is sufficient as Eq. (24) gives us v= 0 and b= 0 everywhere. From Eq. (25), it can be shown that the general solu- tion can be obtained by applying to the static solution both giving boost in the spatial direction and precession around the easy-axis: ˆs= (ˆxcos Ω t+ˆysin Ωt)sech( x−V t) +Qˆztanh( x−V t). (26) In deriving the domain-wall velocity ˆxVand the preces- sion rate Ω, it is convenient to note that Eq. (26) also satisfies Eq. (25) as the latter equation involves no time derivatives. Also, as Eq. (26) is obtained from boost and precession, ∂tˆs= (−V ∂x+ˆzΩ×)ˆs. The velocity Vand the precession rate Ω therefore can be obtained from [v∂x+ (1 + αˆs×)(−V ∂x+ˆzΩ×)]ˆs= 0, (27) where we set the background superflow to be perpendic- ular to the domain wall without any loss of generality, v=vˆx. By taking the scalar product of the above equa- tion with ˆzandˆz׈swe obtain v−V=QαΩ,Ω =−QαV6 (a)<latexit sha1_base64="mvV1IC9/vAkApJWzfWyUx3QadJw=">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVpiZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBSnJC1</latexit> (b)<latexit sha1_base64="Lyp4rKMmNObzzWYxIHNaXwltWlk=">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVoKzibdQtEtuzPQZeJlpEgy1LqFr04v4kkICrlkxrQ9N0Y/ZRoFlzDJdxIDMeMjNoC2pYqFYPx0dvGEnlqlR/uRtqWQztTfEykLjRmHge0MGQ7NojcV//PaCfYv/VSoOEFQfL6on0iKEZ2+T3tCA0c5toRxLeytlA+ZZhxtSHkbgrf48jJpnJe9SvnqrlKsXmdx5MgxOSEl4pELUiW3pEbqhBNFnskreXOM8+K8Ox/z1hUnmzkif+B8/gBUIpC2</latexit> (c)<latexit sha1_base64="eT4ibbwfOOWkpDCTAUUKdbMXNmE=">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVriZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBVqJC3</latexit> FIG. 2. (a) A series of snapshots of a precessing domain wall moving to the right, where xandtare the spatial and the temporal coordinates, respectively. The red arrows, blue arrows, and green arrows represent ˆs,ˆu, and ˆv, respectively. The domain-wall position is denoted by the gray dot. The domain-wall angle, which is the azimuthal angle of ˆsat the center of the domain wall, changes from φ0= 0 to φ0=−2πgradually with increasing time from bottom to top. On the left end, ˆu(green arrow) rotates by −2πabout ˆs(red arrow), whereas on the right end, ˆurotates by 2 πabout ˆs. This process of a domain-wall precession can be considered as a nonsingular 4 πphase slip since these opposite 2 πrotations of ˆuaround ˆsat the left and the right ends induces a finite voltage across the wire. (b) Mapping of the instantaneous configuration of the two vectors ˆs (red arrows) and ˆu(green arrows) onto the unit sphere with the normal vector identified with ˆs. The yellow line represents a spatial dimension of the system. (c) Collection of the mapping of the configuration of ˆsandˆuonto the unit sphere with ˆs identified with the normal vector for all the snapshots shown in (a). Note that the unit tangent vector field ˆuis not uniquely determined at the north and the south poles as dictated by the Poincar´ e-Hopf theorem [39]. Rather, ˆurotates once around ˆscounterclockwise (counterclockwise) at the north (south) pole as the domain wall completes one cycle of rotation, which is consistent with the Euler number 2 of the sphere. See the main text for further detailed discussions. respectively, giving us V=1 1 +α2v ,Ω =−Qα 1 +α2v; (28) Note that in absence of the Gilbert damping, α= 0, there would have been no precession and the domain wall would have remained static with respect to the background su- perflow. See Fig. 2 for the illustration of the domain-wall dynamics with precession. From the above solution, it is straightforward to con- firm that all work done by the emergent electric field is dissipated through the Gilbert damping. The work done by the emergent electric field is −v·e=vjˆs·(∂jˆs×∂tˆs) =QvΩ[1−(ˆs·ˆz)2],(29) which gives the total energy input of W=Z dx(−v·e) = 2 QvΩ. (30) The energy dissipation rate per unit density is given by α(∂tˆs)2=α(V2+ Ω2)[1−(ˆs·ˆz)2]. (31) This energy dissipation through the spin dynamics and the work rate done by the emergent electric field on the superflow are the same since QvΩ =α 1 +α2v2(32) and α(V2+ Ω2) =α 1 +α2v2. (33)B. 4πphase slips from a domain-wall dynamics Due to the U(1) ϕ+sorder parameter redundancy, the energy dissipation from the damping-induced precession in the domain-wall motion can be regarded as an equiva- lent of 4 πphase slips. As shown in Fig. 2, ˆuandˆvrotate around ˆsby±2πby adiabatically following the dynamics of the local spin direction ˆsin one cycle of precession. To see this, note that if we adopt the condition ˆv·ˆz= 0 in defining ˆv, Eq. (26) will give us ˆv=−ˆxsin Ωt+ˆycos Ω t. But given the U(1) ϕ+sredundancy, this is equivalent to the±2πphase twist on the left and the right end, re- spectively. The voltage arising from this precession can be under- stood either as arising from the emergent electric field, or equivalently, arising from the constant rate of 4 πphase slips. When φincreases at the rate ˙ φ= Ω, the emergent scalar potential at the two ends of the wire, x=±∞, is given by ˜Ve=−ˆs·(ˆu×∂tˆu) =( −QΩ at x=−∞, QΩ at x=∞,(34) which is exactly the Josephson voltage for the 4 Qπphase slip occurring at the rate of Ω /2π. Then, to maintain the finite superflow, there must be work given by W=v[˜Ve(x=∞)−˜Ve(x=−∞)] = 2 QvΩ,(35) by external reservoirs on the system, matching the work7 [Eq. (30)]. Figure 2(a) shows the time evolution of the triad{ˆs,ˆu,ˆv}associated with the domain-wall that both moves and precesses. Note that ˆu(green arrow) at the left end ( x→ −∞ ) and the right end ( x→ ∞ ) ro- tates clockwise and counterclockwise, respectively, about ˆs(red arrow), engendering the nonsingular phase slips across the xdirection. In a nutshell, the domain-wall an- gular dynamics produces the nonsingular 4 πphase slips that give rise to a finite voltage difference between the two ends of the superconducting wire, which constitutes our first main result. There is a topological reason why one precession of a magnetic domain wall induces a 4 πphase slip, which can be derived from the Poincar´ e-Hopf theorem or Poincar´ e- Brower theorem [39]. For concrete discussion on this, we will consider in the following the case with Q= 1 as shown in Fig. 2(b) and (c). At a given time, the instan- taneous configuration of ˆsandˆucan be mapped onto a line connecting the north pole [ ˆs(x→ ∞ )] and the south pole [ ˆs(x→ −∞ )] on the unit sphere by identifying ˆs with the surface normal as shown in Fig. 2(b). When we consider the collection of the configuration of {ˆs,ˆu}onto the unit sphere during one complete precession of the do- main wall, i.e., φ0→φ0+2π,ˆucan be regarded as a unit tangent vector field on the sphere since it is perpendic- ular to ˆs, i.e., the surface normal as shown in Fig. 2(c). Here, note that ˆuis not uniquely determined at the north pole and the south pole, which is consistent with the well- known topological property of the sphere that the unit tangent vector field cannot be defined without a singu- larity on it. Instead of being uniquely determined, ˆu rotates by 2 πaround the north pole and rotates by −2π around the south pole as the domain wall completes one cycle of precession, which gives rise to a 4 πphase slip across the wire as discussed above [Eq. (34)]. This can be understood by applying the Poincar´ e-Hopf theorem to the unit tangent vector field ˆuon the sphere. The Euler number of the unit sphere is 2, meaning that the sum of the indices of the isolated singularities of the unit tan- gent vector field on the sphere must be 2. In our case, the indices of the north pole and the south pole associ- ated with ˆuare both 1, adding up to 2, agreeing with the Euler number of the unit sphere. IV. MERON We now consider ferromagnetic superconductors with easy-plane spin anisotropy ( ν= sgn( D) =−1). For easy- plane spin systems, a meron, with its one-half skyrmion charge, is a generically stable topological defect [42– 45]. Therefore, we consider the rotational Cooper pair velocity in absence of the applied magnetic field i.e. ∇×v=−b/˜mwith h= 0. See Fig. 3 for the schematic illustration of a meron. (a)<latexit sha1_base64="mvV1IC9/vAkApJWzfWyUx3QadJw=">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVpiZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBSnJC1</latexit> (b)<latexit sha1_base64="Lyp4rKMmNObzzWYxIHNaXwltWlk=">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVoKzibdQtEtuzPQZeJlpEgy1LqFr04v4kkICrlkxrQ9N0Y/ZRoFlzDJdxIDMeMjNoC2pYqFYPx0dvGEnlqlR/uRtqWQztTfEykLjRmHge0MGQ7NojcV//PaCfYv/VSoOEFQfL6on0iKEZ2+T3tCA0c5toRxLeytlA+ZZhxtSHkbgrf48jJpnJe9SvnqrlKsXmdx5MgxOSEl4pELUiW3pEbqhBNFnskreXOM8+K8Ox/z1hUnmzkif+B8/gBUIpC2</latexit> (c)<latexit sha1_base64="eT4ibbwfOOWkpDCTAUUKdbMXNmE=">AAAB8XicbVDLSgNBEJz1GeMr6tHLYBDiJexKQL0FvXiMYB6YLGF20kmGzM4uM71iWPIXXjwo4tW/8ebfOEn2oIkFDUVVN91dQSyFQdf9dlZW19Y3NnNb+e2d3b39wsFhw0SJ5lDnkYx0K2AGpFBQR4ESWrEGFgYSmsHoZuo3H0EbEal7HMfgh2ygRF9whlZ66CA8YVriZ5NuoeiW3RnoMvEyUiQZat3CV6cX8SQEhVwyY9qeG6OfMo2CS5jkO4mBmPERG0DbUsVCMH46u3hCT63So/1I21JIZ+rviZSFxozDwHaGDIdm0ZuK/3ntBPuXfipUnCAoPl/UTyTFiE7fpz2hgaMcW8K4FvZWyodMM442pLwNwVt8eZk0zstepXx1VylWr7M4cuSYnJAS8cgFqZJbUiN1wokiz+SVvDnGeXHenY9564qTzRyRP3A+fwBVqJC3</latexit> FIG. 3. (a) An illustration of the meron with polarity p= 1 and vorticity n= 1, which is a nonsingular topological de- fect of ferromagnetic superconductors with easy-plane spin anisotropy. The red, the green, and the blue arrows repre- sentˆs,ˆu, and ˆv, respectively. The meron core is denoted by the gray dot. The local spin direction ˆsrotates by 2 πcoun- terclockwise about the zaxis when we follow the infinitely distant trajectory encircling the meron core counterclockwise (and thus the vorticity n= 1). Note that ˆuandˆvalso change spatially to keep their orthonormality to ˆs; they rotate by 2 π clockwise about the local spin direction ˆswhen we enclose the meron center counterclockwise. (b) Mapping of the con- figuration of ˆs(red arrows) and ˆu(green arrows) along the infinitely distant circle in (a) onto the equator with the surface normal identified with ˆs. (c) Mapping of the configuration of ˆsand ˆuof the entire system onto the northern hemisphere with ˆsidentified with the surface normal. Note that ˆuis a well-defined unit tangent vector field on the northern hemi- sphere without any singularity. Since the Euler number of the hemisphere is 1, if the unit tangent vector field defined on the northern hemisphere has no singularity, it should rotate around the surface normal by 2 πalong the equator according to the Poincar´ e-Hopf theorem [39], which is exactly what ˆu (unit tangent vector) does around ˆs(surface normal). A. Static solution Analogous to the case of the domain wall in the previ- ous section, a straightforward construction of the meron motion solution here in the background of the constant and uniform background superflow is possible from the static meron solution in absence of any background su- perflow for the incompressible limit, η→ ∞ [32, 46]. The static solution can be obtained from the following8 two equations; ˜m(∇×v) =−b, (v·∇)ˆs=∂i[(ˆs×∂iˆs)]−(ˆs·ˆz)ˆs׈z.(36) It is important to note here that while we are dealing with a static configuration we still have v̸= 0 due to the intrinsic emergent magnetic flux of the meron. In addition, this solution would possess the axial symmetry, i.e. ˆs0= (sin θcosφ,sinθsinφ,cosθ), (37) with θ=θ(r) and φ=nχ+ Φ where ( r, χ) are polar coordinates for the two-dimensional system, and follow the universal boundary conditions for merons are given by θ(r= 0) = (1 −p)π 2, θ(r→ ∞ ) =π 2; p=±1 here is the polarity, which is the z-component of the local spin direction ˆsat the meron center, and n∈Z the vorticity, which counts how many times ˆswinds in the easy plane along the closed trajectory encircling the meron center. For our purpose, obtaining the differential equation for θ(r) is sufficient for showing Eq. (37) to be the solution of Eq. (36). We start by noting that the emergent magnetic field is aligned entirely along the z- axis and bz=−ˆs0·(∂xˆs0×∂yˆs0) =−nsinθ rdθ dr(38) is function only for θ, which gives us the well-known re- sult Z dxdyb z=Z rdχdr −nsinθ rdθ dr =−2πpn (39) for the total emergent magnetic flux. With this emergent magnetic field, the first equation of Eq. (36) requires the circulating velocity v=ˆφv(r) around the meron with 1 rd(rv) dr=nsinθ ˜mrdθ dr. Inserting this relation into the second equation of Eq. (36) gives us sinθ(1−cosθ)1 ˜mr2=1 rd dr rdθ dr +cos θsinθ 1−1 r2 , from which θ(r) can be obtained numerically. To find explicit expressions for ˆuandˆv, note that the following three unit vectors form an orthonormal triad: ˆs0=ˆer≡(sinθcosφ,sinθsinφ,cosθ), ˆeθ≡(cosθcosφ,cosθsinφ,−sinθ), ˆeφ≡(−sinφ,cosφ,0),which gives us ˆu0(r, χ) = cos φ(χ)ˆeθ(r, χ)−sinφ(χ)ˆeφ(r, χ),(40) ˆv0(r, χ) = sin φ(χ)ˆeθ(r, χ) + cos φ(χ)ˆeφ(r, χ).(41) The local configuration of the triad ( ˆs0,ˆu0,ˆv0) for a meron with p= 1 and n= 1 is shown in Fig. 3. Note that it is nonsingular, differing from a conventional vor- tex of a s-wave superconductor [10]. An analogous non- singular topological defects that give rise to 4 πnonsin- gular phase slips has been discussed by Anderson and Toulouse [47] and has been termed the skyrmion solu- tion in the more recent literature [28, 32]. By contrast, our solution {ˆs0,ˆu0,ˆv0}[Eqs. (37,40,41)] represents an explicit solution for the nonsingular topological defect in the easy-plane case that gives rise to 2 πnonsingular phase slip, as can be seen from Eqs (36) and (39): it harbors the emergent magnetic flux −2πpnand thus its motion gives rise to the emergent electric field, i.e., phase slips perpendicular to its motion. The non-trivial emergent gauge field ai, and thus the quantized non-zero emergent magnetic flux, of our non- singular topological defect can be understood from the rotation of ˆuaround ˆsas stated in Eq. (4). There exists a topological constraint dictating that a meron texture ofˆsshould trap a quantized non-zero emergent magnetic flux, which can be understood by invoking the Poincar´ e- Hopf theorem [27, 39] as follows. For the given meron configuration with p= 1, let us consider the mapping of two unit vectors ˆsandˆuonto the northern hemisphere such that ˆsis identified with the surface normal. Then, ˆubecomes a unit tangent vector field on the hemisphere since it is perpendicular to ˆs, i.e., the surface normal as shown in Fig. 3(c). The Euler number of the hemisphere is 1, and thus the Poincar´ e-Hopf theorem dictates that, if the unit tangent vector field is nonsingular on the north- ern hemisphere, it should rotate exactly one time about the surface normal while traversing the equator, which is exactly what ˆudoes around ˆsin Fig. 3(b). Therefore, the one-time rotation of ˆuaround ˆsalong the closed loop that contains, but is also infinitely far from, the meron core as shown in Fig. 3(a), which gives rise to the quantized emergent magnetic flux, can be regarded as the physical manifestation of the topological constraint that should be satisfied by the nonsingular unit tangent vector field defined on the hemisphere. B. Dynamics with a background superflow Analogous to the domain wall motion, it is straight- forward to work out the spin equation of motion for the meron motion driven by a uniform constant background superflow v0if we assume the meron motion to be rigid, i.e.ˆs(r, t) =ˆs0(r−Vt) (the same holds for ˆuandˆv). The Cooper pair velocity would then be given by v=vm+v0, where vmis the Cooper pair velocity around a static meron. We can therefore employ the collective coordi- nate approach to describe the dynamics of a meron, and9 FIG. 4. Snapshots of a meron moving in the ydirection in the increasing time from (a) to (d), where the meron core is depicted by a gray dot. At x→ −∞ ,ˆu(green arrow) rotates about ˆs(red arrow) counterclockwise, whereas at x→ ∞ ,ˆurotates about ˆsclockwise, inducing phase slips and thereby generating a finite voltage in the xdirection. use the fact that ˆs0(r−Vt) should satisfy the original spin equation of motion of Eq. (20) with, as we are in the incompressible limit, the constant ˜ ρwhile ˆs0(r) also satisfies the equation for the static meron of Eq. (36). Subtraction between the two equations, together with ∂tˆs0(r−Vt) =−V·∇ˆs0(r−Vt) gives us −(1 +αˆs0×)V·∇ˆs0=−v0·∇ˆs0. Taking the scalar product with s0×∂is0on both sides gives us −Vj[s0·(∂iˆs0×∂jˆs0)]−αVj(∂iˆs0·∂jˆs0) =−v0,j[s0·(∂iˆs0×∂jˆs0)].(42) Strictly speaking, this result shows that whereas we have an exact rigid motion solution with V=v0in absence of damping, no rigid motion solution can be exact in presence of damping. Yet to the zeroth order in v0and also in the spirit of collective coordinate, we can average out the effect of the spin texture, i.e.defining Gij=Z dxdys0·(∂iˆs0×∂jˆs0), D ij=Z dxdy∂ iˆs0·∂jˆs0, known respectively as the gyrotropic coefficients and the dissipation coefficients [48–50] ( Dij=DδijandGxy= −Gyx≡G= 2πpnfor a meron). By integrating Eq. (42), we have (αD+Gˆz×)V=Gˆz×v0, (43) which yields the following solution for the velocity V: V=G G2+α2D2(G+αDˆz×)v0. (44) To consider a concrete example, we will hereafter restrict the discussion to the case where the background super- flow flows in the xdirection: v0= (v0,0). In this case,we have  Vx Vy =G G2+α2D2 Gv0 −αDv 0 . (45) Note that the presence of damping gives rise to the com- ponent of the meron velocity transverse to the uniform superflow in proportion to the skyrmion number G/2πof the meron, Vy=−α[GD/(G2+α2D2)]v0, exhibiting the so-called skyrmion Hall effect [16, 51–53]. This transverse motion of the meron with respect to the superflow gives rise to the finite voltage in the direction of the superflow via the 2 πphase slips, which we turn our attention now. C. 2πphase slips from a meron motion Again analogous to the domain wall motion, the U(1) ϕ+sorder parameter redundancy allows the energy dissipation due to the meron motion as equivalent to the 2πphase slips. This can be seen from the emergent elec- tric field e=−ˆs·(∇ˆs×∂tˆs) =ˆs0·(∇ˆs0×V·∇ˆs0) =−V×b, (46) which is in the same form as the Josephson electric field arising from the vortex motion [10]. The same can natu- rally be said about the input power density required for driving the uniform constant background superflow −v0·e=v0,xˆs·(∂xˆs×∂tˆs), =−v0,xVyˆs·(∂xˆs×∂yˆs). It can be checked explicitly that the total energy rate for driving the superflow −Z dxdyv0·ˆe=−v0,xVyG=αG2D G2+α2D2v2 0,10 is equal to the energy dissipation rate Z dxdyα (∂tˆs)2=Z dxdyαV iVj(∂iˆs0·∂jˆs0) =αDV2=αG2D G2+α2D2v2 0. This energy must come from external reservoirs through the boundary of the system, meaning that there should be a development of a finite voltage across the system in the xdirection. This can be explicitly seen from ˜Ve=−ˆs·(ˆu×∂tˆu) = (1 −cosθ)∂tφ . (47) To see how a finite voltage is generated by the motion of a vortex, let us assume that a vortex moves in the y direction, V=Vyˆy. Then for a given point at large x, Z∞ −∞dtˆs·(ˆu×∂tˆu) =−Z∞ −∞dt∂tφ=nπ , forx→+∞. (48) The vector ˆu(x→ ∞ ) rotates by nπaround the ˆs. Also, Z∞ −∞dtˆs·(ˆu×∂tˆu) =−Z∞ −∞dt∂tφ=−nπ , forx→ −∞ . (49) The vector ˆu(x→ −∞ ) rotates by −nπaround the ˆs. This indicates that the motion of a meron in the ydirection induces a nonsingular 2 πphase slips across thexdirection. To keep the superflow in the xdirec- tion constant, we need to counteract the effect of these phase slips. The corresponding work on the system is dissipated to external baths such as quasiparticles or phonons through the Gilbert damping. Figure 4 shows the schematic illustration of the process. As the vor- tex moves in the positive ydirection from Fig. 4(a) to Fig. 4(d), ˆuatx→ ∞ andx→ −∞ rotates about the local spin direction ˆscounterclockwise and clockwise, re- spectively, producing phase slips in the xdirection. This is our second main result: The dynamics of a meron en- genders the nonsingular 2 πphase slips perpendicular to its motion through the generation of the emergent elec- tric field, showcasing the intertwined dynamics of spin and charge degrees of freedom of ferromagnetic super- conductors. V. DISCUSSION Within the phenomenological framework for the dy- namics of the order parameter of ferromagnetic super- conductors, we have shown that the current-induced dy- namics of magnetic defects in the presence of the spin dissipation, i.e., the Gilbert damping, give rise to nonsin- gular phase slips via the emergent electromagnetic fields. The input power, which is the product of the applied current that drives the magnetic defects and the voltage generated by the phase slip, is shown to be equivalent tothe dissipated energy in the form the Gilbert damping into the baths of quasiparticles or phonons. Our work on the dynamics of magnetic defects showcases the intrinsic interplay of spin and charge dynamics of ferromagnetic superconductors. A few remarks are on order about the limitations of our work. First, we did not include the effects of the non-adiabatic spin-transfer torque by a supercurrent on the dynamics of magnetic defects, which is expected to be present on general grounds whenever the Gilbert damp- ing is present [9, 40, 54, 55]. While we believe that the inclusion of the non-adiabatic spin-transfer torque in our model would not qualitatively change the relations that we have found between the dynamics of magnetic defects and nonsingular phase slips, it will certainly enrich the physics of the interplay of spin and charge dynamics in ferromagnetic superconductors. Secondly, in this work, a ferromagnetic superconductor is assumed to be a fully spin-polarized triplet superconductor as in Ref. [12], by leaving the generalization to a partially spin-polarized case as future work. Thirdly, our analysis shows that the assumption of rigid motion is not exact for merons in presence of damping. The coupling between current and magnons bound to meron cores may be a relevant topic for future study. Lastly, the dynamics of magnetic defects has been discussed in the incompressible limit, where the dynamics of the order-parameter amplitude is frozen. Releasing this assumption would allow us to study the interplay of spin dynamics and longitudinal order-parameter dynamics, which is beyond the scope of the current work. ACKNOWLEDGMENTS We thank Mike Stone, Grigori Volovik, Daniel Agterberg and Jim Sauls for useful discussions. S.K.K. was supported by Brain Pool Plus Program through the National Research Foundation of Korea funded by the Ministry of Science and ICT (NRF- 2020H1D3A2A03099291) and by the National Research Foundation of Korea funded by the Korea Government via the SRC Center for Quantum Coherence in Con- densed Matter (NRF-RS-2023-00207732). S.B.C. was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIT) (NRF-2023R1A2C1006144, NRF-2020R1A2C1007554, and NRF-2018R1A6A1A06024977).11 Appendix A: Details of the order parameter The multicomponent superconducting gap is given by ˆ∆ = ∆↑↑∆↑↓ ∆↓↑∆↓↓ ≡ −˜dx+i˜dy˜dz ˜dz˜dx+i˜dy =i(d·σ)σy. (A1) Then, ˆ∆ˆ∆†= ∆↑↑∆↑↓ ∆↓↑∆↓↓∆∗ ↑↑∆∗ ↓↑ ∆∗ ↑↓∆∗ ↓↓ =|∆↑↑|2+|∆↑↓|2∆↑↑∆∗ ↓↑+ ∆↑↓∆∗ ↓↓ ∆↓↑∆∗ ↑↑+ ∆↓↓∆∗ ↑↓|∆↓↓|2+|∆↓↑|2 = −˜dx+i˜dy˜dz ˜dz˜dx+i˜dy−˜d∗ x−i˜d∗ y˜d∗ z ˜d∗ z˜d∗ x−i˜d∗ y =|˜dx|2+|˜dy|2+|˜dz|2+i(˜dx˜d∗ y−˜d∗ x˜dy) ( −˜dx˜d∗ z+i˜dy˜d∗ z) + c.c. (−˜d∗ x−i˜d∗ y)˜dz) + c.c. |˜dx|2+|˜dy|2+|˜dz|2−i(˜dx˜d∗ y−˜d∗ x˜dy) =|d|2σ0+i(d×d∗)·σ, (A2) where d= (˜dx,˜dy,˜dz). The total number density of the Cooper pairs is given by |∆↑↑|2+|∆↑↓|2+|∆↓↑|2+|∆↓↓|2= Tr[ ˆ∆ˆ∆†] = 2d·d∗= 2|d|2. (A3) The expected spin angular momentum is given in units of ℏby Tr[ˆ∆ˆ∆†σ] = Tr[ i(d×d∗)·σσ] = 2id×d∗. (A4) For a fully spin-polarized triplet superconductor, we can use d=deiϕ(ˆu+iˆv), (A5) with ˆu⊥ˆv. Then, we have d·d∗= 2d2. Therefore, d=√ρ/2. The spin polarization is then given by s= 2id×d∗= 4d2(ˆu׈v) ≡4d2ˆs=ρˆs. (A6) To see if the result makes sense, let us consider ˆs=ˆzwith ˆu=ˆxandˆv=ˆy. Then, d=deiϕ(ˆx+iˆy). Then, ˜dx=deiϕand˜dy=deiϕi. Then, ∆ ↑↑=−2d,∆↓↓= ∆↑↓= ∆↓↑= 0. The condensate density (of the Cooper pairs) is given by ρ=|∆↑↑|2= 4d2and the spin density is given by s= 4d2ˆs=ρˆs. Appendix B: Kinetic term of the Lagrangian The kinetic term of the Lagrangian density is given by LK= 2iℏd∗·∂td =iℏ 2√ρe−iϕ(ˆu−iˆv)·∂t√ρeiϕ(ˆu+iˆv) =iℏ 2√ρe−iϕ(ˆu−iˆv)·∂tρ 2√ρeiϕ(ˆu+iˆv) + (i∂tϕ)√ρeiϕ(ˆu+iˆv) +√ρeiϕ(∂tˆu+i∂tˆv) =iℏ 2√ρ(ˆu−iˆv)·∂tρ 2√ρ(ˆu+iˆv) + (i∂tϕ)√ρ(ˆu+iˆv) +√ρ(∂tˆu+i∂tˆv) =iℏ 2∂tρ 22 + (i∂tϕ)ρ2 +ρ(iˆu·∂tˆv−iˆv·∂tˆu) =−ℏρ(∂tϕ+ˆu·∂tˆv) =−ℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)]. (B1)12 The factor of 2 in front is to ensure the commutation relation, [ d∗ i(r), dj(r′)] = 2 iℏδ(r−r′). Then, the total Lagrangian density is given by L=−ℏρ[∂tϕ−ˆs·(ˆu×∂tˆu)] −ρAc 2h ∂iϕ+q ℏAi+ˆs·(ˆu×∂iˆu)i2 +As 2(∂iˆs)2−D 2(ˆs·ˆz)2−Hsz+qV −Ac 16(∇ρ)2+U 2(ρ−ρ0)2 .(B2) From this, we can read the emergent scalar potential Ve=−ℏ qˆs·(ˆu×∂tˆu), (B3) and the emergent vector potential ai=ℏ qˆs·(ˆu×∂iˆu). (B4) Appendix C: Equations of motion The dynamics of the order parameter can be uniquely characterized by the dynamics of three variables, the con- densate number density of the Cooper pairs ρ, the phase of the order parameter ϕ, and the spin direction ˆs. First, the equation of motion for ϕcan be obtained by δL δρ= 0, ⇒ −ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =δF δρ, ⇒ −ℏ[∂tϕ−ˆs·(ˆu×∂tˆu)] =Ac 2h ∂iϕ−q ℏAi−ˆs·(ˆu×∂iˆu)i2 +As 2(∂iˆs)2−D 2(ˆs·ˆz)2−Hsz+qV −Ac 8∇2ρ+U(ρ−ρ0). (C1) Second, the equation of motion for ρcan be obtained by d dtδL δ˙ϕ −δL δϕ= 0, ⇒ −ℏ˙ρ=−δF δϕ, ⇒ −ℏ˙ρ=∂in ρAch ∂iϕ−q ℏAi−ˆs·(ˆu×∂iˆu)io , ⇒˙ρ=−1 q∂iJi, (C2) where Jiis the charge current density. This is nothing but the continuity equation. Third, to obtain the equation of motion for ˆs, by considering infinitesimal variations of the three vectors that maintain the orthonormality conditions, ˆs=ˆs0+δˆs=ˆs0+aˆu0+bˆv0, ˆu=ˆu0−aˆs0, ˆv=ˆv0−bˆs0, (C3) we observe that, to zeroth order in aandb, δ δˆs(ˆs·(ˆu×∂tˆu)) = ˆu0∂ ∂a(ˆs·(ˆu×∂tˆu)) +ˆv0∂ ∂b(ˆs·(ˆu×∂tˆu)) =−ˆu0(ˆs0·(ˆu0×∂tˆs0)) +ˆv0(ˆv0·(ˆu0×∂tˆu0)) =ˆs0×∂tˆs0. (C4)13 Also, by the analogous steps, δ δˆs(ˆs·(ˆu×∂iˆu)) =ˆs×∂iˆs. (C5) Then, from the Lagrangian, we obtain δL δˆs= 0, ⇒ℏρˆs×∂tˆs=δF δˆs, ⇒ℏρ∂tˆs=−ℏJi q∂iˆs+∂i[ρAs(ˆs×∂iˆs)] +ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z. (C6) By using ˙ ρ=−(∂iJi)/q, the last equation can be recast into ∂t(ℏρˆs) =−∂iℏJi qˆs−[ρAs(ˆs×∂iˆs)] +ρD(ˆs·ˆz)ˆs׈z+ρHˆs׈z. (C7) The left-hand side is the spin density, s=ℏρˆs. By identifying the first term on the right-hand side as the negative divergence of the spin-current density, we obtain the expression for the spin current density: Js i=ℏJi qˆs−ρAs(ˆs×∂iˆs). 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2023-05-23

Recent years have seen a number of instances where magnetism and superconductivity intrinsically coexist. Our focus is on the case where spin-triplet superconductivity arises out of ferromagnetism, and we make a hydrodynamic analysis of the effect of a charge supercurrent on magnetic topological defects like domain walls and merons. We find that the emergent electromagnetic field that arises out of the superconducting order parameter provides a description for not only the physical quantities such as the local energy flux density and the interaction between current and defects but also the energy dissipation through magnetic dynamics of the Gilbert damping, which becomes more prominent compared to the normal state as superconductivity attenuates the energy dissipation through the charge sector. In particular, we reveal that the current-induced dynamics of domain walls and merons in the presence of the Gilbert damping give rise to the nonsingular $4\pi$ and $2\pi$ phase slips, respectively, revealing the intertwined dynamics of spin and charge degrees of freedom in ferromagnetic superconductors.

Current-driven motion of magnetic topological defects in ferromagnetic superconductors

2305.13564v1

arXiv:1907.02041v3 [cond-mat.mes-hall] 7 Dec 2020Anisotropy of spin-transfer torques and Gilbert damping in duced by Rashba coupling I.A. Ado,1P.M. Ostrovsky,2,3and M. Titov1,4 1Radboud University, Institute for Molecules and Materials , NL-6525 AJ Nijmegen, The Netherlands 2Max Planck Institute for Solid State Research, Heisenbergs tr.1, 70569 Stuttgart, Germany 3L.D. Landau Institute for Theoretical Physics RAS, 119334 M oscow, Russia 4ITMO University, Saint Petersburg 197101, Russia Spin-transfer torques (STT), Gilbert damping (GD), and effe ctive spin renormalization (ESR) are investigated microscopically in a 2D Rashba ferromagnet wi th spin-independent Gaussian white- noise disorder. Rashba spin-orbit coupling-induced aniso tropy of these phenomena is thoroughly analysed. For the case of two partly filled spin subbands, a re markable relation between the anisotropic STT, GD, and ESR is established. In the absence o f magnetic field and other torques on magnetization, this relation corresponds to a current-i nduced motion of a magnetic texture with the classical drift velocity of conduction electrons. Fina lly, we compute spin susceptibility of the system and generalize the notion of spin-polarized current . Possibility to efficiently manipulate magnetic order by meansofelectriccurrenthasgainedalotofattentionover the past decades1,2. Potential applications include race trackmemory3,4, spin torquemagnetization switching5,6, skyrmion-based technology7,8, and other promising con- cepts. Spintronic logic and memory devices based on current-driven magnetization dynamics are believed to achieve high speed, low volatility, outstanding durabil- ity, and low material costs with promises to outperform charge-trapping solid-state memory devices9. In the light of recent detection of fast domain wall (DW) motion in magnetic films10,11and predictions of evenhigherDWvelocitiesinantiferromagnets12, current- induced dynamics of domain walls, skyrmions, and other magnetic textures remain an important research subject in the field of spintronics. Such dynamics is mainly de- termined by the interplay of the two phenomena: Gilbert damping (GD) and spin torques13–16. In the absence of spin-orbit coupling (SOC), spin torques emerge only in the systems with nonuniform magnetization profiles and are most often referred to as spin-transfer torques (STT). At the same time, the clas- sification of spin torques usually gets more complicated if coupling between spin and orbital degrees of freedom becomes pronounced. Moreover, the debate on the mi- croscopic origin of spin torques in the latter case remains ongoing17,18. Below, we regard STT, in the continuum limit, as a contribution to the total torque on magnetiza- tion that is linearwith respect to both the electricfield E and the first spatial derivatives of the unit vector of mag- netization direction n. We note that, in the absence of SOC, physics of STT is well understood15,16. In a similar fashion, Gilbert damping may be gener- ally associated with the terms of the Landau-Lifshitz- Gilbert (LLG) equation that are odd under time reversal and linear with respect to the time derivative of n. In the most simplistic approach, GD is modeled by a sin- gle phenomenological term αn×∂tnthat corresponds to “isotropic” damping. However, it has been known for quite a while that GD may exhibit anisotropic behaviour19–27. Or, to be more precise, that the scalar damping constant α, in general,should be replaced by a damping matrix with the com- ponents depending on the orientation of n. These two manifestations of anisotropy may be referred to as rota- tional and orientational anisotropy, respectively22. Ex- perimental observation of the orientational anisotropy of Gilbert damping has been reported very recently in a metal ferromagnet (FM)/semiconductor interface of Fe/GaAs(001)28and in epitaxial CoFe films29. The au- thors of Ref. [28] argued that the measured anisotropy rooted in the interplay of interfacial Rashba and Dressel- haus spin-orbit interaction. Given the equal importance of GD and STT in the context of current-induced magnetization dynamics and the significant progressmade in the understanding of the anisotropic nature of Gilbert damping, we find it surpris- ing that the anisotropyof spin-transfer torques has so far only been addressed phenomenologically24,30. In the present paper, we consider a 2D Rashba FM withspin-independent electronscattering. Amicroscopic analysis, performed for an arbitrarymagnetization direc- tion, allows us to quantify the rotational as well as the orientationalanisotropyofboth STT and GD induced by Rashba SOC. Our results indicate that, for a Rashba FM system, spin-transfer torques TSTTand Gilbert damp- ingTGDentering the LLG equation ∂tn=γn×Heff+TSTT+TGD+... (1) naturally acquire the following forms: TSTT=ξ0∂vn−ξ/bardbl[n×∂vn/bardbl]−ξ⊥[n×∂vn⊥],(2a) TGD=ξ0∂tn−ξ/bardbl[n×∂tn/bardbl]−ξ⊥[n×∂tn⊥],(2b) whereξi=ξi(n), the operator ∂v= (vd·∇) is expressed via the classical electron drift velocity vd=eE/planckover2pi1τ/m, andn/bardbl/⊥stands for the in-plane/perpendicular-to-the- plane component of the vector field n: n=n/bardbl+n⊥,n⊥=eznz=ezcosθ.(3) For convenience, we have included the term ξ0∂tninto the definition of TGD. This term, being even under time reversal, leads to a renormalization of spin in the LLG2 equation16and does not contribute to damping. In what follows, we refer to such renormalization as effective spin renormalization (ESR). The rotational and orientational anisotropy arising in Eqs. (2) appear to be a natural consequence of the fact that the Rashba spin-orbit interaction singles out the di- rection perpendicular to the electron 2D plane. The ori- entationalanisotropyofthedimensionlessfunctions ξi(n) is determined by all space symmetries of the system and, for a general Rashba FM, may turn out to be rather complex. However, for the particular interface model of theC∞vsymmetry class, which we consider below, one simply finds ξi=ξi(n2 z). Before we proceed, let us describe at least two impor- tant outcomes of Eqs. (2). First, according to the usual convention, STT consist of two contributions: the adi- abatic torque ∝(js·∇)nand the nonadiabatic torque ∝n×(js·∇)n, wherejsdenotes a spin-polarized cur- rent. For vanishing SOC, the adiabatictorque has aclear physical meaning. As far as spins of conduction elec- trons adiabatically follow local magnetization direction, the corresponding change of their angular momentum is transferred to the magnetic texture. Since ↑and↓spins point in the opposite directions along n, the transfer rate is proportional to ( js·∇)n, where js=j↑−j↓. In the presence of SOC, however, conduction spins are no longer aligned with the direction of nand, thus, the en- tire concept of spin-polarized current becomes somewhat vague. For the particular Rashba model, our results re- veal an important relation between the adiabatic torque and ESR, providing steps toward better understanding of the former for systems with SOC. Another remarkable property of Eqs. (2) is a simple and exact relation between the nonadiabatic torque and GD, which has an important implication for current- induced motion of magnetic textures (e.g., domain walls or skyrmions). Indeed, by transforming Eq. (1) into the moving reference frame31r′=r−vdt, one immedi- ately observes that both components of the nonadiabatic torqueareexactlycancelledbythe correspondingGilbert damping terms. Therefore, if the effect of other driving torques on the motion of a magnetic texture is negligible, then its terminal velocity, in the moving reference frame, shall vanish for mediate currents32,33(in the absence of magnetic field). This implies that, in the laboratory ref- erence frame, the texture moves with the universal elec- tron drift velocity vd. Certainly, in the presence of, e.g., spin-orbit torques, which can assist motion of domain walls and skyrmions10,34, the resulting dynamics might differ. In any case, the analysisofsuch dynamics can still be performed in the moving reference frame, where the effect of the nonadiabatic spin-transfer torque is conve- niently absent. Having outlined our main results, we skip further dis- cussion until Sec VII. The rest of the paper is organized as follows. In Sec. I we introduce the model and use an expansion in spatial gradients to reduce the analysis to a study of a homogeneous system. Self-energy andKubo formulas are addressed in Sec. II. A general re- lation between STT, GD, and ESR (in the considered model) is obtained in Sec. III, while in Sec. IV we estab- lish the exact vector structures of these quantities. Some analytical insight into our general results is provided in Sec. V and Sec. VI. An extensive Discussion of Sec VII is followed by Conclusions (and seven Appendices). I. MODEL A. Generalized torque in s-dmodel In whatfollows, weadoptthe ideologyofthe s-dmodel by performing a decomposition of a FM into a system of localized spins Siand a system of noninteracting con- duction electrons. Despite being rather simplistic, this approach has proven to describe very well the key prop- erties of current-induced magnetization dynamics in fer- romagnetic systems35–38. If the value of |Si|=Scan be assumed sufficiently large, then it is natural to treat the localized spins clas- sically by means of the unit vector n(ri) =Si/S, which points in the opposite to local magnetization direction. In this case, the s-d-like local exchange interaction be- tween the localized spins and conduction electrons is given, in the continuum limit, by Hsd=JsdSn(r,t)·σ, (4) withJsdquantifying the strength of the exchange and Pauli matrices σrepresenting the spins of conduction electrons. It is known16that interaction of the form of Eq. (4), leads to the following LLG equation for the dynamics of the vector n: ∂tn=γn×Heff+JsdA /planckover2pi1[s(r,t)×n(r,t)],(5) whereγisthebaregyromagneticratio, Heffdescribesthe effective magnetic field, Adenotes the areaof the magnet unit cell, and s(r,t) stands for the nonequilibrium spin density of conduction electrons39. The second term on the right hand side of Eq. (5) represents the generalized torque on magnetization T=JsdA /planckover2pi1[s(r,t)×n(r,t)]. (6) Assuming slow dynamics of n(r,t) on the scale of elec- tron scattering time and smoothness of magnetization profile on the scale of electron mean free path, one may expand the generalized torque in time and space gradi- ents ofn. In this paper, we consider twoparticularterms of such expansion, T=TSTT+TGD+..., (7) ignoring all other contributions (such as, e.g., spin-orbit torques). In Eq. (7) and below, we identify spin-transfer3 torquesTSTTas a double response of Tto the electric fieldEandtothespatialgradientsof n, whiletheGilbert damping vector TGD(which also includes the ESR term) is defined as a response to the time derivative of n, TSTT α=/summationdisplay βγδTSTT αβγδEβ∇γnδ, (8a) TGD α=/summationdisplay δTGD αδ∂tnδ. (8b) Microscopic analysis of the tensors TSTTandTGDis the main subject of the present work. B. Single particle problem According to Eqs. (8), the vectors TSTTandTGDrep- resent linear response to the time derivative of magne- tization direction and to the time derivative of vector potential, respectively. Hence, computation of both vec- tors can be performed with the help of Kubo formulas that make use of Green’s functions of the correspond- ing time-independent problem. We choose the latter to originate in the 2D Rashba model40with the effective s-d-type term of Eq. (4), H=p2/2m+αR[p×σ]z+JsdSn(r)·σ,(9) whereαRcharacterizes the strength of Rashba coupling andmis the effective electron mass. The Hamiltonian of Eq. (9) should be supplemented with a momentum relaxation mechanism since both STT andGDtensors,similarlytotheconductivitytensor,con- tain essentially dissipative components. We assume that momentum relaxation in the system is provided by scat- tering on a spin-independent Gaussian white-noise dis- order potential Vdis(r). Thus, the full Hamiltonian of a single conduction electron reads Hdis=H+Vdis(r), (10) where the disorder potential is characterized by the zero average∝an}b∇acketle{tVdis(r)∝an}b∇acket∇i}ht= 0 and the pair correlator ∝an}b∇acketle{tVdis(r)Vdis(r′)∝an}b∇acket∇i}ht= (/planckover2pi12/mτ)δ(r−r′).(11) The angular brackets in Eq. (11) stand for the averaging over the disorder realizations, τis the mean scattering time measured in the inverse energy units. One can readily observe from Eq. (6) that the general- ized torque Tcan be understood as a spatial density of a disorder-averagedmeanvalueoftheoperator( JsdA//planckover2pi1)ˆT, where we refer to ˆT=σ×n(r), (12) as the dimensionless torque operator.C. Expansion in spatial gradients Computation of STT involves the expansion of the Hamiltonian Hof Eq. (9) and the corresponding Green’s function GR,A= (ε−H±i0)−1(13) in the first spatial gradients of nup to the linear terms. We obtain the latter utilizing the Taylor expansion n(r) =n(r∗)+/summationdisplay γ(r−r∗)γ∇γn(r∗),(14) at some particular point r∗. With the help of Eq. (14), Hcan be, then, approxi- mated as H=H+JsdS/summationdisplay γ(r−r∗)γ∇γn(r∗)·σ,(15) where the Hamiltonian H=p2/2m+αR[p×σ]z+JsdSn(r∗)·σ(16) describes the homogeneouselectronic system with a fixed direction of magnetization set by n(r∗). Similarly, we approximate the Green’s function GR,A, employing the Dyson series GR,A(r,r′) =GR,A(r−r′)+JsdS/integraldisplay d2r′′GR,A(r−r′′) ×/bracketleftig/summationdisplay γ(r′′−r∗)γ∇γn(r∗)·σ/bracketrightig GR,A(r′′−r′) (17) and the Green’s function GR,A= (ε−H±i0)−1(18) that corresponds to the homogeneous system. Note that, in Eq. (17), we kept only the terms that are linear in the gradients of n, as prescribed. D. Spectrum of the homogeneous system The spectrum of Hincorporates two spectral branches ε±(p) =p2/2m±/radicalig ∆2 sd+(αRp)2−2ςαR∆sdpsinθsinϕ, (19) where the angle θstands for the polar angle of nwith respecttothe zaxis[seealsoEq.(3)], while ϕisthe angle between the momentum pand the in-plane component of the vector n:ϕ=φp−φn. We have also introduced the notations ∆sd=|Jsd|S, ς = signJsd, (20) where ∆ sdhas a meaning of half of the exchange interaction-induced splitting (in the absence of SOC).4 FIG. 1. Guide for an eye: spectrum of the homogeneous sys- tem of conduction electrons with a fixed direction of magne- tization. Note that the actual spectrum is not isotropic, an d the two subbands may even touch each other. We restrict the analysis to the case of ε >∆sd. For the latter, both subbands are always partly filled. If the chemical potential εexceeds the value of ∆ sd, both subbands are always partly filled41. Below, we fo- cus solely on the latter case, which is schematically illus- trated in Fig. 1. Note that the spectrum is not isotropic. Moreover, for finite values of sin θ, separation of the two subbands diminishes and they may even touch each other. In what follows, we also find it convenient to intro- duce the energy scale ∆ so=|αR|√ 2mε, which is equal to half of the spin-orbit coupling-induced splitting of the branches (for vanishing ∆ sd). E. Roots of dispersion relation Now let us analyze the roots of the dispersion of Eq. (19). Using, for example, Ref. [42], one can show that, under the assumption ε >∆sd, the quartic func- tion (ε+(p)−ε)(ε−(p)−ε) of the absolute value of mo- mentum palways has four real roots: two positive and twonegative. The former twodefine the angle-dependent Fermi momenta p±corresponding to ε±branches. The four roots are distinct in all cases, except one. Namely, whenn⊥= 0 (i.e., when sin θ= 1) and ∆ so= ∆sd, the subbands touch each other. We will not consider this particular case. Using the notation p±,negfor the negative roots, we have p−> p+>0> p+,neg> p−,neg, (21) where p∓=1 2/parenleftbigg√ 2u±/radicalig −2u−2q−r/radicalbig 2/u/parenrightbigg ,(22a) p±,neg=1 2/parenleftbigg −√ 2u±/radicalig −2u−2q+r/radicalbig 2/u/parenrightbigg ,(22b) u >0 is the largest root of the resolvent cubic u3+qu2−(s−q2/4)u−r2/8, (23)while the parameters q,s, andrare given by q=−4m(ε+mα2 R), s= (2m)2(ε2−∆2 sd),(24a) r= 8m2αRς∆sdsinθsinϕ. (24b) It is straightforward to see, from Eqs. (24), that the dependence on the momentum angle enters Eq. (23) only via the parameter r2. As a result, the quantity umay only depend on sin2ϕand other parameters of the model that areϕindependent. This will play an important role below. ForαR= 0 (vanishing SOC), ∆ sd= 0 (nonmagnetic limit), or n=n⊥(perpendicular-to-the-plane magneti- zation) situation with the rootsbecomes less complex. In these cases, ( ε+(p)−ε)(ε−(p)−ε) is biquadratic (with respect to p) andp±=−p±,neg, as one can also see di- rectly from Eqs. (22). Furthermore, the Fermi momenta p±, then, are angle independent, while their values yield the relations p2 ±= 2m[ε∓∆sd],forαR= 0,(25a) p2 ±= 2m/bracketleftbig ε+mα2 R∓λ(0)/bracketrightbig ,for ∆sd= 0,(25b) p2 ±= 2m/bracketleftbig ε+mα2 R∓λ(∆sd)/bracketrightbig ,forn=n⊥,(25c) whereλ(Υ) =/radicalbig Υ2+2εmα2 R+m2α4 R. II. DISORDER AVERAGING Having analysed the spectrum of the “clean” homoge- neous system, we can proceed with the inclusion of the disorder. In what follows, we assume ε0τ≫1, where ε0is the difference between the Fermi energy εand the closest band edge. We start with a calculation of the self-energy in the first Born approximation. A. Self-energy According to Eq. (11), the self-energy is defined as ΣR,A(r) = (/planckover2pi12/mτ)GR,A(r,r), (26) with the Green’s function GR,Aof Eq. (13). It should be explicitly pronounced that ΣR,A(r) may have a spatial dependenceoriginatinginthespatialdependenceof n(r). However, as we are about to see, the first spatial gradi- ents of magnetization do not affect the self-energy in the model under consideration. Disregarding the “real” part of the self-energy that should be included in the renormalized value of the chemical potential, we focus only on the calculation of ImΣ(r) =−i[ΣR(r)−ΣA(r)]/2. By substituting the expansion of Eq. (17) into Eq. (26), switching to mo- mentum representation, and symmetrizing the result we obtain ImΣ(r) = Σ(0)+/summationdisplay γδ/braceleftig (r−r∗)γΣ(1) δ+Σ(2) γδ/bracerightig ∇γnδ(r∗), (27)5 with Σ(0)=1 2imτ/integraldisplayd2p (2π)2/parenleftbig GR−GA/parenrightbig ,(28a) Σ(1) δ=ς∆sd 2imτ/integraldisplayd2p (2π)2/parenleftig GRσδGR−GAσδGA/parenrightig ,(28b) Σ(2) γδ=ς∆sd/planckover2pi1 4mτ/integraldisplayd2p (2π)2/parenleftig GRσδGRvγGR− GRvγGRσδGR+h.c./parenrightig ,(28c) where “h.c.” denotes Hermitian conjugate, GR,Ais the Green’s function of Eq. (18) in momentum representa- tion, GR,A=ε−p2/2m+αR[p×σ]z+ς∆sdn(r∗)·σ (ε−ε+(p)±i0)(ε−ε−(p)±i0),(29) andv=∂H/∂pis the velocity operator. In Eqs. (28), Σ(0)defines the scattering time (for uniform magneti- zation), Σ(1)corresponds to the renormalization of the gradient term on the right hand side of Eq. (15), while Σ(2)determinesthe possible dependence ofthe scattering time on the first spatial gradients of magnetization. To proceed, we take advantage of the additional sym- metrization of the integrands with respect to the trans- formation43ϕ→π−ϕand observe that, in the first Born approximation, integration over the absolute value of momentum, in Eqs. (28), is reduced to a calculation of residues at p=p±. Using Eqs. (22), we, then, get Σ(0)=−1 2τ/integraldisplay2π 0dϕ 2π/bracketleftbig 1+rW1+rW2n(r∗)·σ +W3n/bardbl(r∗)·σsinϕ/bracketrightbig ,(30) whereWi=Wi/parenleftbig r2,u(r2)/parenrightbig are some functions of the pa- rameterr2andϕ-independent parameters of the model. Sincer∝sinϕand, obviously, all integrals of the form/integraltext2π 0W(sin2ϕ)sinϕdϕvanish for arbitrary function W, we obtain a particularly simple result for the constant part of the self-energy, Σ(0)=−1/2τ. (31) Similar, but more lengthy, analysis shows that each component of Σ(1)and Σ(2)is equal to zero. Therefore, thereexistsnorenormalizationofthegradienttermofthe Hamiltonian Has well as no scattering time dependence on the first magnetization gradients. The self-energy, in the first Born approximation, is found as ΣR,A(r) =∓i/2τ. (32) B. Kubo formula for STT As was outlined in Sec. IB, the generalized torque T(r0) of Eq. (6), at a certain position r0in space, is de- fined as a disorder-averaged mean value of the operatorFIG.2. Diagrammatic representationoftheSTTtensor TSTT αβγδ of Eq. (34). Solid lines correspond to the disorder-average d Green’s functions gR,A. Vertex corrections (impurity ladders) are represented by green fillings. (JsdA//planckover2pi1)δr0ˆT, whereδr0=δ(r−r0). At zero tempera- ture, thelinearresponse44ofTα(r0)tothezerofrequency electric field Eis given by the standard Kubo expression e/planckover2pi1 2πJsdA /planckover2pi1/angbracketleftig Tr/bracketleftig GAδr0ˆTαGRv/bracketrightig E/angbracketrightig ,(33) wherev=∂H/∂pis the velocity operator, Tr stands for the operator trace, and angular brackets represent the disorder averaging. From Eq. (33), we can further deduce the Kubo for- mula for spin-transfer torques. In order to do that, we substitutetheexpansionofEq.(17)intoEq.(33)andcol- lect all terms proportional to ∇γnδ(r∗). Then we switch tomomentum representationandperformspatialaverag- ing of torque on the scale of transport mean free path in the vicinity of r=r0. In the noncrossingapproximation, this leads to the general formula for the STT tensor, TSTT αβγδ=e∆2 sdA 2π/planckover2pi1S/integraldisplayd2p (2π)2 ×itr/bracketleftig gAσδgAvγgAˆTvc αgRvvc β−h.c./bracketrightig ,(34) where the superscript “vc” marks the vertices corrected with the impurity ladders, the notation tr refers to the matrix trace, and gR,A=∝an}b∇acketle{tGR,A∝an}b∇acket∇i}ht= (ε−H±i/2τ)−1(35) is the disorder-averaged Green’s function of the homoge- neous system. In Eq. (35), we have used the result for the self-energy obtained in Sec. IIA. The expression of Eq. (34) is represented diagrammat- ically in Fig. 2. We note that similar diagrams have been used in Ref. [45] to compute STT in a 3D FM, in the absence of SOC, and in Ref. [46] to study STT for the model of massive Dirac fermions. C. Kubo formula for GD and ESR Similarly, from the zero frequency linear response44of Tα(r0) to the time derivative of n, JsdS/planckover2pi1 2πJsdA /planckover2pi1/angbracketleftig Tr/bracketleftig GAδr0ˆTαGRσ/bracketrightig ∂tn/angbracketrightig ,(36)6 onemayderivethe formulaforthe GDtensorofEq.(8b), TGD αδ=∆2 sdA 2π/planckover2pi12S/integraldisplayd2p (2π)2tr/bracketleftig gAˆTvc αgRσδ/bracketrightig ,(37) where, according to the definition of TGD, spatial depen- dence of nis completely disregarded. Note that n,∇γnδ, and∂tnin Eqs. (8), (34), and (37) are all taken at r=r0. From now on, we consistently omit the argument of all these functions. D. Relation between TGDand vertex corrections to the torque operator ˆT Vertex corrected torque operator that enters both Eqs. (34) and (37) can be expressed with the help of vertex corrected Pauli matrices. One can infer the latter from the “matrix of one dressing” M, whose elements Mij=1 2mτ/integraldisplayd2p (2π)2tr/bracketleftig gAσigRσj/bracketrightig (38) arethe coordinates(in the basis {σx,σy,σz}) ofthe oper- atorσidressed with a single impurity line. We note that, in the model considered, vertex corrected Pauli matrices σvc iappear to have zero trace if ε >∆sd. This is a direct consequence of the fact that the self-energy in Eq. (32) is scalar. Hence, {σx,σy,σz}is, indeed, a proper basis for the operators σvc i. Matrixrepresentationofthe operator ˆT=σ×n, with respect to this basis, is defined as ˆTi=/summationdisplay jUijσj, U= 0nz−ny −nz0nx ny−nx0 .(39) Since, obviously, ˆTvc i=/summationdisplay jUijσvc j, (40) we can see, from Eq. (38), that the geometric series T=U(M+M2+···) =UM(I−M)−1,(41) provides the matrix representation of vertex corrections to the torque operator. Moreover, from Eq. (37), it is evident that the GD tensor is, in fact, determined by the same matrix T, TGD αδ=∆2 sdAmτ π/planckover2pi12STαδ. (42) E. Crossing diagrams It has been demonstrated recently that the diagrams with two crossing impurity lines may contribute to such quantitiesasthe anomalousHall effect47–49, the spinHalleffect50, and the Kerr effect51in the same leading or- der with respect to the small parameter ( ε0τ)−1, as the conventionalnoncrossingapproximationdoes. Scattering mechanisms associated with these diagrams, in general, should affect spin torques and damping as well. In the presentstudy we, however,completely disregard the crossing diagrams, as being significantly more diffi- cult to calculate. At the same time, preliminary anal- ysis shows that the related additional contributions to STT, GD, and ESR are parametrically different from the present resultsand that, for ε≫∆sd, they are negligible. III. RELATION BETWEEN STT, GD, AND ESR A. Symmetrization of STT diagrams Calculation of spin-transfer torques can be performed with the help of Eq. (34) directly. Such brute-force cal- culation has been originally performed by us. We have, however, subsequently found a shortcut that makes it possible not only to obtain the same results in a much more concise manner but also to establish a general re- lation between TSTTandTGDtensors. This alternative approach takes a reformulation of the result of Eq. (34) in a more symmetric form. We apply the identity gAvγgA=∂gA/∂pγin Eq. (34) and perform integration by parts. Then, we take a half- sum of the result obtained and the original expression of Eq. (34). This leads to the formula TSTT αβγδ=δTSTT αβγδ+e∆2 sdA 2π/planckover2pi1S/integraldisplayd2p (2π)2 ×i 2tr/bracketleftbigg −gAσδgAˆTvc αgR∂vvc β ∂pγ−h.c./bracketrightbigg ,(43) where the first term on the right-hand side δTSTT αβγδ=e∆2 sdA 2π/planckover2pi1S/integraldisplayd2p (2π)2 i 2tr/bracketleftig gAσδgAvγgAˆTvc αgRvvc β−gAvγgAσδgAˆTvc αgRvvc β −gAσδgAˆTvc αgRvγgRvvc β−h.c./bracketrightig .(44) is illustrated schematically in Fig. 3 by a group of en- circled diagrams. The remaining two diagrams in Fig. 3 correspond to the second term on the right-hand side of Eq.(43). We will seebelowthat, in fact, the entiretensor δTSTTdoes vanish. B. Relation between TSTTand vertex corrections to the torque operator ˆT As was argued in Ref. 52 on the basis of perturbative expansions, the velocity operator v=p/m−αR[ez×σ],7 FIG. 3. Another diagrammatic representation of the STT tens orTSTT αβγδ, given by Eq. (43). Six diagrams encircled by the dashed line define the δTSTT αβγδtensor of Eq. (44) that vanishes for any direction of nprovided ε >∆sd. Solid lines correspond to the disorder-averaged Green’s functions gR,A. Vertex corrections (impurity ladders) are represented by green fillings. correctedbyanimpurityladder,hasaparticularlysimple form in the present model, vvc=p/m. (45) A formal proof of this statement that does not refer to any perturbative expansion is presented in Appendix A. Interestingly, Eq. (45) also allows to make a spin-orbit torque (SOT) calculation extremely concise. We provide a brief discussion of this matter in the same Appendix A. It is important that the momentum operator p, as well asvvc, commutes with the Green’s function gR,A. In Ap- pendix B, we demonstrate that this is sufficient for the entire tensor δTSTTto vanish. As a result, TSTTis de- termined by the second term on the right hand side of Eq. (43) alone. Computation of the this term is facili- tated by the relation ∂vvc β/∂pγ=δβγ/m, (46) whereδq1q2is Kronecker delta. With the help of the above, the STT tensor of Eq. (43) readily simplifies to TSTT αβγδ=δβγe∆2 sdA 2π/planckover2pi1Sm/integraldisplayd2p (2π)2 ×i 2tr/bracketleftig −gAσδgAˆTvc αgR−h.c./bracketrightig ,(47) since, as we have mentioned, δTSTT= 0. Employing the Hilbert’s identity for the Green’s func- tions of Eq. (35), gA−gR=gR(i/τ)gA, (48) we can further reduce44Eq. (47) to the formula TSTT αβγδ=δβγe∆2 sdAτ 2π/planckover2pi1Sm/integraldisplayd2p (2π)2tr/bracketleftig gAˆTvc αgRσδ/bracketrightig ,(49)which resembles very closely the formula of Eq. (37) for the GD tensor. The result of Eq. (49) can also be ex- pressed in terms of the matrix Tas TSTT αβγδ=δβγe∆2 sdAτ2 π/planckover2pi1STαδ, (50) where we have again used the argumentationof Sec. IID. C. Relation between TSTTandTGD It can now be seen that both TSTTandTGDvectors turn out to be fully defined by the matrix of vertex cor- rectionsTto the torque operator. Moreover, comparison of Eq. (42) and Eq. (50) reveals a remarkable direct con- nection between the STT and GD tensors, TSTT αβγδ=δβγe/planckover2pi1τ mTGD αδ, (51) which is one of the central results of the paper. Accordingtothe definitionsofEqs.(8), the established relation between the two tensors indicates that all quan- tities of interest (STT, GD, and ESR) may be related to the action of a single linear operator Ξ, TSTT= Ξ[∂vn],TGD= Ξ[∂tn],(52) on one of the vectors, ∂vnor∂tn. We remind here the short-handed notations for the directional spatial deriva- tive53∂v= (vd·∇) and for the classical drift velocity of conduction electrons vd=eE/planckover2pi1τ/m. The matrix of the operator Ξ coincides with the ma- trixTGD, being also proportional to the matrix T[see Eqs. (8b), and (42)]. In the next section we obtain the general form of the latter and then use it to derive the exact vector forms of TSTTandTGD.8 IV. VECTOR FORMS A. Matrix gauge transformation In order to establish the structure of the operator Ξ, it should be first noted that the constraint n2≡1 is responsibleforanessentialfreedominthedefinition of T. For an arbitrary operator of differentiation ∂, we have 1 2∂n2=/summationdisplay δnδ∂nδ= 0. (53) Therefore, the left hand sides of TSTT α=∆2 sdAmτ π/planckover2pi12S/summationdisplay δTαδ∂vnδ,(54a) TGD α=∆2 sdAmτ π/planckover2pi12S/summationdisplay δTαδ∂tnδ,(54b) remain invariant under the addition of the matrix row R= (nx,ny,nz), with an arbitrary coefficient, to any of therowsofthematrix T. Inotherwords,thetransforma- tionT → T Xdoes not change TSTTandTGD, provided TX=T+XR, (55) with any matrix column X= (X1,X2,X3)T. B. Vector structure of TSTTandTGD The matrix Tis defined in Eq. (41) with the help of the matrix M. The latter is determined by the disorder- averaged Green’s function which, in momentum repre- sentation, takes the form gR,A=ε±i/2τ−p2/2m+αR[p×σ]z+ς∆sdn·σ (ε−ε+(p)±i/2τ)(ε−ε−(p)±i/2τ). (56) Using Eq. (56), one can prove that M, in general, is expressed as a linear combination of six matrices, I, P, U, U2, P UP, P U2P, (57) whereUis introducedin Eq.(39) and P= diag(1 ,1,0)is a diagonal matrix. In Appendix C, we demonstrate how the components of this decomposition can be calculated forn∝ne}ationslash=n⊥. Then, in Appendix D, we show that any power of M retains the same structure. It immediately follows that the matrix T=U(M+M2+···) can be represented as T=c1U+c2UP+c3U2+c4U3+c5UP UP+c6UP U2P, (58) whereciare some dimensionless scalar functions. The representation of Eq. (58) can be substantially simplified with the use of the matrix gauge transforma- tion described in the previous section. Namely, by takingadvantage of the directly verifiable relations U2=RTR−I, U3=−U, (59a) UP UP= (I−P)RTR−n2 zI, (59b) UP U2P=UPRTR−UP+n2 zU(I−P) (59c) we find that the choice of the gauge /tildewideX=−[c3I+c5(I−P)+c6UP]RT,(60) for the transformation T → T /tildewideX≡/tildewideT, leads to /tildewideT=t0I+t/bardblUP+t⊥U(I−P),(61) or, more explicitly, to /tildewideT= t0nzt/bardbl−nyt⊥ −nzt/bardblt0nxt⊥ nyt/bardbl−nxt/bardblt0 , (62) where the quantities tiare related to the matrix Tby means of the relations t0=−c3−c5n2 z, (63a) t/bardbl=c1+c2−(c4+c6), (63b) t⊥=c1−c4+c6n2 z. (63c) Replacing Twith/tildewideTin Eqs. (54), TSTT α=∆2 sdAmτ π/planckover2pi12S/summationdisplay δ/tildewideTαδ∂vnδ,(64a) TGD α=∆2 sdAmτ π/planckover2pi12S/summationdisplay δ/tildewideTαδ∂tnδ,(64b) we observe that the operator Ξ in Eq. (52) is represented by three dimensionless quantities ξ0,ξ/bardbl,ξ⊥, such that ξi=∆2 sdAmτ π/planckover2pi12Sti, (65) while the vector structure of TSTTandTGDis, indeed, provided by the formulas TSTT=ξ0∂vn−ξ/bardbl[n×∂vn/bardbl]−ξ⊥[n×∂vn⊥], TGD=ξ0∂tn−ξ/bardbl[n×∂tn/bardbl]−ξ⊥[n×∂tn⊥], announcedin the introductorypart. With someremarks, they remain valid for n=n⊥as well. We consider this specific case separately, in Sec. VB. In the next section, we derive closed-form results for ξ0,ξ/bardbl, andξ⊥, in two particular regimes. Afterwards, we find asymptotic expansions of these functions in either smallαRor in small ∆ sd. All the obtained results are collected in Table I and represented in Fig. 4 alongside with the corresponding numerical curves.9 V. CLOSED-FORMS The analysis of TSTTandTGDtensors, as has been pointed out, reduces to integration in Eq. (38) and sub- sequent matrix arithmetics. Unfortunately, for arbitrary direction of magnetization, the results cannot be ex- pressed in terms of elementary functions. For example, forn⊥= 0, Eq. (38) already involves elliptic integrals. The complexity is caused, primarily, by the angle de- pendence of the dispersion relation roots p±,p±,negof Eqs. (22). Additional complications arise due to the fact that all four roots are distinct. On the other hand, if the parameter rdefined in Eq. (24b) vanishes, then the angle dependence of p±, p±,negis absent and, furthermore, p±=−p±,neg(see also Sec. IE). In this case, angle integration in Eq. (38) is trivial, while integration over the absolute value pof momentum can be replaced with an integration over p2. For such integrals, we can extend the integration contour to−∞and close it through the upper half-plane. Then the value of the integral is given by a sum of residues at thep2 ±poles of Eqs. (25) that acquire finite imaginary parts due to a ε→ε+i/2τshift. Hence, computation of the matrix Mis straightfor- ward when αR= 0, ∆ sd= 0, orn=n⊥. In this section, we calculate ξ0,ξ/bardbl, andξ⊥, for the first and third cases. In the next section, we use the first two cases as reference points for perturbative analysis of these functions. A. Vanishing spin-orbit coupling Wewillstudythecaseof αR= 0first. Inthe absenceof SOC, conservation of spin brings a technical difficulty to the calculation of T. Namely, at zero frequency and zero momentum, the matrix of disorder-averaged advanced- retarded spin-spin correlators M(I− M)−1that enters Eq. (41) cannot be finite. Indeed, using the formulas of Appendix C with αR= 0, one finds M=I−2ςτ∆sd 1+(2τ∆sd)2U(I−2ςτ∆sdU),(66) so thatI− Mis proportional to U. But det U= 0 and, therefore, M(I− M)−1=∞. Physically, this di- vergence is caused by the absence of linear response of electron spins polarized along nto time-dependent ho- mogeneous perturbations of Jsd(cf. Sec. 8.3 in Ref. 54). Nevertheless, even in the limit of zero momentum and zero frequency, STT, GD, and ESR remain finite, since the series T=UM+UM2+UM3+... (67) actually converges. The sum in Eq. (67) is most easily calculated in the diagonal representation of U, U=VUdiagV†, U diag= diag(i,−i,0),(68)which is defined by the unitary matrix V= iny−nxnz√ 2(n2x+n2y)−iny+nxnz√ 2(n2x+n2y)nx −inx+nynz√ 2(n2x+n2y)inx−nynz√ 2(n2x+n2y)ny√ n2x+n2y√ 2√ n2x+n2y√ 2nz .(69) Introducing MU=V†MVand making use of the rela- tionUdiag=UdiagP, to take care of the potential diver- gence, we can rewrite Eq. (67) as T=VUdiag(PMU+PM2 U+PM3 U+...)V†,(70) where, according to Eqs. (66) and (68), PMk U= diag/parenleftig [1+2iςτ∆sd]−k,[1−2iςτ∆sd]−k,0/parenrightig . (71) Summation in Eq. (70) is trivially performed, leading to T=−ς 2τ∆sdVU2 diagV†=−ς 2τ∆sdU2= ς 2τ∆sd/parenleftbig I−RTR/parenrightbig =/tildewideT −/tildewideXR,(72) where/tildewideT= (ς/2τ∆sd)Irepresents the gauge of Eq. (61) and we have used the first identity of Eq. (59a). The above result clearly corresponds to t0=ς/2τ∆sd andt/bardbl=t⊥= 0, or ξ0=ς∆sdAm 2π/planckover2pi12S, ξ /bardbl=ξ⊥= 0. (73) Hence, Gilbert damping and the nonadiabatic spin- transfertorqueareboth absentwhen αR= 0, asit should be in the model with no SOC, spin-dependent disorder, or other sources of spin relaxation. The parameter ξ0defines the effective spin renormal- ization(duetoconductionelectrons)intheLLGequation as16ξ0=−δSeff/S. In fact, for αR= 0, the effective spin renormalization coincides with actual spin renormaliza- tion. Indeed, without SOC, all electrons are polarized along±n, and, for the calculation of the total electron spin in a unit cell, δS=δS↑−δS↓=ς 2(N+−N−) = ςA 8π2/planckover2pi12 /integraldisplay ε+(p)≤εpdpdφ p−/integraldisplay ε−(p)≤εpdpdφ p ,(74) one may use ε±(p)≤ε⇔p2≤2m(ε∓∆sd) to obtain δS=−ς∆sdAm 2π/planckover2pi12. (75) Thus,δS=−ξ0S=δSeffin this case. In Appendix E, we compute spin susceptibility of the system for αR∝ne}ationslash= 0 and demonstrate that the spin renor- malization does not depend on the SOC strength. At the10 sametime, the effectivespinrenormalizationdoes. More- over, the identity δSeff=δSis, in fact, a very specific case. It holds either for vanishing spin-orbit interaction, oratsomeparticularvalueof∆ so≈∆sd, asonecanlearn from Table I and Fig. 4 (we recall that ∆ so=|αR|√ 2mε characterizes the SOC-induced splitting of the spectral branches). B. Perpendicular-to-the-plane magnetization Now we turn to the n=n⊥regime. The formulas of Appendix C arenot applicable in this case. Nevertheless, one can perform the integration in Eq. (38) directly, uti- lizing the expression for the Green’s function of Eq. (56) with sinθ= 0 (and n=ezcosθ). It follows that M=/bracketleftbig 1+4τ2(∆2 sd+∆2 so)/bracketrightbig−1/parenleftig/bracketleftbig 1+2(τ∆so)2/bracketrightbig P +/bracketleftbig 1+4(τ∆sd)2/bracketrightbig (I−P)−2ςτ∆sdP UP/parenrightig (76) and, after some arithmetic, T=ς 2τ∆sd/bracketleftigg 1−/parenleftbig τ∆2 so/parenrightbig2 ∆2 sd+τ2(2∆2 sd+∆2so)2/bracketrightigg P +1 2/bracketleftigg ∆2 so/bracketleftbig 1+2τ2(2∆2 sd+∆2 so)/bracketrightbig ∆2 sd+τ2(2∆2 sd+∆2so)2/bracketrightigg P UP.(77) Substitution of this result into Eqs. (54) shows that, in this case, both TSTTandTGDare represented as linear combinations of two vector forms: ∂n/bardblandn⊥×∂n/bardbl. Sincen=n⊥and, thus, ∂n⊥= 0, the coefficients in front of these forms should be recognized as t0andt/bardbl, respectively. With the help of Eq. (75), we, therefore, find ξ0=−δS S/bracketleftigg 1−/parenleftbig τ∆2 so/parenrightbig2 ∆2 sd+τ2(2∆2 sd+∆2so)2/bracketrightigg ,(78a) ξ/bardbl=/vextendsingle/vextendsingle/vextendsingleδS S/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg ∆2 so/bracketleftbig 1+2τ2(2∆2 sd+∆2 so)/bracketrightbig ∆2 sd+τ2(2∆2 sd+∆2so)2/bracketrightigg .(78b) For a fixed n=n⊥, however, one cannot directly de- fineξ⊥. Indeed, the latter function, in this case, is a prefactor in front of the vanishing vector form n×∂n⊥ and, in principle, can be even taken arbitrary. The only way to assign a clear meaning to ξ⊥, here, is to consider itsasymptoticbehaviouratsmallvaluesofsin θ. Namely, one should expand the integrands in Eq. (38) up to sin2θ and, after the integration, compute the coefficients of the decomposition of Eq. (58) with the same accuracy. Ap- plication of a sin θ→0 limit in Eq. (63c), afterwards, will lead to ξ⊥=/vextendsingle/vextendsingle/vextendsingleδS S/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg 1 2∆2 so/bracketleftbig 1+(2τ∆sd)2/bracketrightbig ∆2 sd+τ2(2∆2 sd+∆2so)2/bracketrightigg .(79)One may use Eqs. (78b) and (79) to evaluate the strengthoftherotationalanisotropyofGDandthenona- diabatic STT, given n≈n⊥. We see, for example, that, for small sin θ, the ratio ξ/bardbl/ξ⊥= 2+∆2 so ∆2 sd+(1/2τ)2+O(sin2θ),(80) exceeds 2, making the rotational anisotropy considerable even if SOC is weak. At the same time, for strong spin- orbit coupling, ξ/bardblcan potentially be orders of magnitude larger than ξ⊥(see also Fig. 4). For the perpendicular-to-the-plane magnetization, GD was analyzed previously in Ref. [55] under an additional assumption of large chemical potential. Our result for the Gilbert damping coefficient ξ/bardbl, given by Eq. (78b), coincides with the expression on the right hand side of Eq. (25) of Ref. [55], to an overall factor that we were unable to identify (most likely, it is equal to 4). The τ→ ∞limit of the same expression was derived recently in Ref. [56] (with another overall factor). This paper also mentions the role of the diagonal terms of the GD tensor on ESR. A separate study of the nonadiabatic STT (also lim- ited to the n=n⊥case) was reported in Ref. [57]. As we have shown above, this torque should be fully de- termined by the very same function ξ/bardblas is GD. The authors, however, ignored vertex corrections, and, as it seems, overlooked this fact. In any case, their results differ from those of Eq. (78b). VI. ASYMPTOTIC EXPANSIONS We proceed with a calculation of the ξiexpansions in either small αRor small ∆ sd. To perform such cal- culation, one should expand the integrands in Eq. (38) or, alternatively, in Eqs. (C2), with respect to the corre- spondingvariable. Thentheresultcanbeintegratedover the poles, provided by Eqs. (25a) and (25b), respectively (whereεshould be replaced with ε+i/2τ). A. Weak spin-orbit coupling Keeping the notation of Sec. VA for the matrices M andTin the absence of SOC, below we use the symbols δMandδTtorepresenttherespectivecontributionspro- vided by finite αR. SinceδM ∝ne}ationslash= 0, the result of matrix inversion in T+δT=U(M+δM)(I−M−δM)−1(81) is finite, making the analysis straightforward yet rather cumbersome. Retaining only proportionalto α2 Rterms in δM(see Appendix F for explicit formulas), we obtain δT=δc2P+δc3U+δc4U2+..., (82)11 ξ0/(−δS S) orδSeff/δS ξ/bardbl/(|δS S|τ∆sd) ξ⊥/(|δS S|τ∆sd) αR= 0 1 0 0 O(∆2 so)1+2(τ∆so)2 1+(2τ∆sd)21−n2 z 1+n2z(∆so/∆sd)2 1+(2τ∆sd)2/bracketleftbigg (2τ∆sd)2+2 1+n2z/bracketrightbigg(∆so/∆sd)2 1+(2τ∆sd)21+(2nzτ∆sd)2 1+n2z ∆sd→0/parenleftbigg∆sd ∆so/parenrightbigg2/bracketleftbigg 4n2 z+1+n2 z 2(τ∆so)2/bracketrightbigg 2+1 (τ∆so)21 2(τ∆so)2 n=n⊥1−(τ∆2 so)2 ∆2 sd+τ2(2∆2 sd+∆2so)2∆2 so/bracketleftbig 1+2τ2(2∆2 sd+∆2 so)/bracketrightbig ∆2 sd+τ2(2∆2 sd+∆2so)21 2∆2 so/bracketleftbig 1+(2τ∆sd)2/bracketrightbig ∆2 sd+τ2(2∆2 sd+∆2so)2 TABLE I. Closed-form results and asymptotic expansions for the dimensionless functions ξ0,ξ/bardbl, andξ⊥that define anisotropic spin-transfer torques, Gilbert damping, and effective spin renormalization. The results are expressed in terms of the e nergy scales ∆ sd=|Jsd|Sand ∆ so=|αR|√ 2εmthat describe, respectively, the exchange and spin-orbit- induced splitting. The second row shows the expansion up to the second order in ∆ so. The third row provides the leading order terms of the expans ion with respect to small ∆ sd. Spin renormalization is defined in Eq. (75) by δS=−JsdSAm/2π/planckover2pi12. where dots represent terms that do not contribute to the δ/tildewideTgauge in the α2 Rorder and δc2=∆2 so 2∆2 sd1 1+n2z, (83a) δc3=−τ∆2 so ς∆sd/bracketleftig 1+(2τ∆sd)2/bracketrightig1−n2 z 1+n2z,(83b) δc4=−∆2 so 2∆2 sd1+(2nzτ∆sd)2 1+(2τ∆sd)21 1+n2z.(83c) Then, utilizingEqs.(63)with cireplacedby δci,wearrive at the second-orderexpansionsin small SOC strength for the functions ξi. Those are collected in the second row of Table I. We may again use the obtained results to quantify the rotational anisotropy of GD and the nonadiabatic STT by computing the ratio ξ/bardbl/ξ⊥= 2+1−n2 z n2z+1/(2τ∆sd)2+O(∆2 so).(84) For weak spin-orbit coupling, the rotational anisotropy is minimal when magnetization is perpendicular to the plane and increases for the magnetization approaching the in-plane direction. We also note that the asymptotic expansions up to the orderα2 Rallowustoestimatetheorientationalanisotropy ofξi. Employing the notation ξi=ξi(n2 z), we find ξ0(0)−ξ0(1) =2(τ∆so)2 1+(2τ∆sd)2, (85a) ξ/bardbl(0)−ξ/bardbl(1) =1 1+(2τ∆sd)2∆2 so ∆2 sd, (85b) ξ⊥(0)−ξ⊥(1) =1−(2τ∆sd)2 1+(2τ∆sd)2∆2 so 2∆2 sd,(85c) for weak SOC. Clearly, ξ0andξ/bardblare both maximal for n⊥= 0. On the other hand, the expression on the righthand side of Eq. (85c) can change sign, depending on the value of τ∆sd. Therefore, the orientational anisotopy of ξ⊥in a “clean” system ( τ∆sd≫1) differs from that in a “dirty” one (Fig. 4 corresponds to the case of a “clean” system). Interestingly, at αR= 0 the matrix function δTturns out to be discontinuous. Namely, its elements have fi- nite limits for αR→0. This discontinuity has, however, no physical consequences, since the matrix δTitself is not gauge invariant. In the δ/tildewideTgauge, the discontinuity is removed and, thus, it does not affect the physically relevant quantities ξ0,ξ/bardbl, andξ⊥. This property demon- strates the importance of full analysis of all components of the STT and GD tensors. B. Weak exchange interaction Up to the linear order in ∆ sd, we have M=I+2(τ∆so)2P 1+4(τ∆so)2−2ςτ∆sdU+4(τ∆so)2P UP [1+4(τ∆so)2]2. (86) This corresponds to the following coefficients of the de- composition of Eq. (58), c1=1 4(τ∆so)2, c2= 1+1 4(τ∆so)2, (87a) c3=−ςτ∆sd 4(τ∆so)4, c4= 0, (87b) c5=−ςτ∆sd/bracketleftbig 1+8(τ∆so)2/bracketrightbig 4(τ∆so)4, c6= 0.(87c) Substituting the latter expressions into Eqs. (63), one obtains the leading-order contributions to ξiin the limit of small ∆ sd. The respective results are presented in the third row of Table I. Using them, we can find yet another expression for the ratio ξ/bardbl/ξ⊥= 2+(2τ∆so)2+O(∆2 sd).(88)12 FIG. 4. Dimensionless functions ξ0,ξ/bardbl, andξ⊥that define anisotropic spin-transfer torques, Gilbert dam ping, and effective spin renormalization as functions of the spin-orbit coupling st rengthαRfor four different polar angles of magnetization ( nz= cosθ). The notations coincide with those of Table I. We use the dimen sionless combinations ετ= 50,τ∆sd= 10. Since for θ= 0 it is impossible to compute ξ⊥numerically, only analytical result is shown. The O/parenleftbig 1/∆4 so/parenrightbig expansion is addressed in Appendix G. Remarkably, the rotational anisotropy of GD and the nonadiabatic STT, ξ/bardbl/ξ⊥= 2, persists to both limits ∆sd≪∆so≪1/τand ∆ so≪∆sd≪1/τ,(89) in which the Fermi surfaces defined in Eq. (19) are not only essentially isotropic but, at the same time, do get strongly broadened by the disorder (the broadening 1 /τexceeds the splitting of the subbands). It is also interesting to mention that, for small values of ∆sd, the nonadiabatic spin-transfer torque dominates over the adiabatic one: ξ/bardbl,⊥/ξ0∝1/∆sd. This agrees withtheintuitivelogicthat, foraweakexchangebetween conduction and localized spins, the former would rather not adiabatically follow the direction of the latter.13 VII. DISCUSSION A. Role of vertex corrections We would like to begin this final section by stressing that it is the accurate consideration of vertex corrections that is responsible for the established vector structures of anisotropic STT, GD, and ESR, as well as for the relation between them. Practically none of this would be seen from an uncontrolled analysis that ignores vertex corrections. For example, if one does not apply the disorder dress- ing to the current vertex v, the relation of Eq. (50) will no longer be valid. Instead, the STT tensor, in this case, will contain 18 additional nonzero components of differ- ent symmetries, which one might by mistake interpret as physical torques. B. Renormalization of spin In Sec. VA, we have demonstrated that, in the limit of vanishing SOC, the ESR factor δSeff=−ξ0Sdoes coincide with the actual total electron spin in a unit cell δS=−JsdSAm/2π/planckover2pi12. On the other hand, this equality breaksdown forfinite αR, and the ratio δSeff/δSstarts to depend on all of the parameters of the system, including scattering time (see Table I and Fig. 4). Forlargevaluesofspin-orbit-inducedsplitting∆ so, the quantity ξ0(which determines ESR) understandably de- cays due to the effective randomization of the electron spin direction induced by SOC. What is, however, rather interesting, is that, for relatively small values of αR, the ESRfactor δSeffexceedsδS,reachingthemaximumvalue at ∆so≈∆sd. We do not have an intuitive explanation for such behaviour. C. LLG equation It is instructive to compare the microscopic LLG Eq.(1)toitsconventionalphenomenologicalcounterpart. In the absence of spin-orbit, thermal, and other torques that we do not consider in this study, the latter equation reads ∂tn=γn×Heff+(js·∇)n −α[n×∂tn]−β[n×(js·∇)n],(90) where the vector quantity jsis interpreted as the phe- nomenological spin-polarized current, while the param- etersαandβdefine Gilbert damping and the nonadi- abatic spin-transfer torque, respectively. The latter is alsocommonlyreferredtoasthe β-torque. Theadiabatic spin-transfertorqueisrepresentedbytheterm( js·∇)n, whileHeffstands for effective field contributions. First, taking into account Eqs. (2), we can rewrite the microscopic LLG Eq. (1) in a form which is similar tothat of Eq. (90), ∂tn= ¯γn×Heff+(js·∇)n −α/bardbl[n×∂tn/bardbl]−β/bardbl[n×(js·∇)n/bardbl] −α⊥[n×∂tn⊥]−β⊥[n×(js·∇)n⊥],(91) where js=vdξ0 1−ξ0=−vdδSeff S+δSeff,(92a) α/bardbl,⊥=ξ/bardbl,⊥ 1−ξ0, β/bardbl,⊥=ξ/bardbl,⊥ ξ0,¯γ=γ 1−ξ0(92b) and each of the quantities js,α/bardbl,⊥,β/bardbl,⊥, ¯γdepend on the orientation of the vector n. For the particular 2D Rashba FM model system considered in this paper, js=js(n2 z), α /bardbl,⊥=α/bardbl,⊥(n2 z),(93a) β/bardbl,⊥=β/bardbl,⊥(n2 z),¯γ= ¯γ(n2 z). (93b) We see that the microscopic LLG Eq. (91) is essentially anisotropic, in contrast with the phenomenological LLG Eq. (90). Namely, the coefficients αandβgot split into two components each. Moreover, the new coefficients α/bardbl,⊥andβ/bardbl,⊥as well as the other parametersof the LLG equation became dependent on the direction of magneti- zation. We note that the splitting of the GD coefficient α has been reported, for a Rashba FM, in Ref. [58]. Next, let us comment on the microscopic definiton of the spin-polarized current formulated in Eq. (92a). Nor- mally, ifspins ofconductionelectrons(travellingwith the characteristic velocity v) adiabatically follow the direc- tion ofn, one assumes js=−vδS/(S+δS), where δS is a contribution from conduction electrons to the total spin of the system. In this case, Eq. (90) can be simply viewedasamanifestationofthetotalangularmomentum conservation (for n×Heff= 0), (S+δS)∂tn+δS(v·∇)n= 0. (94) where−δS(v·∇)nis the rate of angular momentum transfer from conduction to total spin. The definition of the vector quantity js, given by Eq. (92a), provides a perfect generalization of the above logic for a system with finite Rashba SOC. Indeed, con- duction spins no longer follow the direction of n(due to, e.g., nonzero damping). Nevertheless, −δSeff(vd·∇)n still has a meaning of the rate of “angular momentum transfer” from the effective conduction spin δSeffto the totalS+δSeff. Importantly, it was a fully controllable accurate microscopic treatment of the problem that led us to Eq. (92a). (We identified the drift velocity vd as a “proportionality coefficient” between the STT and GD tensors and observedthat the adiabatic spin-transfer torque and ESR are described by the same quantity ξ0.) Finally, for the sake of historical integrity, let us also mention that the equalities α/bardbl=β/bardblandα⊥=β⊥, in this system, are equivalent59to the relation δSeff=−S/2, (95) which appears to be rather unphysical.14 D. Material derivative and moving reference frame In the presence of the anisotropic STT and GD of Eqs. (2), it is natural to analyse the microscopic LLG Eq. (1) in such a frame, where the effect of the nonadia- baticspin-transfertorqueisabsent. Namely, inthe frame that moves with the classical drift velocity of conduction electrons vd. One may use a nice analogy to continuum mechanics as an illustration of this fact. Indeed, despite the essentially anisotropic character of bothTSTTandTGD, their sum is conveniently expressed in the LLG Eq.(1) viathe operatorofmaterialderivative Dt=∂t+(vd·∇) as (1−ξ0)Dtn=γn×Heff+(vd·∇)n−ξ/bardbl/bracketleftbig n×Dtn/bardbl/bracketrightbig −ξ⊥[n×Dtn⊥]+...,(96) where we have moved the term ξ0Dtnto the left hand side and added ( vd·∇)nto both sides. By considering conduction electrons as a “fluid” flowing with the drift velocity vd, one may interpret the material derivatives of Eq. (96) as the change rates of components of nthat are associated with the electronic “fluid parcels”. Thus, in the moving (“flowing”) frame, r′=r−vdt, the ma- terial derivatives Dtare automatically replaced31by the ordinary time derivatives ∂t. In other words, in the movingreferenceframe, Eq. (96) takes the form of the LLG equation (1−ξ0)∂tn=γn×Heff+(vd·∇)n−ξ/bardbl/bracketleftbig n×∂tn/bardbl/bracketrightbig −ξ⊥[n×∂tn⊥]+...(97) that comprises the analogue of the adiabatic torque (vd·∇)n, two components of damping, and (repre- sentedherebydots)allotherpossibletorques. As longas the latterareabsent, the dynamicsofamagnetictexture, governed by such equation (under mediate currents and in the absence of magnetic field), is likely to be a motion with zero terminal velocity (as it is32,33, in the isotropic case, for domain walls). For a general situation, current- induced magnetic dynamics can differ significantly. Nev- ertheless, it should still be more convenient to perform the analysis once the effect of the nonadiabatic STT has been accounted for by switching to the “flowing” frame. Interestingly, any “propagating” texture of the form n(r,t)=ζ(r−vdt)=ζr(t)nullifiesthesum TSTT+TGD. Hence, for such textures, the LLG Eq. (1) reads dζr/dt=γζr×Heff+..., (98) wherercan be regardedas aparameter. Ifone takesinto accountonlyspin-transfertorquesandfieldlike spin-orbit torque, solutions of this equation will have an oscillatory character. Note that Eq. (98) is different from the LLG equation 0 =γζr×Heff (99) that describes the uniform motion of the ground state in the presence of the Galilean invariance [the case α=β in Eq. (90)]13,15,16,60.E. Response to electric current So far, we have computed spin-transfer torques as a linear response of the system to the external electric field E. In experiment, however, it is not the electric field but rather the electric current jwhich is externally ap- plied. To relate spin torques to the latter, one should compute the conductivity tensor ˆ σand, afterwards, use the identity E= ˆσ−1j (100) to replace Ewithj. Importantly, the conductivity ten- sor has to be computed up to the linear order in first magnetization gradients ∇αnβ. F. Relation to Edelstein effect It is worth noting that some of our results can be inde- pendently benchmarked. As it was suggested in Ref. 36, thereexistsaconnectionbetweensomeparticularpairsof quantities in the model of Eq. (9), as, e.g., between the Dzyaloshinskii-Moriya interaction strength and the ex- change stiffness, or between spin-orbit torques and spin- transfer torques. The latter relation is relevant to our study. A general interpretation of the approach described in Ref. 36 would be the following. Suppose there exists a quantity F(αR) which, for the model with αR= 0, depends on the gradients of n, such that F(0) =F(∇xn,∇yn). (101) Then, up to the linear order with respect to αR, one would obtain61 F(αR) =F(0)+αR/bracketleftbigg∂ ∂αRF(/tildewide∇xn,/tildewide∇yn)/bracketrightbigg αR=0,(102) where /tildewide∇in=∇in+2mαR /planckover2pi1[n×[ez×ei]].(103) Let us now choose three functions Fi(αR) to be the components of the vector TSTT. Using the expression for the quantity ξ0in the limit αR= 0 (see Table I), we can write TSTT=eA 2π/planckover2pi1Jsdτ(E·∇)n. (104) From Eq. (102) we, then, find another contribution to the generalized torque in the ∝αRorder TSOT=2mαR /planckover2pi1eA 2π/planckover2pi1Jsdτ[n×[ez×E]],(105) whichis preciselythe expressionfor the Edelsteineffect62 inaformofafieldliketorqueonmagnetization. Inasimi- larway,vanishingofthefunctions ξ/bardblandξ⊥whenαR= 015 can be translated into the absence52of the antidamping SOT in the model of Eq. (9). TheresultofEq.(105)coincideswith thedirectderiva- tion of SOT, for the model of Eq. (9), that has been re- ported previously52. A more compact and accurate form of this derivation is also presented in Appendix A. Such independent consistency check adds to the credibility of our results. CONCLUSIONS We have presented a thorough microscopic analysis of STT, GD, and ESR, for the particular 2D FM system with Rashba spin-orbit coupling and spin-independent Gaussian white-noise disorder. Assuming arbitrary di- rection of magnetization, we have established the ex- act relation between these effects. We have intro- duced the notion of the matrix gauge transformation for magnetization-dependent phenomena and used it to express spin-transfer torques, Gilbert damping, and ef- fective spin renormalization in terms of meaningful vec- tor forms. The latter allowed us to quantify the SOC- induced anisotropy of the former. We have analysed, both analytically and numerically, three dimensionless functions that fully define anisotropic STT, GD, and ESR. We have also generalized the concept of spin- polarized current, computed spin susceptibility of the system, and obtained a number of other results. It would be an interesting challenge to observe the anisotropy of STT experimentally. It might be possi- ble to do this by measuring current-induced corrections to the magnon spectrum asymmetry that is normally as- sociated with the Dzyaloshinskii-Moriya interaction. We also believe that, to some extent, the anisotropy of STT and GD might explain the differences in dynamics of do- mainwalls(andskyrmions)with differentcharacteristics. ACKNOWLEDGMENTS We would like to thank Jairo Sinova for pointing out a number of flaws in the original version of the manuscript. We are also grateful to Artem Abanov, Arne Brataas, Sergey Brener, Ivan Dmitriev, Rembert Duine, Olena Gomonay, Andrew Kent, Alessandro Principi, Alireza Qaiumzadeh, and Yaroslav Tserkovnyak for helpful dis- cussions. This research was supported by the JTC- FLAGERAProjectGRANSPORTandbytheDutch Sci- ence Foundation NWO/FOM 13PR3118. M.T. acknowl- edges the support from the Russian Science Foundation under Project 17-12-01359.Appendix A: Vertex corrections to velocity operator; spin-orbit torque In order to compute vertex corrections to the velocity operator v=p/m−αR[ez×σ], we first apply a single impurity line to the scalar part of the latter, (p/m)1×dr=1 mτ/integraldisplayd2p (2π)2gR(p/m)gA.(A1) Duetothefactthatthemomentumoperator pcommutes with the Green’s functions gR,A, the above relation can be equivalently written as (p/m)1×dr=i m/integraldisplayd2p (2π)2(p/m)/parenleftbig gR−gA/parenrightbig ,(A2) where we have used the Hilbert’s identity of Eq. (48). The subsequent analysis follows the route of Sec. IIA. Integration over the absolute value of momentum in Eq. (A2) is performed by computing residues at p=p±. Symmetrization of the obtained result, with respect to the transformation43ϕ→π−ϕ, leads to (p/m)1×dr=/integraldisplay2π 0dϕ 2π/parenleftig αR(1+rW4)[ez×σ] +(αR+rW5)/braceleftbig n/bardbl[σ×n]z−/parenleftbig n/bardbl·σ/parenrightbig [ez×n]/bracerightbig cos2ϕ +(W6+W7n·σ)[ez×n]sinϕ/parenrightig ,(A3) whereWi=Wi/parenleftbig r2,u(r2)/parenrightbig are some functions of the pa- rameterr2andϕ-independent parameters of the model. Again, all terms that contain Wivanish identically after integration over the angle and we conclude that (p/m)1×dr=αR[ez×σ]. (A4) Next, we observe that the corrected by an impurity ladder velocity operator vvccan be recast in the form vvc=/braceleftbig p/m−αR[ez×σ]/bracerightbigvc= p/m+/braceleftbig (p/m)1×dr−αR[ez×σ]/bracerightbigvc.(A5) According to Eq. (A4), expression inside the brackets on the second line vanishes, leading us to the desired result, vvc=p/m, (A6) which coincides with Eq. (45) of the main text. Note that, since the momentum operator commutes with the Green’s functions, Eq. (A6) determines both advanced- retardedand retarded-advancedvertexcorrectionsto the velocity operator. One immediate consequence of Eqs. (A4) and (A6) is a trivial form of spin-orbit torque in the considered inter- face Rashba model. Indeed, it was conjectured in Ref. 52 that the antidamping SOT, in this model, is identically16 absent, while the field-like SOT is entirely isotropic. To prove the conjecture, we use the Kubo formula for SOT TSOT=eJsdA 2π/planckover2pi12/integraldisplayd2p (2π)2tr/braceleftig ˆTgR(vvc·E)gA/bracerightig .(A7) Substituting vvc=p/mand using Eq. (A1), we immedi- ately find TSOT=eJsdAmτ 2π/planckover2pi12tr/braceleftig ˆT/parenleftig (p/m)1×dr·E/parenrightig/bracerightig ,(A8) Finally, with the help of Eqs. (12) and (A4), we obtain the expression for spin-orbit torque, TSOT=eJsdAmτα R 2π/planckover2pi12tr/braceleftbig [σ×n]([ez×σ]·E)/bracerightbig = eJsdAmτα R π/planckover2pi12[n×[ez×E]],(A9) which coincides with that of Eq. (105), as expected. Appendix B: Vanishing of δTSTT We will now prove that the absence of the spin compo- nent inthe vertexcorrectedvelocityoperator vvcnullifies the contribution δTSTTto the STT tensor of Eq. (44). Using cyclic permutations under the matrix trace and the fact that vvc=p/mcommutes with any function of momentum, one can rewrite Eq. (44) as δTSTT αβγδ=−e∆2 sdA 2π/planckover2pi1S/integraldisplayd2p (2π)2pβτ 2mtr[Λ1+Λ2] (B1) with Λ1=/parenleftig vγgAˆTvc αgRσδ−σδgAˆTvc αgRvγ/parenrightiggRgA iτ,(B2a) Λ2=/parenleftig σδgAvγgAˆTvc α−vγgAσδgAˆTvc α/parenrightiggRgA iτ −/parenleftig ˆTvc αgRvγgRσδ−ˆTvc αgRσδgRvγ/parenrightiggRgA iτ.(B2b) In Eq. (B2a), we employ the Hilbert’s indentity of Eq. (48) to replace the factor gRgA/iτwithgR−gAand again use cyclic permutations to obtain Λ1=ˆTvc αgRσδgRvγgA−ˆTvc αgRvγgRσδgA −ˆTvc αgRσδgAvγgA+ˆTvc αgRvγgAσδgA.(B3) A similar procedure is performed to simplify the expres- sion for Λ 2. We note, however, that terms with only re- tarded or only advanced Green’s functions, in Eq. (B2b), should be disregarded44. Hence, gRgA/iτis replaced withgRin the first line of Eq. (B2b) and with −gAin the second line. After moving the torque operator to the first place in each term, Λ2=ˆTvc αgRσδgAvγgA−ˆTvc αgRvγgAσδgA +ˆTvc αgRvγgRσδgA−ˆTvc αgRσδgRvγgA,(B4) we conclude that Λ 1+Λ2= 0 and, therefore, δTSTT= 0 as well.Appendix C: Structure of M Using Green’s function of Eq. (56) we compute the matrix trace in Eq. (38) and further symmetrize the in- tegrandswith respect to the transformation43ϕ→π−ϕ. This results in the decomposition M=γ1I+γ2P+γ3U+γ4U2+γ5P UP+γ6P U2P(C1) where the coefficients are given in the integral form, γ1= 2/bracketleftig/parenleftig ∆2 sd+|ε+i/2τ|2/parenrightig I0−2(ε+δso)I1+I2/bracketrightig , (C2a) γ2=−4/bracketleftigg 2δson2 z 1−n2zI1+/parenleftbig 1+n2 z/parenrightbig ς∆sd/radicalbig 1−n2zJ1−1+n2 z 1−n2zJ2/bracketrightigg , (C2b) γ3=−2 τ/bracketleftigg ς∆sdI0−1/radicalbig 1−n2zJ1/bracketrightigg , (C2c) γ4= 4ς∆sd/bracketleftigg ς∆sdI0−1/radicalbig 1−n2zJ1/bracketrightigg ,(C2d) γ5=−2 τ/radicalbig 1−n2zJ1, (C2e) γ6=−4/bracketleftigg 2δso 1−n2zI1+ς∆sd/radicalbig 1−n2zJ1−2 1−n2zJ2/bracketrightigg ,(C2f) withδso=mα2 Rand Ik=/integraldisplayd2p (2π)2(2mτ)−1/parenleftbig p2/2m/parenrightbigk |ε−ε+(p)+i/2τ|2|ε−ε−(p)+i/2τ|2, (C3a) Jk=/integraldisplayd2p (2π)2(2mτ)−1(αRpsinϕ)k |ε−ε+(p)+i/2τ|2|ε−ε−(p)+i/2τ|2. (C3b) SomeofEqs.(C2)formallybecomeinvalidwhen n=n⊥. However,structureof MandTintherespectivecasewas analysed directly in Sec. VB. Appendix D: Structure of Mk We have already demonstrated that M ∈spanL,L={I, P, U, U2, P UP, P U2P},(D1) Let us now prove that any natural power of Mbelongs to the same linear span, Mk∈spanL,∀k∈N. (D2) Theoperationofmatrixproduct, byitself, isnotclosed on spanL. Moreover, 14 of 36 elements of L×Ldo not17 belong to span L. On the other hand, a combination of two such elements (matrices P UandUP), P U+UP={P,U}=U+P UP, (D3) obviously does. Similarly, the remaining 12 “unsuit- able” elements of L × Ldo form 6 pairs, such that the corresponding anticommutators (namely, {P,U2}, {P UP,U},{P U2P,U},{P UP,U2},{P U2P,U2}, and {P UP,P U2P}) belong to span L. In general, the following statement holds: operation of matrix anticommutation sends elements of L × Lto a linear span of L, {,}:L×L → spanL. (D4) Taking into account the fact that anticommutator is a bilinear map, we deduce from Eq. (D4): {,}: spanL×spanL →spanL.(D5) Finally, since for arbitrary kwe have Mk=1 2{M,Mk−1}, (D6) the desired result, Mk∈spanL, is proven by induction. Appendix E: Spin susceptibility in the presence of SOC In this Appendix, the total spin δSof conduction elec- tronsin a unit cell ofthe area Ais computed fora general case ofαR∝ne}ationslash= 0. We use the following standard definition: δS=A 2πi/integraldisplay dǫf(ǫ)/integraldisplayd2p (2π/planckover2pi1)2tr/bracketleftigσ 2/parenleftbig GA−GR/parenrightbig/bracketrightig ,(E1) wherefstands for the Fermi-Dirac distribution, f(ǫ) = (1+exp[( ǫ−ε)/T])−1, (E2) andGA,Rrefers to the momentum-dependent Green’s function of Eq. (29). We will first consider the in-plane component of δS. Matrix trace calculation followed by an integration overǫ, in Eq. (E1), gives δSx=A/integraldisplayd2p (2π/planckover2pi1)2ς∆sdnx−αRpy ε+(p)−ε−(p)(f+−f−),(E3a) δSy=A/integraldisplayd2p (2π/planckover2pi1)2ς∆sdny+αRpx ε+(p)−ε−(p)(f+−f−),(E3b) wheref±=f(ε±(p)). It is convenient to introduce the quantity δS+=δSx+iδSy. For the latter, we find δS+=A 4αR/integraldisplayd2p (2π/planckover2pi1)2(f+−f−) ×/parenleftbigg i∂ ∂px−∂ ∂py/parenrightbigg [ε+(p)−ε−(p)],(E4)where we took advantage of the fact that the fractions in Eqs. (E3) can be expressed as the derivatives with respect to the components of momentum. In the zero- temperaturelimit, onecanuseGreen’stheoremtoreduce the double integrals in Eq. (E4) to the integrals over the closed curves C±={p|ε±(p) =ε}, δS+=δS+ ++δS− +, (E5a) δS± +=±A 4αR/integraldisplay C±dpx+idpy (2π/planckover2pi1)2[ε+(p)−ε−(p)].(E5b) Next, we follow the approach used by K.-W. Kim et al. in Ref. 41. Using the variable w=px+ipyand the relationε±(p) =p2/2m±[ε+(p)−ε−(p)]/2, we find δS± +=A 16π2/planckover2pi12αR/integraldisplay C±dw/parenleftbigg 2ε−w∗w m/parenrightbigg ,(E6) wherew∗w=p2andC±={w|ε±(w,w∗) =ε}are now regarded as contours in the complex w-plane. Since the contours are closed, Eq. (E6) is further simplifed to δS± +=−A 16π2/planckover2pi12mαR/integraldisplay C±dww∗w. (E7) In order to perform integration in Eq. (E7), we solve the equation ε±(w,w∗) =εforw∗and express the result as a function of w∈C±, w∗=2m w2/parenleftig w/bracketleftbig ε+mα2 R/bracketrightbig −imαRς∆sdn+±√ R/parenrightig ,(E8) wheren+=nx+inyandRis a cubic function of w. Different signs in front of the square root in Eq. (E8) correspond to two different functions w∗=w∗ ±(w) of w∈C±, respectively. We do not specify which sign correspondsto which function. Such ambiguity, however, does not affect the final result for δS+. Indeed, it can be proven41that all three zeroes of Rare of the form wk=irkn+with real rk. Then, from the general relation [ε−ε+(w,w∗)][ε−ε−(w,w∗)] =−R +/parenleftbiggw∗w 2m−/bracketleftbig ε+mα2 R/bracketrightbig +imαRς∆sdn+ w/parenrightbigg2 ,(E9) we learn that [ε−ε+(wk,w∗ k)][ε−ε−(wk,w∗ k)]≥0 (E10) and, thus, ε−(wk,w∗ k)< ε⇒ε+(wk,w∗ k)≤ε. Hence, all the singularities of w∗ −that lie inside the contour C− are, in fact, located inside or, at most, on the contour C+(note that C+is inside C−). Disregarding the case63 wk∈C±and using Cauchy integral theorem, we can shrink64C−in Eq. (E7) to obtain δS+=−A 16π2/planckover2pi12mαR/integraldisplay C+dw/parenleftbig w∗ ++w∗ −/parenrightbig w,(E11)18 so that the terms ±√ R, in Eq. (E8), do not contribute toδS+. The only remaining singularity of the integrand is located at the origin and, by the residue theorem, δS+=−ς∆sdAm 2π/planckover2pi12n+orδS/bardbl=−ς∆sdAm 2π/planckover2pi12n/bardbl,(E12) which completes the computation of the in-plane compo- nent ofδS. In order to calculate δSz, it is useful to introduce the “magnetization”vector M=ς∆sdn. Intermsof M,one can straightforwardlyestablish the “thermodynamic” re- lationδSi=∂Ω/∂Mi, where Ω has a meaning of the electronic grand potential in a unit cell, Ω =−TA 2πi/integraldisplay dǫg(ǫ)/integraldisplayd2p (2π/planckover2pi1)2tr/bracketleftbig GA−GR/bracketrightbig ,(E13a) g(ǫ) = log(1+exp[( ε−ǫ)/T]). (E13b) We further note that, according to Eq. (E12), δSxand δSydo not depend on Mz. Therefore, equating the sec- ond derivatives, we find ∂δSz ∂Mα=∂2Ω ∂Mα∂Mz=∂δSα ∂Mz= 0,(E14) whereα=x,y. As a result, δSzdoes not depend on Mx andMyand, thus, can be computed for Mx=My= 0 (or, equivalently, for nx=ny= 0). From Eq. (E1) we obtain δSz=A/integraldisplayd2p (2π/planckover2pi1)2ς∆sdnz ε+(p)−ε−(p)(f+−f−),(E15)which, for nx=ny= 0, can be integrated over the mo- mentum angle with the result δSz=Aς∆sdnz 4π/planckover2pi12∞/integraldisplay 0pdpf+−f−/radicalig ∆2 sd+(αRp)2.(E16) Atzerotemperature, theintegrationdomaininEq.(E16) is reduced to a finite interval p+< p < p −, wherep±are given by Eq. (25c). After some algebraic practice, we finally arrive at δSz=Aς∆sdnz 4π/planckover2pi12α2 R/radicalig ∆2 sd+(αRp)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglep− p+=−ς∆sdAm 2π/planckover2pi12nz. (E17) Combining the results of Eqs. (E12) and (E17) into a single vector form δS=−ς∆sdAm 2π/planckover2pi12n, (E18) weseethat, onaverage,evenforfinite valuesofspin-orbit coupling strength αR, spins of conduction electrons, in theequilibrium, arealignedwiththe localmagnetization. Moreover, the spin susceptibility tensor is fully isotropic and is expressed by a single scalar parameter δS=−|δS|=−ς∆sdAm 2π/planckover2pi12, (E19) which coincides with that given by Eq. (75) of the main text. Appendix F: Expansion of Mup toα2 R Expansion of Eqs. (C2) up to α2 R= ∆2 so/2εmprovides us with the coefficients δγ1=−/bracketleftbigg2τ∆so 1+(2τ∆sd)2/bracketrightbigg2/bracketleftbig 1+(2nzτ∆sd)2/bracketrightbig , δγ 2= 2/bracketleftbiggτ∆so 1+(2τ∆sd)2/bracketrightbigg2/bracketleftbig 1−(1+2n2 z)(2τ∆sd)2/bracketrightbig ,(F1a) δγ3=/bracketleftbigg4τ∆so 1+(2τ∆sd)2/bracketrightbigg21+(2nzτ∆sd)2 1+(2τ∆sd)2ςτ∆sd, δγ 4=−2/bracketleftbigg4τ2∆so∆sd 1+(2τ∆sd)2/bracketrightbigg21+(2nzτ∆sd)2 1+(2τ∆sd)2, (F1b) δγ5=−2/bracketleftbigg2τ∆so 1+(2τ∆sd)2/bracketrightbigg2 ςτ∆sd, δγ 6=−/bracketleftbigg4τ2∆so∆sd 1+(2τ∆sd)2/bracketrightbigg2 (F1c) of the decomposition that we refer to in Sec. VIA: δM=δγ1I+δγ2P+δγ3U+δγ4U2+δγ5P UP+δγ6P U2P. Appendix G: O(1/∆4 so)expansion of ξi(limit of strong SOC) The quantities ξiare shown in the plots of Fig. 4 as functions of the spin-orbit coupling strength αR(while keeping bothmandεconstant). Therefore, the right “tails” of the curves can be pro perly fit using the asymptotic expansion with respect to the parameter 1 /∆so. Such expansion can be obtained indirectly, from the expansion in sm all ∆sd. Below, for consistency with the results of Sec. VIB, we list all the co ntributions to ξithat do not exceed the fourth19 order in 1 /∆so, ξ0=−δS S/bracketleftigg/parenleftbigg∆sd ∆so/parenrightbigg2/bracketleftigg 4n2 z+1+n2 z 2(τ∆so)2/bracketrightigg +6/parenleftbigg∆sd ∆so/parenrightbigg4/bracketleftbig 1−3n2 z/bracketrightbig n2 z/bracketrightigg , (G1a) ξ/bardbl=/vextendsingle/vextendsingle/vextendsingleδS S/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg 2+1 (τ∆so)2−/parenleftbigg∆sd ∆so/parenrightbigg2/bracketleftigg 4n2 z−1−7n2 z (τ∆so)2/bracketrightigg −4/parenleftbigg∆sd ∆so/parenrightbigg4/bracketleftbig 1−3n2 z/bracketrightbig n2 z/bracketrightigg , (G1b) ξ⊥=/vextendsingle/vextendsingle/vextendsingleδS S/vextendsingle/vextendsingle/vextendsingleτ∆sd/bracketleftigg 1 2(τ∆so)2+/parenleftbigg∆sd ∆so/parenrightbigg2/bracketleftigg 2n2 z+1−5n2 z 2(τ∆so)2/bracketrightigg +2/parenleftbigg∆sd ∆so/parenrightbigg4/bracketleftbig 1−5n2 z/bracketrightbig n2 z/bracketrightigg . 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Dmitriev, P. M. Ostrovsky, and M. Titov, Phys. Rev. B 96, 235148 (2017). 50M. Milletar` ı and A. Ferreira, Phys. Rev. B 94, 134202 (2016). 51E. J. K¨ onig and A. Levchenko, Phys. Rev. Lett. 118, 027001 (2017). 52I. A. Ado, O. A. Tretiakov, and M. Titov, Phys. Rev. B 95, 094401 (2017). 53In fact,∂vis proportional to a directional derivative, with a prefactor equal to |vd|−1. 54J. Rammer, Quantum Transport Theory (Perseus Books, New York, 1998). 55I. Garate and A. MacDonald, Phys. Rev. B 79, 064404 (2009). 56A. A. Pervishko, M. I. Baglai, O. Eriksson, and D. Yudin, Sci. Rep. 8, 17148 (2018). 57I. Garate, K. Gilmore, M. D.Stiles, andA. H. MacDonald, Phys. Rev. B 79, 104416 (2009). 58K.-W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee, Phys. Rev. Lett. 108, 217202 (2012). 59As it follows from Eq. (92b) and the relation δSeff=−ξ0S. 60S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005). 61According to the definition of Eq. (9), the spin-orbit cou- pling term has an opposite sign as compared to that used in Ref. 36. 62V. M. Edelstein, Solid State Commun. 73, 233 (1990). 63The conditions wk∈C±can only be fulfilled for some particular values of ε. SinceδSis a continuous function ofε, one may just ignore such values. 64See Ref. 41 for important details on branch cuts.

2019-07-03

Spin-transfer torques (STT), Gilbert damping (GD), and effective spin renormalization (ESR) are investigated microscopically in a 2D Rashba ferromagnet with spin-independent Gaussian white-noise disorder. Rashba spin-orbit coupling induced anisotropy of these phenomena is thoroughly analysed. For the case of two partly filled spin subbands, a remarkable relation between the anisotropic STT, GD, and ESR is established. In the absence of magnetic field and other torques on magnetization, this relation corresponds to a current-induced motion of a magnetic texture with the classical drift velocity of conduction electrons. Finally, we compute spin susceptibility of the system and generalize the notion of spin-polarized current.

Anisotropy of spin-transfer torques and Gilbert damping induced by Rashba coupling

1907.02041v3

arXiv:1203.0607v1 [cond-mat.mtrl-sci] 3 Mar 2012Scaling of intrinsic Gilbert damping with spin-orbital cou pling strength P. He1,4, X. Ma2, J. W. Zhang4, H. B. Zhao2,3, G. L¨ upke2, Z. Shi4, and S. M. Zhou1,4 1Surface Physics State Laboratory and Department of Physics , Fudan University, Shanghai 200433, China 2Department of Applied Science, College of William and Mary, Williamsburg, Virginia 23185 3Key Laboratory of Micro and Nano Photonic Structures (Minis try of Education) and Department of Optical Science and Engineering, Fudan Unive rsity, Shanghai 200433, China and 4Shanghai Key Laboratory of Special Artificial Microstructu re Materials and Technology &Physics Department, Tongji University, Shanghai 200092, C hina (Dated: November 6, 2018) We have experimentally and theoretically investigated the dependence of the intrinsic Gilbert damping parameter α0on the spin-orbital coupling strength ξby using L1 0ordered FePd 1−xPtx ternary alloy films with perpendicular magnetic anisotropy . With the time-resolved magneto-optical Kerr effect, α0is found to increase by more than a factor of ten when xvaries from 0 to 1.0. Since changes of other leading parameters are found to be neglecte d, theα0has for the first time been proven to be proportional to ξ2. PACS numbers: 75.78.Jp; 75.50.Vv; 75.70.Tj; 76.50.+g Magnetization dynamics has currently become one of the most popular topic in modern magnetism due to its crucial importance in information storage. Real space trajectory of magnetization processional switching triggered by magnetic field pulses, fs laser pulses, and spin-polarized current1–6, can be well described by the phenomenological Landau-Lifshitz-Gilbert (LLG) equation that incorporates the Gilbert damping term7 which controls the dissipation of magnetic energy towards the thermal bath. The time interval from the non-equilibrium magnetization to the equilibrium state is governed by the Gilbert parameter α. It has very recently been shown that the laser-induced ultrafast demagnetization is also controlled by the α8. The intrinsic Gilbert damping α0has been exten- sively studied in theory9–15, and in general believed to arise from the spin orbital coupling (SOC). In the SOC torque-correlation model proposed by Kambersk´ y, contributions of intraband and interband transitions are thought toplay adominant rolein the α0at lowand high temperatures Tand are predicted to be proportional to ξ3(ξ=the SOC strength) and ξ2, respectively10,14. Up to date, however, no experiments have been reported to demonstrate the quantitative relationship between α0andξalthough many experimental attempts have been made to study the α0in various metallic and alloy films16–23. It is hard to rule out effects other than the SOC because α0is also strongly related to parameters such as the electron scattering time and density of state D(EF) at Fermi surface EF21,23,24which in turn change among various metals and alloys. In order to rigorously address the ξdependence of α0in experiments, it is therefore essential to find magnetic alloys in which the ξcan be solely adjusted while other parameters almost keep fixed. In this Letter, we elucidate the ξdependence of α0by using L1 0FePd1−xPtx(=FePdPt) ternary alloy films. Here, only ξis modulated artificially by the Pt/Pd concentration ratio because heavier atoms are expectedto have a larger ξ27–29and parameters other than ξare theoretically shown to be almost fixed. Experimental results have shown that α0is proportional to ξ2. It is therefore the first time to have given the experimental evidence of the ξ2scaling law. This work will also facili- tate exploration of new magnetic alloys with reasonably large perpendicular magnetic anisotropy (PMA) and low α. L10FePdPt ternary alloy films with 0 ≤x≤1.0 were deposited on single crystal MgO (001) substrates by magnetron sputtering. The FePdPt composite target was formed by putting small Pt and Pd pieces on an Fe target. During deposition, the substrates were kept at 500◦C. After deposition, the samples were annealed in situ at the same temperature for 2 hours. The base pressure of the deposition system was 1 ×10−5Pa and the Ar pressure was 0.35 Pa. Film thickness was deter- mined by X-ray reflectivity (XRR) to be 12 ±1 nm. In order to measure the Gilbert damping parameter α25,26, time-resolved magneto-optical Kerr effect (TRMOKE) measurements were performed in a pump-probe setup using a pulsed Ti:sapphire laser in the wavelength of 400 nm (800 nm) for pump (probe) pulses with a pulse duration of 200 fs and a repetition rate of 250 kHz. An intense pump pulse beam with a fluence of 0.16 mJ/cm2 was normally incident to excite the sample, and the transient Kerr signal was detected by a probe pulse beam which is time-delayed with respect to the pump. The intensity ratio of the pump to probe pulses was set to be about 6:1, and their respective focused spot diameters were 1 mm and 0.7 mm. A variable magnetic fieldHup to 5 T was applied at an angle of 45 degrees with respect to the film normal using a superconducting magnet. TRMOKE measurements were performed at 200 K and other measurements were performed at room temperature. Microstructural analysis was accomplished with the aid of X-ray diffraction (XRD). Figures 1(a)-1(c) show the XRD patterns for L1 0FePdPt films with x= 1,2 /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 /s32/s32 /s40/s97/s41 /s70/s101/s80/s116/s40/s100/s41 /s32/s32 /s32/s79/s85/s84 /s32/s73/s78 /s40/s98/s41 /s70/s101/s80/s100 /s48/s46/s53/s80/s116 /s48/s46/s53 /s32 /s70/s101/s80/s100 /s48/s46/s53/s80/s116 /s48/s46/s53/s40/s101/s41 /s32 /s70/s101/s80/s100 /s48/s46/s55/s53/s80/s116 /s48/s46/s50/s53/s40/s99/s41 /s50 /s32/s40/s100/s101/s103/s114/s101/s101/s41 /s32/s73/s110/s116/s101/s114/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41 /s109/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s115/s41 /s40/s102/s41 /s70/s101/s80/s100 /s48/s46/s55/s53/s80/s116 /s48/s46/s50/s53 /s70/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41 /s32 FIG. 1: XRD patterns(a, b, c), out-of-plane and in-plane hysteresis loops (d,e,f)for L1 0FePd1−xPtxfilms with x= 1 (a,d),x= 0.5 (b,e) and x= 0.25 (c,f). x= 0.5, andx= 0.25, respectively. The films are of the L1 0ordered structure in the presence of (001) superlattice peak. The chemical ordering degree Scan be calculated with the intensity of the (001) and (002) peaks and found to be 0 .7±0.1 for all samples. Since no other diffraction peaks exist except for (001) and (002) ones, all samples are of L1 0single phase with c axis perpendicular to the film plane. Here, c= 3.694˚A. Magnetization hysteresis loops were measured by vibrat- ing sample magnetometer. Figures 1(d)-1(f) display the corresponding out-of-plane and in-plane magnetization hysteresis loops. As shown in Fig.1(d), for x= 1 (L10FePt) the out-of-plane hysteresis loop is almost square-shaped with coercivity HC= 3.8 kOe, indicating the establishment of high PMA. With decreasing x, the HCdecreases. For x= 0.25 in Fig. 1(f), HCapproaches zero and the out-of-plane and in-plane loops almost overlap with each other, indicating a weak PMA. Ap- parently, the PMA increases with increasing x. Similar phenomena have been reported elsewhere28,29. Figure 2(b) displays the typical TRMOKE results for L1 0FePdPt films with x= 0.25 under θH= 45oas shown in Fig.2(a). For the time delay longer than 5.0 ps, damped oscillatory Kerr signals are clearly seen due to the magnetization precession. The precession period becomes short significantly with increasing H. In order to extract the precession frequency, the Kerr signal was fitted by following exponentially damped sine function, a+bexp(−t/t0) +Aexp(−t/τ)sin(2πft+ϕ), where parameters A,τ,fandϕare the amplitude, relaxation time, frequency, and phase of damped magnetization precession, respectively30. Here,a,b, andt0correspond to the background signal owing to the slow recovery process. The experimental data are well fitted by the above equation, as shown in Fig.2(b). Figure 3(a) shows that for all samples studied here, the extracted precession frequency fincreases monotonically as Hincreases. Moreover, fshows an FIG. 2: Schematic illustration of the TRMOKE geometry (a) and TRMOKE results for x= 0.25 under various magnetic fields (b). Here θH= 45◦. Curves are shifted for clarity. The solid lines are fit results. increasing tendency with increasing xat fixed H. For x= 1 (L1 0FePt),fis in a very high frequency range of 180-260 GHz due to the high PMA. Figure 3(b) shows that the relaxation time τdisplays a decreasing trend with increasing H. Moreover, τincreases by two orders of magnitude when Pd atoms are replaced by Pt ones. In particular, we observed the short relaxation time τ= 3 ps for x= 1 (L1 0FePt). When the oscillation period is longer than the relaxation time for low Hthe precession cannot be excited for x= 131. Withα≪1.0, one can obtain the follow- ing dispersion equation, 2 πf=γ√H1H2, where H1=Hcos(θH−θ) +HKcos2θandH2= Hcos(θH−θ)+HKcos2θ, whereHK= 2KU/MS−4πMS with uniaxial anisotropy constant KU. The equilibrium magnetization angle θis calculated from the following equation sin2 θ= (2H/HK)sin(θH−θ), which is derived by taking the minimum of the total free energy. The measured Hdependence of fcan be well fitted, as shown in Fig.3(a). With the measured MSof 1100 emu/cm3, theKUcan be calculated. The gfactor is equal to 2.16 forx= 1, 0.7, and 0.5, and to 2.10 and 2.03 for x= 0.25 and 0.15, respectively. A small fraction of the orbital3 /s48/s56/s48/s49/s54/s48/s50/s52/s48/s51/s50/s48/s52/s48/s48 /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 /s120/s61/s48/s46/s53/s32 /s120/s61/s48/s46/s50/s53/s32/s32 /s120/s61/s48/s46/s49/s53/s102/s32/s40/s71/s72/s122/s41/s40/s97/s41 /s40/s98/s41/s32/s40/s110/s115/s41 /s72/s32/s40/s107/s79/s101/s41 FIG. 3: Uniform magnetization precession frequency f(a) and relaxation time τ(b) as a function of Hfor all samples studied here. Solid lines refer to fit results. /s48/s46/s48/s48/s46/s49/s48/s46/s50/s49/s50/s51/s52 /s32/s77/s101/s97/s115/s117/s114/s101/s100 /s32/s67/s97/s108/s99/s117/s108/s97/s116/s101/s100/s40/s98/s41 /s32/s32/s40/s97/s41/s75 /s85 /s32/s40/s49/s48/s55 /s32/s101/s114/s103/s47/s99/s109/s51 /s41 /s32 /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 /s40/s99/s41 /s32/s32 /s32/s40/s101/s86/s41 /s120 FIG. 4: Measured KU(a), measured (solid box) and calcu- lated (solid circles) α0(b),ξcalculated in this work (solid circles) and elsewhere38(open ones) (c) as a function of x. The lines serve as a visual guide in (b) and refer to the fit results in (c). angular momentum is therefore restored by the SOC10 and close to results reported elsewhere32. The measured Hdependence of τcan be well fitted byτ= 2/αγ(H1+H2) with the fitted values of g andHKforα≪1.0. Here, the Gilbert damping αis an adjustable parameter. As shown in Fig.3(b), the experimental and fitted data coincide with each other at highHand exhibit significant deviation from each other at low H. It is therefore illustrated that the extrinsic magnetic relaxation contributes to the αat lowHand becomes weak at high H. This is because the extrinsic magnetic relaxation may arise from the inhomogeneous PMA distribution and the interfacial effect and is greatly suppressed under high H33–35. The intrinsic α0therefore plays a dominant role at high H, that is to say, α0is fitted in Fig. 3(b). To determine the SOC strength ξand intrinsicdamping parameter α0in L1 0ordered FePd 1−xPtx ternary alloys, we perform spin dependent first prin- ciples calculations based on linear muffin-tin orbital density functional theorem, where the lattice constants area= 3.86˚Aandc= 3.79˚Afor L1 0ordered FePt. The D(EF) is 2.55, 2.47, 2.43, and 2.39 per atom per eV for xvarying from 0, 0.5, 0.75, to 1.0, respectively. The α0was achieved by using spin-orbital torque-correlation model based on spin dependent electron band structures obtained above9,13. It is significant to compare variations of the PMA andα0. Figures 4(a) & 4(b) show the KUandα0both decrease with decreasing x. Similar variation trends of KUandα0have been observed for perpendicularly mag- netized Pt/Co/Pt multilayers30. When the ξis smaller than the exchange splitting, the magnetic anisotropy is thought to come from the second order energy correction of the SOC in the perturbation treatment and is roughly proportional to both the ξand the orbital angular momentum. The orbital momentum in 3 dmagnetic metallic films restored by the SOC is also proportional to theξand the PMA therefore is proportional to ξ2/W with the bandwidth of 3 delectrons W36. Since the W does not change much with x, the enhanced PMA at highxis attributed to a larger ξof Pt atoms compared with that of Pd atoms27,37. Our calculations show ξ change from 0 .20, 0.26, 0.41 to 0.58 (eV) when xvarying from 0, 0 .5, 0.75, to 1.0, as shown in Fig. 4(c). This is because the ξis 0.6, 0.20, and 0.06 (eV) for Pt, Pd, and Fe atoms, respectively27,38and the effect of Fe atoms is negligible compared with those of Pd and Pt atoms. The present results of ξare in good agreement with previous ab initio calculations38. Apparently, the PMA behavior arises from the increase of ξat highx. As shown in Fig. 4(b), measured and calculated results of α0are in a good agreement. Since the lattice constant, D(EF), the Curie temperature, the gyromagnetic ratio, and the averaged spin are experimentally and theoretically shown to almost not change with x, the enhanced α0is mainly attributed to the ξincrease with increasing x. Figure 5 showsthat the α0is approximatelyproportional toξ2, where the ξvalues at other xare exploited from the fitted curve in Fig. 4(c). Since for the present L10ordered FePd 1−xPtxternary alloy films only ξis tuned with x, the present work has rigorously proven the theoretical prediction about the ξ2scaling of α09. It is indicated that the α0at 200 K is mainly contributed by the interband contribution10,12,14. The electronic- scattering-based model of ferromagnetic relaxation is therefore proved to be applicable for the α0in L10 FePdPt ternary alloys9. In order to further verify the ξ3dependence of α014, measurements of magnetization precession at low temperatures need to be accomplished. In summary, we have investigated the magneti- zation dynamics in L1 0FePdPt ternary alloy films using TRMOKE. The intrinsic α0can be continuously tuned, showing a decrease with increasing Pd content due to smaller ξcompared with that of Pt atoms. In4 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50 /s32/s32/s48/s32 /s32/s40/s101/s86/s50 /s41 FIG. 5: The measured (solid square) and calculated (solid circles)α0versusξ2as a function of x. The dashed curve refers to the linear fit results.particular, the ξ2dependence of α0has been rigorously demonstrated in experiments. The experimental results deepen the understanding the mechanism of α0in magnetic metallic materials and provide a new clue to explore ideal ferromagnets with reasonably low α0and high PMA as storage media for the next generation microwave-assisted magnetic recording. Acknowledgements This work was supported by the MSTC under grant No. 2009CB929201, (US) DOE grant No. DE-FG02-04ER46127, NSFC under Grant Nos. 60908005, 51171129 and 10974032, and Shanghai PuJiang Program (10PJ004). 1Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ ar, and Th. Rasing, Nature (London) 418, 509 (2002). 2H.W. Schumacher, C. 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2012-03-03

We have experimentally and theoretically investigated the dependence of the intrinsic Gilbert damping parameter $\alpha_0$ on the spin-orbital coupling strength $\xi$ by using L1$_{\mathrm{0}}$ ordered FePd$_{\mathrm{1-x}}$Pt$_{\mathrm{x}}$ ternary alloy films with perpendicular magnetic anisotropy. With the time-resolved magneto-optical Kerr effect, $\alpha_0$ is found to increase by more than a factor of ten when $x$ varies from 0 to 1.0. Since changes of other leading parameters are found to be neglected, the $\alpha_0$ has for the first time been proven to be proportional to $\xi^2$.

Scaling of intrinsic Gilbert damping with spin-orbital coupling strength

1203.0607v1

arXiv:1506.05622v2 [cond-mat.str-el] 23 Oct 2015The absence of intraband scattering in a consistent theory o f Gilbert damping in metallic ferromagnets D M Edwards Department of Mathematics, Imperial College London, Londo n SW7 2BZ, United Kingdom Damping of magnetization dynamics in a ferromagnetic metal , arising from spin-orbit coupling, is usually characterised by the Gilbert parameter α. Recent calculations of this quantity, using a formula due to Kambersky, find that it is infinite for a perfec t crystal owing to an intraband scattering term which is of third order in the spin-orbit par ameterξ. This surprising result conflicts with recent work by Costa and Muniz who study damping numeric ally by direct calculation of the dynamical transverse susceptibility in the presence of spin-orbit coupling. We resolve this inconsistency by following the approach of Costa and Muniz f or a slightly simplified model where it is possible to calculate αanalytically. We show that to second order in ξone retrieves the Kambersky result for α, but to higher order one does not obtain any divergent intrab and terms. The present work goes beyond that of Costa and Muniz by pointi ng out the necessity of including the effect of long-range Coulomb interaction in calculating damping for large ξ. A direct derivation of the Kambersky formula is given which shows clearly the res triction of its validity to second order inξso that no intraband scattering terms appear. This restrict ion has an important effect on the damping over a substantial range of impurity content and tem perature. The experimental situation is discussed. I. INTRODUCTION Magnetization dynamics in a ferromagnetic metal is central to the fi eld of spintronics with its many applications. Damping is an essential feature of magnetization dynamics and is usu ally treated phenomenologically by means of a Gilbert term in the Landau-Lifshitz-Gilbert equation [1, 2]. For a system with spin-rotational invariance the uniform precession mode of the magnetization in a uniform external magnetic field is undamped and the fundamental origin of damping in ferromagnetic resonance is spin-orbit coupling (S OC). Early investigations of the effect include those of Kambersky [3–5] and Korenman and Prange [6]. Kambersky ’s [4] torque-correlation formula for the Gilbert damping parameter αhas been used by several groups [7–14], some of whom have given alt ernative derivations. However the restricted validity of this formula, as discussed below, has not been stressed. In this torque-correlation model contributions to αof both intraband and interband electronic transitions are usually c onsidered. The theory is basically developed for a pure metal with the effect of defects and /or phonons introduced as phenomenological broadening of the one-electron states. Assuming that the electr on scattering-rate increases with temperature T due to electron-phonon scattering the intraband and interband tran sitions are found to play a dominant role in low and high T regimes, respectively. The intraband(interband) term is pre dicted to decrease(increase) with increasing T and to be proportional to ξ3(ξ2) whereξis the SOC parameter. Accordingly αis expected to achieve a minimum at an intermediate T. This is seen experimentally in Ni and hcp Co [15] but not in Fe [15] and FePt [16]. The ξ2dependence ofαis well-established at high T [17, 18] but there seems to be no experim ental observation of the predicted ξ3 behaviourat lowT. The interband ξ2term in Kambersky’stheory canbe givenaverysimple interpretation in termsof second-orderperturbation theory [5]. A quite different phenomen ologicalapproach, not applicable in some unspecified low scattering-rate regime, has been adopted to try and find a phy sical interpretation of the intraband term [5, 8]. No acceptable theoretical treatment of damping in this low scatter ing regime is available because the intraband term of Kambersky’s theory diverges to infinity in the zero-scattering lim it of a pure metal with translational symmetry at T=0 [9, 11, 13]. We consider it essential to understand the pure met al limit before introducing impurity and phonon scattering in a proper way. Costa and Muniz [19] recently studied damping numerically in this limit by direct calculation of the dynamical spin susceptibility in the presence of SOC within the random phase app roximation (RPA). They determine αfrom the linewidth of the uniform (wave-vector q= 0) spin-wave mode which appears as a resonance in the transvers e susceptibility. One of the main objects of this paper is to establish so me degree of consistency between the work of Kambersky and that of Costa and Muniz. We follow the approach of t he latter authors for a slightly simplified model where it is possible to calculate αanalytically. We show that to second order in ξone retrieves the Kambersky result, but to higher order no intraband terms occur, which removes the p roblem of divergent α. To confirm this point, in Appendix A we derive the Kambersky formula directly in a way that mak es clear its restriction to second order in ξto which order the divergent terms in αarising from intraband transitions do not appear. This throws open the interpretation of the minimum observed in the temperature depend ence ofαfor Ni and Co. At this point we may mention an alternative theoretical approach to the calculation of Gilbert damping using2 scattering theory [20, 21]. Starikov et al [21] find that, for Ni 1−xFexalloys at T=0, αbecomes large near the pure metal limits x=0,1. They attribute this to the Kambersky intraband c ontribution although no formal correspondence is made between the two approaches. The work of Costa and Muniz [19] follows an earlier paper [22] where it is shown that SOC has the effect of coupling the transverse spin susceptibility to the longitudinal spin su sceptibility and the charge response. It is known that a proper calculation of these last two quantities in a ferromagn et must take account of long-range Coulomb interactions [23–27]. The essential role of these interactions is to e nsure conservation of particle number. Costa et al [19, 22] do not consider such interactions but we show here that this neglect is not serious for calculating αwith sufficiently small SOC. Howeverin the wider frameworkof this paper, where mixed charge-spinresponse is also readily studied, long-rangeinteractions are expected to sometimes play a role. They also come into play, even to second order inξ, when inversion symmetry is broken. In section II we establish the structure of spin-density response theory in the presence of SOC by means of exact spin-density functional theory in the static limit [28]. In section III w e introduce a spatial Fourier transform and an approximation to the dynamical response is obtained by introducing the frequency dependence of the non-interacting susceptibilities. The theory then has the same structure as in the R PA. Section IV is devoted to obtaining an explicit expression for the transverse susceptibility in terms of the non-in teracting susceptibilities. Expressions for these, in the presence of SOC, are obtained within the tight-binding approxim ation in section V. In section VI we consider the damping of the resonance in the q=0 transverse susceptibility a nd show how the present approach leads to the Kambersky formula for the Gilbert damping parameter αwhere this is valid, namely to second order in the SOC parameter ξ. We do not give an explicit formula for αbeyond this order but it is clear that no intraband terms appear. In section VII some experimental aspects are discussed with sugg estions for future work. The main conclusions are summarized in section VIII. II. SPIN-DENSITY FUNCTIONAL THEORY WITH SPIN-ORBIT COUPLI NG The Kohn-Sham equation takes the form /summationdisplay σ′[−δσσ′(/planckover2pi12/2m)∇2+Veff σσ′(r)+Hso σσ′]φnσ′(r) =ǫnφnσ(r) (1) with the spin index σ=↑,↓corresponding to quantization along the direction of the ground-s tate magnetization in a ferromagnet. This may be written in 2 ×2 matrix form with eigenvectors ( φn↑,φn↓)T. The density matrix is defined in terms of the spin components φnσ(r) of the one-electron orbitals by nσσ′=/summationdisplay nφnσ(r)φnσ′(r)∗θ(µ0−ǫn) (2) whereθ(x) is the unit step function and µ0is the chemical potential. The electron density is given by ρ(r) =/summationdisplay σnσσ(r) =/summationdisplay nσ|φnσ(r)|2θ(µ0−ǫn) (3) and the effective potential in (1) is Veff σσ′(r) =wσσ′(r)+δσσ′/integraldisplay d3r′ρ(r′)v(r−r′)+vxc σσ′(r) (4) wherewσσ′(r) is the external potential due to the crystal lattice and any magn etic fields and v(r) =e2/|r|is the Coulomb potential. The exchange-correlation potential vxc σσ′(r) is defined as δExc/δnσσ′(r), a functional derivative of the exchange-correlation energy Exc. The term Hso σσ′in (1) is the SOC energy. A small external perturbation δwσσ′, for example due to a magnetic field, changes the effective potential toVeff+δVeff, giving rise to new orbitals and hence to a change in density matrix δnσσ′. The equation δnσσ′(r) =−Ω−1/summationdisplay σ1σ′ 1/integraldisplay d3r1χ0 σσ′σ1σ′ 1(r,r1)δVeff σ1σ′ 1(r1), (5) where Ω is the volume of the sample, defines a non-interacting respo nse function χ0and the full response function χ is defined by δnσσ′(r) =−Ω−1/summationdisplay ττ′/integraldisplay d3r′χσσ′ττ′(r,r′)δwττ′(r′). (6)3 From (4) δVeff σ1σ′ 1(r1) =δwσ1σ′ 1(r1)+/summationdisplay σ2σ′ 2/integraldisplay d3r2[v(r1−r2)δσ1σ′ 1δσ2σ′ 2+δvxc σ1σ′ 1(r1) δnσ2σ′ 2(r2)]δnσ2σ′ 2(r2) (7) and we may write δvxc σ1σ′ 1(r1) δnσ2σ′ 2(r2)=δ2Exc δnσ2σ′ 2(r2)δnσ1σ′ 1(r1)=Kσ1σ′ 1σ2σ′ 2(r1,r2). (8) Combining (5) - (8) we find the following integral equation for the spin -density response function χσσ′ττ′(r,r′): χσσ′ττ′(r,r′) =χ0 σσ′ττ′(r,r′)−(Ω)−1/summationdisplay σ1σ′ 1/summationdisplay σ2σ′ 2/integraldisplay d3r1/integraldisplay d3r2χ0 σσ′σ1σ′ 1(r,r1)[v(r1−r2)δσ1σ′ 1δσ2σ′ 2+Kσ1σ′ 1σ2σ′ 2(r1,r2)] χσ2σ′ 2ττ′(r2,r′). (9) This equation is a slight generalisation of that given by Williams and von Ba rth [28]. In the static limit it is formally exact although the exchange-correlation energy Excis of course not known exactly. In the next section we generalise the equation to the dynamical case approximately by introducing th e frequency dependence of the non-interacting response functions χ0, and also take a spatial Fourier transform. In the case where SOC is absent the result is directly compared with results obtained using the RPA. III. DYNAMICAL SUSCEPTIBILITIES IN THE PRESENCE OF SPIN-OR BIT COUPLING AND LONG-RANGE COULOMB INTERACTION In general the response functions χ(r,r′) are not functions of r−r′and a Fourier representation of (9) for a spatially periodic system involves an infinite number of reciprocal latt ice vectors. There are two cases where this complication is avoided. The first is a homogeneous electron gas and t he second is in a tight-binding approximation with a restricted atomic basis. We may then introduce Fourier trans forms of the form χ(r) =/summationtext qχ(q)eiq·ror χ(q) = (Ω)−1/integraltext d3rχ(r)e−iq·rand write (9) as χσσ′ττ′(q,ω) =χ0 σσ′ττ′(q,ω)−/summationdisplay σ1σ′ 1/summationdisplay σ2σ′ 2χ0 σσ′σ1σ′ 1(q,ω)Vσ1σ′ 1σ2σ′ 2(q)χσ2σ′ 2ττ′(q,ω), (10) where we have also introduced the ωdependence of χas indicated at the end of the last section. Here V(q) is an ordinary Fourier transform, without a factor (Ω)−1, so that Vσ1σ′ 1σ2σ′ 2(q) =v(q)δσ1σ′ 1δσ2σ′ 2+Kσ1σ′ 1σ2σ′ 2, (11) wherev(q) = 4πe2/q2is the usual Fourier transform of the Coulomb interaction and the s econd term is independent ofqsinceKis a short-range spatial interaction. In the gas case it is proportio nal to a delta-function δ(r−r′) in the local-density approximation (LDA) [28] and in tight-binding it can be t aken as an on-site interaction. In both cases Kmay be expressed in terms of a parameter Uas Kσ1σ′ 1σ2σ′ 2=−U[δσ1σ′ 1δσ2σ′ 2δσ1σ2+δσ1σ′ 1δσ2σ′ 2δσ′ 1σ2] (12) whereσ=↓,↑forσ=↑,↓. in the tight-binding case this form of Kcorresponds to a simple form of interaction which leads to a rigid exchange splitting of the bands ( [29], [22]). This is only appropriate for transition metals in a model with d bands only, hybridization with s and p bands being neglect ed. We adopt this model in order to obtain transparent analytic results as far as possible. Although not as re alistic as ”first-principles” models of the electronic structure it has been used, even with some quantitative success, in treating the related problem of magnetocrystalline anisotropy in Co/Pd structures as well as pure metals [30]. In (10) t he response functions χare per unit volume in the gas case but, more conveniently, may be taken as per atom in t he tight-binding case with v(q) modified to v(q) = 4πe2/(q2Ωa) where Ω ais the volume per atom.4 To show how equations (10) - (12) are related to RPA we examine two examples in the absence of SOC. First consider the transverse susceptibility χ↓↑↑↓(q,ω) which is more usually denoted by χ−+(q,ω). Equation (10) becomes χ↓↑↑↓=χ0 ↓↑↑↓−χ0 ↓↑↑↓V↑↓↓↑χ↓↑↑↓ (13) and, from (11) and (12), V↑↓↓↑=K↑↓↓↑=−U. Hence χ↓↑↑↓=χ0 ↓↑↑↓(1−Uχ0 ↓↑↑↓)−1(14) which is just the RPA result of Izuyama et al [31] for a single-orbital Hubbard model and of Lowde and Windsor [32] for a five-orbital d-band model. Clearly in the absence of SOC the Co ulomb interaction v(q) plays no part in the transverse susceptibility, as is well-known. A more interesting case is the longitudinal susceptibility denoted by χmm in the work of Kim et al ( [26], [27]) and in [28]. This involves only the respon se functions χσσττwhich we abbreviate toχστ. In fact [28] χmm=χ↑↑+χ↓↓−χ↑↓−χ↓↑. (15) Without SOC χ0 στtakes the form χ0 σδστand (10) becomes χστ=χ0 σδστ−/summationdisplay σ2χ0 σVσσ2χσ2τ (16) withVσσ2=v(q)−Uδσσ2. On solving the 2 ×2 matrix equation (16) for χστ, and using (15), we find the longitudinal susceptibility in the form χmm=χ0 ↑+χ0 ↓−2[U−2v(q)]χ0 ↑χ0 ↓ 1+(χ0 ↑+χ0 ↓)[v(q)−U]+U[U−2v(q)]χ0 ↑χ0 ↓(17) which agrees with the RPA result that Kim et al ( [26], [27])found for a s ingle-orbitalmodel. The Coulomb interaction v(q) is clearly important, particularly for the uniform susceptibility with q =0, where v→ ∞. It plays an essential role in enforcing particle conservation and hence in obtaining the cor rect result of Stoner theory. In view of the correspondence between our approach and the RPA method it see ms likely that when SOC is included our procedure using equations (10) - (12) should be almost equivalent to that of Co sta and Muniz [19] in the case of a model with d-bands only. However our inclusion of the long-range Coulomb inter action will modify the results. IV. AN EXPLICIT EXPRESSION FOR THE TRANSVERSE SUSCEPTIBILI TY In this section we obtain an explicit expression for the transverses usceptibility χ↓↑↑↓in terms of the non-interacting response functions χ0. We consider equation (10) as an equation between 4 ×4 matrices where σσ′=↓↑,↑↓,↑↑,↓↓ labels the rows in that order and ττ′labels columns similarly. The formal solution of (10) is then χ= (1+χ0V)−1χ0. (18) This expression could be used directly as the basis of a numerical inve stigation similar to that of Costa and Muniz. However we wish to show that the present approach leads to a Gilber t damping parameter αin agreement with the Kambersky formula, to second order in the SOC parameter ξwhere Kambersky’s result is valid. This requires some quite considerable analytic development of (18). First we partition each matrix in (18) into four 2 ×2 matrices. Thus from (11) and (12) V=/parenleftbigg V10 0V2/parenrightbigg (19) with V1=/parenleftbigg 0−U −U0/parenrightbigg , V2=/parenleftbigg v−U v v v−U/parenrightbigg . (20) Also χ=/parenleftbigg χ11χ12 χ21χ22/parenrightbigg (21)5 and similarly for χ0. If we write 1+χ0V=/parenleftbigg 1+χ0 11V1χ0 12V2 χ0 21V11+χ0 22V2/parenrightbigg =/parenleftbigg A B C D/parenrightbigg (22) (18) becomes χ=/parenleftbigg S−1−S−1BD−1 −D−1CS−1D−1+D−1CS−1BD−1/parenrightbigg/parenleftbigg χ0 11χ0 12 χ0 21χ0 22/parenrightbigg (23) where S=A+BD−1C. (24) The transverse susceptibility χ↓↑↑↓in which we are interested is the top right-hand element of χ11so this is the quantity we wish to calculate. From (23) χ11=S−1(χ0 11−BD−1χ0 21) (25) and, from (24) and (22), S= 1+χ0 11V1−χ0 12(V−1 2+χ0 22)−1χ0 21V1. (26) The elements of the 2 ×2 matrix S are calculated by straight-forward algebra and S11= 1−Uχ0 ↓↑↑↓+(U/Λ)[(X+χ0 ↓↓↓↓)χ0 ↑↑↑↓χ0 ↓↑↑↑−(Y+χ0 ↑↑↓↓)χ0 ↓↓↑↓χ0 ↓↑↑↑ −(Y+χ0 ↓↓↑↑)χ0 ↑↑↑↓χ0 ↓↑↓↓+(X+χ0 ↑↑↑↑)χ0 ↓↓↑↑χ0 ↓↑↓↓](27) where X= [v−U]/[U(U−2v)], Y=−v/[U(U−2v)] (28) and Λ = (X+χ0 ↑↑↑↑)(X+χ0 ↓↓↓↓)−(Y+χ0 ↑↑↓↓)(Y+χ0 ↓↓↑↑) (29) The other three elements of Sare given in Appendix B. The transverse susceptibility is obtained fro m (25) as χ↓↑↑↓= [S22(χ0 11−BD−1χ0 21)12−S12(χ0 11−BD−1χ0 21)22]/(S11S22−S12S21) (30) and BD−1=χ0 12(V−1 2+χ0 22)−1. (31) A comparison of the fairly complex equation above for the transver se susceptibility with the simple well-known result (14) showsthe extent ofthe new physicsintroducedby SOC.This is due tothe coupling ofthe transversesusceptibility to the longitudinal susceptibility and the charge response, both of which involve the long-range Coulomb interaction. To proceed further it is necessary to specify the non-interacting response functions χ0 σσ′σ1σ′ 1which occur throughout the equations above. V. THE NON-INTERACTING RESPONSE FUNCTIONS In the tight-binding approximation the one-electron basis function s are the Bloch functions |kµσ∝angb∇acket∇ight=N−1/2/summationdisplay jeik·Rj|jµσ∝angb∇acket∇ight (32) wherejandµare the site and orbital indices, respectively, and Nis the number of atoms in the crystal. The Hamiltonian in the Kohn-Sham equation now takes the form Heff=/summationdisplay kµνσ(Tµν(k)+Veff σδµν)c† kµσckνσ+Hso(33)6 whereTµνcorresponds to electron hopping and Veff σ=−(σ/2)(∆+bex) (34) whereσ= 1,−1 for spin ↑,↓respectively. Here ∆ = 2 U∝angb∇acketleftSz∝angb∇acket∇ight/NwhereSzis the total spin angular momentum, in units of /planckover2pi1, and the Zeeman splitting bex= 2µBBex, whereBexis the external magnetic field and µBis the Bohr magneton. The spin-orbit term Hso=ξ/summationtext jLj·Sjtakes the second-quantized form Hso= (ξ/2)/summationdisplay kµν[Lz µν(c† kµ↑ckν↑−c† kµ↓ckν↓)+L+ µνc† kµ↓ckν↑+L− µνc† kµ↑ckν↓] (35) whereLz µν,L± µνare matrix elements of the atomic orbital angular momentum operat orsLz,L±=Lx±iLyin units of/planckover2pi1. Within the basis of states (32) eigenstates of Hefftake the form |kn∝angb∇acket∇ight=/summationdisplay µσaσ nµ(k)|kµσ∝angb∇acket∇ight, (36) and satisfy the equation Heff|kn∝angb∇acket∇ight=Ekn|kn∝angb∇acket∇ight. (37) Thus c† kµσ=/summationdisplay naσ nµ(k)∗c† kn(38) wherec† kncreates the eigenstate |kn∝angb∇acket∇ight. Thenon-interactingresponsefunction χ0 σσ′σ1σ′ 1(q,ω) isconvenientlyexpressedastheFouriertransformofaretarde d Green function by the Kubo formula χ0 σσ′σ1σ′ 1(q,ω) =/summationdisplay k∝angb∇acketleft∝angb∇acketleft/summationdisplay µc† k+qµσckµσ′;/summationdisplay νc† kνσ1ck+qνσ′ 1∝angb∇acket∇ight∝angb∇acket∇ight0 ω (39) where the right-hand side is to be evaluated using the one-electron Hamiltonian Heff. Consequently, using (38), we have χ0 σσ′σ1σ′ 1(q,ω) =/summationdisplay kµν/summationdisplay mnaσ mµ(k+q)∗aσ′ nµ(k)aσ1nν(k)∗aσ′ 1mν(k+q)∝angb∇acketleft∝angb∇acketleftc† k+qmckn;c† knck+qm∝angb∇acket∇ight∝angb∇acket∇ight0 ω =N−1/summationdisplay kµν/summationdisplay mnaσ mµ(k+q)∗aσ′ nµ(k)aσ1 nν(k)∗aσ′ 1mν(k+q)fkn−fk+qm Ek+qm−Ekn−/planckover2pi1ω+iη.(40) The last step uses the well-known form of the response function pe r atom for a non-interacting Fermi system (e.g. [33]) andηis a small positive constant which ultimately tends to zero. The occup ation number fkn=F(Ekn−µ0) where Fis the Fermi function with chemical potential µ0. Clearly for q= 0 the concept of intraband transitions ( m=n), frequently introduced in discussions of the Kambersky formula, ne ver arises for finite ωsince the difference of the Fermi functions in the numerator of (40) is zero. Equation (40) ma y be written in the form χ0 σσ′σ1σ′ 1(q,ω) =N−1/summationdisplay kmnBσσ′ mn(k,q)Bσ1′σ1 mn(k,q)∗fkn−fk+qm Ek+qm−Ekn−/planckover2pi1ω+iη(41) where Bσσ′ mn(k,q) =/summationdisplay µaσ mµ(k+q)∗aσ′ nµ(k). (42)7 VI. FERROMAGNETIC RESONANCE LINEWIDTH; THE KAMBERSKY FORM ULA We now consider the damping of the ferromagnetic resonance in the q= 0 transverse susceptibility. The present approach, like the closely-related one of Costa and Muniz [19], is valid f or arbitrary strength of the SOC and can be used as the basis of numerical calculations, as performed by the latter authors. However it is important to show analytically that the present method leads to the Kambersky [4] for mula for the Gilbert damping parameter where this is valid, namely to second order in the SOC parameter ξ. This is the subject of this section. It is useful to consider first the case without SOC ( ξ= 0). The eigenstates nofHeffthen have a definite spin and may be labelled nσ. It follows from (40) that χ0 σσ′σ1σ′ 1∝δσσ′ 1δσ′σ1. Henceχ0 12= 0 and, from (31), BD−1= 0. Also, from Appendix B, S12=S21= 0. Thus, (30) reduces to (14) as it should. Considering χ0 ↓↑↑↓(0,ω), given by (40) and (41), we note that state mis pure↓spin, labelled by m↓, andnis pure↑, labelled by n↑. Hence for ξ= 0 B↓↑ mn(k,0) =/summationdisplay µ∝angb∇acketleftkm|kµ∝angb∇acket∇ight∝angb∇acketleftkµ|kn∝angb∇acket∇ight=δmn (43) from closure. Thus χ0 ↓↑↑↓(0,ω) =N−1/summationdisplay knfkn↑−fkn↓ Ekn↓−Ekn↑−/planckover2pi1ω+iη(44) and it follows from (34) that Eknσmay be written as Eknσ=Ekn−(σ/2)(∆+bex). (45) Hence we find from (14) that for ξ= 0 χ↓↑↑↓(0,ω) = (2∝angb∇acketleftSz∝angb∇acket∇ight/N)(bex−/planckover2pi1ω+iη)−1. (46) Thus, as η→0,ℑχ↓↑↑↓(0,ω) has a sharp delta-function resonance at /planckover2pi1ω=bexas expected. When SOC is included /planckover2pi1ωacquires an imaginary part that corresponds to damping. We now pr oceed to calculate this imaginary part to O(ξ2). To do this we can take ξ= 0 in the numerator of (30) so that χ↓↑↑↓(0,ω) =χ0 ↓↑↑↓(0,ω)/(S11−S12S21/S22) (47) In factS12andS21are both O(ξ2) whileS22isO(1). Thus to obtain /planckover2pi1ωtoO(ξ2) we need only solve S11= 0. Furthermore all response functions such as χ0 ↑↑↑↓, with all but one spins in the same direction, are zero for ξ= 0 and need 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 inS11is zero and we only have to solve the equation 1 −Uχ0 ↓↑↑↓= 0 for/planckover2pi1ω. This means that to second order in ξ the shift in resonance frequency and the damping do not depend on the long-range Coulomb interaction. To determine χ0 ↑↑↑↓(0,ω) to first order in ξfrom (40) we notice that states nmust be pure ↑spin, that is |kn∝angb∇acket∇ight= |kn↑∝angb∇acket∇ight, while states mmust be calculated using perturbation theory. The latter states m ay be written |km1∝angb∇acket∇ight=|km↑∝angb∇acket∇ight−ξ/summationdisplay pσ∝angb∇acketleftkpσ|hso|km↑∝angb∇acket∇ight Ekm↑−Ekpσ|kpσ∝angb∇acket∇ight (48) |km2∝angb∇acket∇ight=|km↓∝angb∇acket∇ight−ξ/summationdisplay pσ∝angb∇acketleftkpσ|hso|km↓∝angb∇acket∇ight Ekm↓−Ekpσ|kpσ∝angb∇acket∇ight, (49) where we have put Hso=ξhso, and to first order in ξ, χ0 ↑↑↑↓=1 N/summationdisplay kµν/summationdisplay mn(a↑ m1µ∗anµa∗ nνa↓ m1νfkn↑−fkm↑ Ekm↑−Ekn↑−/planckover2pi1ω+iη+a↑ m2µ∗anµa∗ nνa↓ m2νfkn↑−fkm↓ Ekm↓−Ekn↑−/planckover2pi1ω+iη) (50) withaσ msµ=∝angb∇acketleftkµσ|kms∝angb∇acket∇ight,s= 1,2,andanµ=∝angb∇acketleftkµ|kn∝angb∇acket∇ightis independent of spin. Since a↓ m1ν∼ξwe takea↑ m1µ=amµin the first term of (50). Also/summationtext µa∗ mµanµ=δmnby closure so that the first term of (50) vanishes since the differen ce of Fermi functions is zero. Only the second term of χ0 ↑↑↑↓remains and this becomes, by use of (49), χ0 ↑↑↑↓=−ξ/summationdisplay kµν/summationdisplay mnp∝angb∇acketleftkp↑ |hso|km↓∝angb∇acket∇ight∗ Ekm↓−Ekp↑a∗ pµanµa∗ nνamνfkn↑−fkm↓ Ekm↓−Ekn↑−/planckover2pi1ω+iη. (51)8 Again using closure only terms with p=m=nsurvive and the matrix element of hsobecomes ∝angb∇acketleftkn↑ |/summationdisplay jLj·Sj|kn↓∝angb∇acket∇ight=1 2∝angb∇acketleftkn|L−|kn∝angb∇acket∇ight= 0 (52) due to the quenching of total orbital angular momentum L=/summationtext jLj[30]. Thus, to first order in ξ,χ0 ↑↑↑↓(0,ω), and similar response functions with one reversed spin, are zero. Hence we have only to solve 1 −Uχ0 ↓↑↑↓= 0 to obtain ℑ(/planckover2pi1ω) toO(ξ2). Here we assume the system has spatial inversion symmetry witho ut which the quenching of orbital angular momentum, as expressed by (52), no longer pertains [30]. We briefly discuss the consequences of a breakdown of inversion symmetry at the end of this section. On introducing the perturbed states (48) and (49) we write (41) in the form χ0 ↓↑↑↓(0,ω) =1 N/summationdisplay kmn(|B↓↑ m1n1|2fkn1−fkm1 Ekm1−Ekn1−/planckover2pi1ω+iη+|B↓↑ m1n2|2fkn2−fkm1 Ekm1−Ekn2−/planckover2pi1ω+iη +|B↓↑ m2n1|2fkn1−fkm2 Ekm2−Ekn1−/planckover2pi1ω+iη+|B↓↑ m2n2|2fkn2−fkm2 Ekm2−Ekn2−/planckover2pi1ω+iη).(53) ClearlyB↓↑ m1n1andB↓↑ m2n2are of order ξ,B↓↑ m1n2isO(ξ2) andB↓↑ m2n1isO(1). We therefore neglect the term |B↓↑ m1n2|2 and, using (48) and (49), we find B↓↑ m1n1=−B↓↑ m2n2=ξ 2∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight Ekn↓−Ekm↑. (54) The evaluation of |B↓↑ m2n1|2requires more care. It appears at first sight that to obtain this to O(ξ2) we need to include second order terms in the perturbed eigenstates given by (48) an d (49). However it turns out that these terms do not in fact contribute to |B↓↑ m2n1|2toO(ξ2) so we shall not consider them further. Then we find B↓↑ m2n1=δmn−ξ∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight Ekn−Ekm−ξ2 4/summationdisplay p∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight∝angb∇acketleftkp|Lz|kn∝angb∇acket∇ight (Ekm−Ekp)(Ekn−Ekp)(55) and hence to O(ξ2) |B↓↑ m2n1|2=δmn(1−ξ2 2/summationdisplay p|∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight|2 (Ekm−Ekp)2)+ξ2|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2 (Ekn−Ekm)2(56) The contribution of this quantity to χ0 ↓↑↑↓(0,ω) in (53) may be written to O(ξ2) as 1 N/summationdisplay kmn|B↓↑ m2n1|2fkn1−fkm2 Ekm2−Ekn1−bex+iη(1−bex−/planckover2pi1ω Ekm2−Ekn1−bex+iη). (57) Thisisobtainedbyintroducingtheidentity −/planckover2pi1ω=−bex+(bex−/planckover2pi1ω)intherelevantdenominatorin(53),andexpanding to first order in bex−/planckover2pi1ωwhich turns out to be O(ξ2). The remaining factors of this second term in (57) may then be 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 combining equations (53), (54) and (57), and ignoring some real te rms, we find that the equation 1 −Uχ0 ↓↑↑↓(0,ω) = 0 leads to the relation ℑ(/planckover2pi1ω) =πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay knm[(fkn↑−fkm↓)|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekm↓−Ekn↑−bex) +(1/4)(fkn↑+fkn↓−fkm↑−fkm↓)|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Ekm−Ekn−bex)](58) The Gilbert damping parameter αis given by ℑ(/planckover2pi1ω)/bex(e.g. [39]) and in (58) we note that (fkn↑−fkm↓)δ(Ekm↓−Ekn↑−bex) = [F(Ekn↑−µ0)−F(Ekn↑+bex−µ0)]δ(Ekm↓−Ekn↑−bex) =bexδ(Ekn↑−µ0)δ(Ekm↑−µ0)(59) to first order in bexat temperature T= 0. Similarly (fknσ−fkmσ)δ(Ekm−Ekn−bex) =bexδ(Eknσ−µ0)δ(Ekmσ−µ0). (60)9 Thus from (58) α=πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay knm|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekn↑−µ0)δ(Ekm↓−µ0) +πξ2/(8∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay knmσ|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Eknσ−µ0)δ(Ekmσ−µ0)(61) correct 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- forward to show that to O(ξ2) this is equivalent to the expression α=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay knm/summationdisplay σσ′|Amσ,nσ′(k)|2δ(Ekmσ−µ0)δ(Eknσ′−µ0) (62) where Amσ,nσ′(k) =ξ∝angb∇acketleftkmσ|[S−,hso]|knσ′∝angb∇acket∇ight (63) andS−is the total spin operator/summationtext jS− jwithS− j=Sx j−iSy j. This may be written more concisely as α=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay knm|Amn(k)|2δ(Ekm−µ0)δ(Ekn−µ0) (64) with Amn(k) =ξ∝angb∇acketleftkm|[S−,hso]|kn∝angb∇acket∇ight (65) and the understanding that the one-electron states km,knare calculated in the absence of SOC. Equation (64) is the standard form of the Kambersky formula ( [4], [9]) but in the literatur e SOC is invariably included in the calculation of the one-electronstates. This means that the intraband terms withm=nno longer vanish. They involve the square of a delta-function and this problem is always addressed by invoking t he effect of impurity and/or phonon scattering to replace the delta-functions by Lorentzians of width proportion al to an inverse relaxation time parameter τ−1. Then as one approaches a perfect crystal ( τ→ ∞) the intraband contribution to αtends to infinity. This behaviour is illustrated in many papers ( [7], [10], [13], [14]). In fig. 1 of [14] it is shown c learly that αremains finite if one does not include SOC in calculating the one-electron states. However the effect of not including SOC is not confined to total removal of the intraband contribution. The remaining interb and contribution is increased considerably in the low scattering rate regime, by almost an order of magnitude in the ca se of Fe. This makes αalmost independent of scattering rate in Fe which may relate to its observed temperature independence [15]. The corresponding effect in Co is insufficient to produce the increase of αat low scattering rate inferred from its temperature dependence . The non-inclusion of SOC in calculating the one-electron states used in th e Kambersky formula clearly makes a major qualitative and quantitative change in the results. This occurs as so on as intraband terms become dominant in calculations where they are included. For Fe, Co and Ni this corresp onds to impurity content and temperature such that the scattering rate 1 /τdue to defects and/or phonons is less that about 1014sec−1( [7, 14]). Typically these metals at room temperature find themselves well into the high scatt ering-rate regime where the damping rate can be reliably estimated from the Kambersky interband term, with or wit hout SOC included in the band structure [35]. The physics at room temperature is not particularly interesting. On e needs to lower the temperature into the low scattering-rate regime where intraband terms, if they exist, will d ominate and lead to an anomalous ξ3dependence of the damping on spin-orbit parameter ξ( [4], [8], [14]). The origin of this behaviour is explained in [4], [14]. It arises in theksum of (64) from a striplike region on the Fermi surface around a line where two different energy bands cross each other in the absence of SOC. The strip width is proportional to ξ, or more precisely |ξ|. SinceAnn(k) is of orderξthe contribution of intraband terms in (64) is proportional to |ξ|3. Thus the intraband terms lead to terms inαwhich diverge in the limit τ−1→0 and are non-analytic functions of ξ. The calculation of αin this section can be extended to higher powers of ξthan the second. No intraband terms appear and the result is an an alytic power series containing only even powers of ξ. The interband term in Kambersky’s formula can be given a very simple in terpretation in terms of Fermi’s ”golden rule” for transition probability [5]. This corresponds to second orde r perturbation theory in the spin-orbit interaction. The decay of a uniform mode ( q= 0) magnon into an electron-hole pair involves the transition of an ele ctron from an occupied state to an unoccupied state of the same wave-vecto r. This is necessarily an interband transition and the states involved in the matrix element are unperturbed, that is c alculated in the absence of SOC. A quite different approach has been adopted to try and find a physical interpretat ion of Kambersky’s intraband term ( [5, 8]). This employs Kambersky’s earlier ”breathing Fermi surface” model ( [3, 34]) whose range of validity is uncertain.10 We now briefly discuss the consequences of a breakdown of spatial inversion symmetry so that total orbital angular momentum is not quenched. In general response functions such a sχ0 ↑↑↑↓(0,ω) with one reversed spin are no longer zero to first order in ξ. HenceS11is not given to order ξ2just by the first two terms of (27) but involves further terms which depend explicitly on the long-range Coulomb interaction. Conse quentlyαhas a similar dependence which does not emerge from the torque-correlation approach. In Appe ndix A it is pointed out how the direct proof of the Kambersky formula breaks down in the absence of spatial inversion symmetry. VII. EXPERIMENTAL ASPECTS The inclusion of intraband terms in the Kambersky formula, despite t heir singular nature, has gained acceptance because they appear to explain a rise in intrinsic damping parameter αat low temperature which is observed in some systems [15]. The calculated intraband contribution to αis proportional to the relaxation time τand it is expected that, due to electron-phonon scattering, τwill increase as the temperature is reduced. This is in qualitative agre ement with data [15] for Ni and hcp Co. Also a small 10% increase in αis observed in Co 2FeAl films as the temperature is decreased from 300 K to 80 K [36]. However in Fe the damping αis found to be independent of temperature down to 4 K [15]. Very recent measurements [16] on FePt films, with varying an tisite disorder xintroduced into the otherwise well-ordered structure, show that αincreases steadily as xincreases from 3 to 16%. Hence αincreases monotonically with scattering rate 1 /τas expected from the Kambersky formula in the absence of intraba nd terms. Furthermore forx= 3% it is found that αremains almost unchanged when the temperature is decreased fro m 200 to 20 K. Ma et al [16] therefore conclude that there is no indication of an intraban d term in α. From the present point of view the origin of the observed low temperature increase of αin Co and Ni is unclear. Further experimental work to confirm the results of Bhagat and Lubitz [15] is desirable. The second unusual feature of the intraband term in Kambersky’s formula for αis its|ξ|3dependence on the SOC parameter ξ. This contrasts with the ξ2dependence of the interband contribution which has been observe d in a number of alloys at room temperature [17]. Recently this behaviour has been seen very precisely in FePd 1−xPtxalloys whereξcan be varied over a wide range by varying x[18]. Unfortunately this work has not been extended to the low temperature regime where the |ξ|3dependence, if it exists, should be seen. It would be particularly inte resting to see low temperature data for NiPd 1−xPtxand CoPd 1−xPtxsince it is in Ni and Co where the intraband contribution has been invoked to explain the low temperature behaviour of α. From the present point of view, with the intraband term absent, one would expect ξ2behaviour over the whole temperature range. VIII. CONCLUSIONS AND OUTLOOK In this paper we analyse two methods which are used in the literature to calculate the damping in magnetization dynamics due to spin-orbit coupling. The first common approach is to employ Kambersky’s[4] formula for the Gilbert damping parameter αwhich delivers an infinite value for a pure metal if used beyond second order in the spin-orbit parameter ξ. The second approach [19] is to calculate numerically the line-width of the ferromagnetic resonance seen in the uniform transverse spin susceptibility. This is always found to b e finite, corresponding to finite α. We resolve this apparent inconsistency between the two methods by an analyt ic treatment of the Costa-Muniz approach for the simplified model of a ferromagnetic metal with d-bands only. It is sho wn that this method leads to the Kambersky result correct to second order in ξbut Kambersky’s intraband scattering term, taking the non-analy tic form |ξ|3, is absent. Higher order terms in the present work are analytic even p owers of ξ. The absence of Kambersky’s intraband term is the main result of this paper and it is in agreement with the conc lusion that Ma et al [16] draw from their experiments on FePt films. Further experimental work on the depe ndence of damping on electron scattering-rate and spin-orbit parameter in other systems is highly desirable. A secondaryconclusionis that beyond second orderin ξsome additional physics ariseswhich has not been remarked on previously. This is the role of long-range Coulomb interaction which is essential for a proper treatment of the longitudinal susceptibility and charge response to which the transv erse susceptibility is coupled by spin-orbit interac- tion. Costa and Muniz [19] stress this coupling but fail to introduce t he long-range Coulomb interaction. Generally, however, it seems unnecessary to go beyond second order in ξ[17, 18] and for most bulk systems Kambersky’s for- mula, with electron states calculated in the absence of SOC, should b e adequate. However in systems without spatial inversion symmetry, which include layered structures of practical importance, the Kambersky formulation may be inadequate even to second order in ξ. The long-range Coulomb interaction can now play a role. An important property of ferromagnetic systems without inversio n symmetry is the Dzyaloshinskii-Moriya inter- action (DMI) which leads to an instability of the uniform ferromagnet ic state with the appearance of a spiral spin structure or a skyrmion structure. This has been studied extens ively in bulk crystals like MnSi [37] and in layered11 structures [38]. The spiral instability appears as a singularity in the t ransverse susceptibility χ(q,0) at a value of q related to the DMI parameter. The method of this paper has been u sed to obtain a novel closed form expression for this parameter which will be reported elsewhere. In this paper we have analysed in some detail the transverse spin su sceptibility χ↓↑↑↓but combinations of some of the 15 other response functions merit further study. Mixed char ge-spin response arising from spin-orbit coupling is of particular interest for its relation to phenomena like the spin-Hall effect. Appendix A: A direct derivation of the Kambersky formula In this appendix we give a rather general derivation of the Kambers ky formula for the Gilbert damping parameter αwith an emphasis on its restriction to second order in the spin-orbit in teraction parameter ξ. We consider a general ferromagnetic material described by the ma ny-body Hamiltonian H=H1+Hint+Hext (A1) whereH1is a one-electron Hamiltonian of the form H1=Hk+Hso+V. (A2) HereHkis the total kinetic energy, Hso=ξhsois the spin-orbit interaction, Vis a potential term, Hintis the Coulomb interaction between electrons and Hextis due to an external magnetic field Bexin thezdirection. Thus Hext=−Szbexwherebex= 2µBBex, as in (34), and Szis thezcomponent of total spin. Both HsoandVcan contain disorderalthough in this paper we consider a perfect crystal. Follow ingthe general method of Edwardsand Fisher [40] we use equations of motion to find that the dynamical transverse s usceptibility χ(ω) =χ−+(0,ω) satisfies [39] χ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight /planckover2pi1ω−bex+ξ2 (/planckover2pi1ω−bex)2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) (A3) where χF(ω) =/integraldisplay ∝angb∇acketleft∝angb∇acketleftF−(t),F+∝angb∇acket∇ight∝angb∇acket∇ighte−iωtdt (A4) withF−= [S−,hso]. This follows since S−commutes with other terms in H1and with Hint. For small ω,χis dominated by the spin wave pole at /planckover2pi1ω=bext+/planckover2pi1δωwhereδω∼ξ2, so that χ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight /planckover2pi1(ω−δω)−bex. (A5) Following [39] we compare (A3) and (A5) in the limit /planckover2pi1δω≪/planckover2pi1ω−bexto obtain −2∝angb∇acketleftSz∝angb∇acket∇ight/planckover2pi1δω=ξ2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) =ξ2[ lim /planckover2pi1ω→bexχξ=0 F(ω)−lim ξ→0(1 ξ∝angb∇acketleft[F−,S+]∝angb∇acket∇ight)] (A6) correct to order ξ2. It is important to note that the limit ξ→0 within the bracket must be taken before putting /planckover2pi1ω=bex. If we put /planckover2pi1ω=bexfirst it is clear from (A3) that the quantity in brackets would vanish, giving the incorrect resultδω= 0. Furthermore it may be shown [M. Cinal, private communication] th at the second term in the bracket is real. Hence ℑ(/planckover2pi1ω) =−ξ2 2∝angb∇acketleftSz∝angb∇acket∇ightlim /planckover2pi1ω→bexℑ[χξ=0 F(ω)]. (A7) Kambersky [4] derived this result, using the approach of Mori and K awasaki ( [41] [42]), without noting its restricted validity to second order in ξ. This restriction is crucial since, as discussed in the main paper, it av oids the appearance of singular intraband terms. Oshikawa and Affleck emphasise strong ly a similar restriction in their related work on electron spin resonance (Appendix of [43]). Equation (A7) is an exact result even in the presence of disorder in t he potential and spin-orbit terms of the Hamiltonian. In the following we assume translational symmetry.12 To obtain the expression (61) for α=ℑ(/planckover2pi1ω)/bex, which is equivalent to Kambersky’s result (62), it is necessary to evaluate the response function χξ=0 F(ω) in tight-binding-RPA. Using (35)we find F−=/summationdisplay kµν[Lz µνc† kµ↓ckν↑+(1/2)L− µν(c† kµ↓ckν↓−c† kµ↑ckν↑)]. (A8) Hence χξ=0 F=/summationdisplay µν/summationdisplay αβ[Lz µνLz βαGµ↓ν↑,β↑α↓+(1/4)L− µνL+ βα(Gµ↓ν↓,β↓α↓+Gµ↑ν↑,β↑α↑−Gµ↓ν↓,β↑α↑−Gµ↑ν↑,β↓α↓)] (A9) where Gµσνσ′,βτατ′=∝angb∇acketleft∝angb∇acketleft/summationdisplay kc† kµσckνσ′;/summationdisplay uc† uβτcuατ′∝angb∇acket∇ight∝angb∇acket∇ightω. (A10) The Green function Gis to be calculated in the absence of SOC ( ξ= 0). Within RPA it satisfies an equation of the form Gµσνσ′,βτατ′=G0 µσνσ′,βτατ′−/summationdisplay µ1σ1ν1σ′ 1/summationdisplay µ2σ2ν2σ′ 2G0 µσνσ′,µ1σ1ν1σ′ 1Vµ1σ1ν1σ′ 1,µ2σ2ν2σ′ 2Gµ2σ2ν2σ′ 2,βτατ′ (A11) whereG0is the non-interacting (Hartree-Fock) Green function and Vµ1σ1ν1σ′ 1,µ2σ2ν2σ′ 2=Vσ1σ′ 1σ2σ′ 2(q)δµ1ν1δµ2ν2 (A12) withV(q) given by (11) and (12). Hence Gµσνσ′,βτατ′=G0 µσνσ′,βτατ′−/summationdisplay µ1σ1σ′ 1/summationdisplay µ2σ2σ′ 2G0 µσνσ′,µ1σ1µ1σ′ 1Vσ1σ′ 1σ2σ′ 2Gµ2σ2µ2σ′ 2,βτατ′. (A13) The form of the interaction Vgiven in (A12) is justified by the connection between (A13) and (10) , withq= 0. To see this connection we note that χσσ′ττ′=/summationtext µνGµσµσ′,ντντ′and that (A13) then leads to (10) which is equivalent to RPA. On substituting (A13) into (A9) we see that the contributions from the second term of (A13) contain factors of the form /summationdisplay µνµ1Lz µνG0 µ↓ν↑,µ1↑µ1↓,/summationdisplay µνµ1L− µνG0 µσνσ,µ 1σ1µ1σ1. (A14) We now show that such factors vanish owing to quenching of orbital angular momentum in the system without SOC (ξ= 0). Hence the Green functions Gin (A9) can be replaced by the non-interacting ones G0. The non-interacting Green functions G0are of a similar form to χ0in (40) and for ξ= 0 may be expressed in terms of the quantities anµ=∝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 that (44) emerged, /summationdisplay µνµ1Lz µνG0 µ↓ν↑,µ1↑µ1↓=/summationdisplay µν/summationdisplay knLz µνa∗ nµanνfkn↑−fkn↓ ∆+bex−/planckover2pi1ω+iη. (A15) Also by closure /summationdisplay µνLz µνa∗ nµanν=∝angb∇acketleftkn|Lz|kn∝angb∇acket∇ight= 0, (A16) the last step following from quenching of total orbital angular mome ntum. The proof that the second expression in (A14) vanishes is very similar. Hence we can insert the non-interacting Green functions G0in (A9) and straight-forwardalgebra, with use of (A7), leads to (58). At the end of section VI this is shown to be equivalent t o the Kambersky formula for α. We emphasize again that the present proof is valid only to order ξ2so that the one-electron states used to evaluate the formula should be calculated in the absence of SOC. This proof relies on the quenching of orbital angular momentum which does not occur in the absence of spatial inversion symmetry. When this symmetry is broken it is not difficult to s ee that the second term of (A13) gives a contribution to the first term on the right of (A9) which contains th eq= 0 spin-wave pole and diverges as /planckover2pi1ω→bex. Hence the proof of the torque-correlationformula (A7) collapses . The method of section VI must be used as discussed at the end of that section.13 Appendix B: Elements of S The element S11of the matrix Sis given in (27). The remaining elements are given below. S12=−Uχ0 ↓↑↓↑+(U/Λ)[(X+χ0 ↓↓↓↑)χ0 ↑↑↓↑χ0 ↓↑↑↑−(Y+χ0 ↑↑↓↓)χ0 ↓↓↓↑χ0 ↓↑↑↑ −(Y+χ0 ↓↓↑↑)χ0 ↑↑↓↑χ0 ↓↑↓↓+(X+χ0 ↑↑↑↑)χ0 ↓↓↓↑χ0 ↓↑↓↓](B1) S21=−Uχ0 ↑↓↑↓+(U/Λ)[(X+χ0 ↓↓↓↓)χ0 ↑↑↑↓χ0 ↑↓↑↑−(Y+χ0 ↑↑↓↓)χ0 ↓↓↑↓χ0 ↑↓↑↑ −(Y+χ0 ↓↓↑↑)χ0 ↑↑↑↓χ0 ↑↓↓↓+(X+χ0 ↑↑↑↑)χ0 ↓↓↑↓χ0 ↑↓↓↓](B2) S22= 1−Uχ0 ↑↓↓↑+(U/Λ)[(X+χ0 ↓↓↓↓)χ0 ↑↑↓↑χ0 ↑↓↑↑−(Y+χ0 ↑↑↓↓)χ0 ↓↓↓↑χ0 ↑↓↑↑ −(Y+χ0 ↓↓↑↑)χ0 ↑↑↓↑χ0 ↑↓↓↓+(X+χ0 ↑↑↑↑)χ0 ↓↓↓↑χ0 ↑↓↓↓](B3) Acknowledgement My recent interest in Gilbert damping arose through collaboration wit h O. Wessely, E. Barati, M. Cinal and A. Umerski. I am grateful to them for stimulating discussion and corre spondence. The specific work reported here arose directly from discussion with R.B.Muniz and I am particularly grateful t o him and his colleague A. T. Costa for this stimulation. References [1] Landau L D, Lifshitz E M and Pitaevski L P 1980 Statistical Physics part2 (Oxford: Pergamon) [2] Gilbert T L 1955 Phys. Rev. 1001243 [3] Kambersky V 1970 Can. J. Phys. 482906 [4] Kambersky V 1976 Czech. J. Phys. B261366 [5] Kambersky V 2007 Phys. Rev. B76134416 [6] Korenman V and Prange R E 1972 Phys. Rev. B62769 [7] Gilmore K, Idzerda Y U and Stiles M D 2007 Phys. Rev. Lett. 99027204 [8] Gilmore K, Idzerda Y U and Stiles M D 2008 J. Appl. Phys. 10307D3003 [9] Garate I and MacDonald A 2009 Phys. Rev. B79064403 [10] Garate I and MacDonald A 2009 Phys. Rev. B79064404 [11] Liu C, Mewes CKA, Chshiev M, Mewes T and Butler WH 2009 Appl. Phys. Lett. 95022509 [12] Umetsu N, Miura D and Sakuma A 2011 J. Phys. Conf. Series 266012084 [13] Sakuma A 2012 J. Phys. Soc. Japan 81084701 [14] Barati E, Cinal M, Edwards D M and Umerski A 2014 Phys.Rev. B90014420 [15] Bhagat S M and Lubitz P 1974 Phys. Rev. B10179 [16] Ma X, Ma L, He P, Zhao H B, Zhou S M and L¨ upke G 2015 Phys. Rev. B91014438 [17] Scheck C, Cheng L, Barsukov I, Frait Z and Bailey WE 2007 Phys. Rev. Lett. 98117601 [18] He P, Ma X, Zhang JW, Zhao HB, L¨ upke G, Shi Z and Zhou SM 201 3Phys. Rev. Lett. 110077203 [19] Costa A T and Muniz R B 2015 Phys. Rev. B92014419 [20] Brataas A, Tserkovnyak Y and Bauer GEW 2008 Phys. Rev. Lett. 101037207 [21] Starikov A, Kelly PJ, Brataas A, Tserkovnyak Y and Bauer GEW 2010 Phys. Rev. Lett. 105236601 [22] Costa A T, Muniz R B, Lounis S, Klautau A B and Mills D L 2010 Phys. Rev. B82014428 [23] Rajagopal A K, Brooks H and Ranganathan N R 1967 Nuovo Cimento Suppl. 5807 [24] Rajagopal A K 1978 Phys. Rev. B172980 [25] Low G G 1969 Adv. Phys. 18371 [26] Kim D J, Praddaude H C and Schwartz B B 1969 Phys. Rev. Lett. 23419 [27] Kim D J, Schwartz B B and Praddaude H C 1973 Phys. Rev. B7205 [28] Williams A R and von Barth U 1983, in Theory of the Inhomogeneous Electron Gas eds. Lundqvist S and March N H (Plenum)14 [29] Edwards D M 1984, in Moment Formation in Solids NATO Advanced Study Institute, Series B: Physics Vol. 117, e d. Buyers W J L (Plenum) [30] Cinal M and Edwards D M 1997 Phys. Rev B553636 [31] Izuyama T, Kim D J and Kubo R 1963 J. Phys. Soc. Japan 181025 [32] Lowde R D and Windsor C G 1970 Adv. Phys. 19813 [33] Doniach S and Sondheimer E H 1998 Green’s Functions for Solid State Physicists Imperial College Press [34] Kunes J and Kambersky V 2002 Phys. Rev B65212411 [35] Gilmore K, Garate I, MacDonald AH and Stiles MD Phys. Rev B84224412 [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. 105072413 [37] Grigoriev SV, Maleyev SV, Okorokov AI, Chetverikov Yu. O, B¨ oni P, Georgii R, Lamago D, Eckerlebe H and Pranzas K 2006Phys. Rev. B74214414 [38] von Bergmann K, Kubetzka A, Pietzsch O and Wiesendanger R 2014J. Phys. Condens. Matter 26394002 [39] Edwards D M and Wessely O 2009 J. Phys. Condens. Matter 21146002 [40] Edwards D M and Fisher B 1971 J. Physique 32C1 697 [41] Mori H 1965 Prog. Theor. Phys. 33423 [42] Mori H and Kawasaki K 1962 Prog. Theor. Phys. 27529 [43] Oshikawa M and Affleck I 2002 Phys. Rev. B65134410

2015-06-18

Damping of magnetization dynamics in a ferromagnetic metal is usually characterized by the Gilbert parameter alpha. Recent calculations of this quantity, using a formula due to Kambersky, find that it is infinite for a perfect crystal owing to an intraband scattering term which is of third order in the spin-orbit parameter xi This surprising result conflicts with recent work by Costa and Muniz who study damping numerically by direct calculation of the dynamical transverse spin susceptibility in the presence of spin-orbit coupling. We resolve this inconsistency by following the Costa-Muniz approach for a slightly simplified model where it is possible to calculate alpha analytically. We show that to second order in the spin-orbit parameter xi one retrieves the Kambersky result for alpha, but to higher order one does not obtain any divergent intraband terms. The present work goes beyond that of Costa and Muniz by pointing out the necessity of including the effect of long-range Coulomb interaction in calculating damping for large xi. A direct derivation of the Kambersky formula is given which shows clearly the restriction of its validity to second order in xi so that no intraband scattering terms appear. This restriction has an important effect on the damping over a substantial range of impurity content and temperature. The experimental situation is discussed.

The absence of intraband scattering in a consistent theory of Gilbert damping in metallic ferromagnets

1506.05622v2

Nanomagnet coupled to quantum spin Hall edge: An adiabatic quantum motor Liliana Arrachea Departamento de F sica, FCEyN, Universidad de Buenos Aires and IFIBA, Pabell on I, Ciudad Universitaria, 1428 CABA and International Center for Advanced Studies, UNSAM, Campus Miguelete, 25 de Mayo y Francia, 1650 Buenos Aires, Argentina Felix von Oppen Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit at Berlin, 14195 Berlin, Germany Abstract The precessing magnetization of a magnetic islands coupled to a quantum spin Hall edge pumps charge along the edge. Conversely, a bias voltage applied to the edge makes the magnetization precess. We point out that this device realizes an adiabatic quantum motor and discuss the eciency of its operation based on a scattering matrix approach akin to Landauer-B uttiker theory. Scattering theory provides a microscopic derivation of the Landau-Lifshitz-Gilbert equation for the magnetization dynamics of the device, including spin-transfer torque, Gilbert damping, and Langevin torque. We nd that the device can be viewed as a Thouless motor, attaining unit eciency when the chemical potential of the edge states falls into the magnetization-induced gap. For more general parameters, we characterize the device by means of a gure of merit analogous to the ZT value in thermoelectrics. 1. Introduction Following Ref. [1], Meng et al. [2] recently showed that a transport current owing along a quantum spin Hall edge causes a precession of the magneti- zation of a magnetic island which locally gaps out the edge modes (see Fig. 1 for a sketch of the de- vice). The magnetization dynamics is driven by the spin transfer torque exerted on the magnetic island by electrons backscattering from the gapped region. Indeed, the helical nature of the edge state implies that the backscattering electrons reverse their spin polarization, with the change in angular momen- tum transfered to the magnetic island. This eect is not only interesting in its own right, but may also have applications in spintronics. Current-driven directed motion at the nanoscale has also been studied for mechanical degrees of free- dom, as motivated by progress on nanoelectrome- chanical systems. Qi and Zhang [3] proposed that a conducting helical molecule placed in a homo- geneous electrical eld could be made to rotate around its axis by a transport current and pointedout the intimate relations with the concept of a Thouless pump [4]. Bustos-Marun et al. [5] devel- oped a general theory of such adiabatic quantum motors, used it to discuss their eciency, and em- phasized that the Thouless motor discussed by Qi and Zhang is optimally ecient. It is the purpose of the present paper to em- phasize that the current-driven magnetization dy- namics is another { perhaps more experimentally feasible { variant of a Thouless motor and that the theory previously developed for adiabatic quan- tum motors [5] is readily extended to this device. This theory not only provides a microscopic deriva- tion of the Landau-Lifshitz-Gilbert equation for the current-driven magnetization dynamics, but also al- lows one to discuss the eciency of the device and to make the relation with the magnetization-driven quantum pumping of charge more explicit. Specically, we will employ an extension of the Landauer-B uttiker theory of quantum transport which includes the forces exerted by the electrons on a slow classical degree of freedom [6, 7, 8, 9]. Preprint submitted to Elsevier October 11, 2018arXiv:1507.06505v2 [cond-mat.mes-hall] 8 Sep 2015



exeyezeθFigure 1: (Color online) Schematic setup. A nanomagnet with magnetic moment Mcouples to a Kramers pair of edge states of a quantum spin Hall insulator. The eective spin current produces a spin-transfer toque and the magnetic mo- ment precesses. Markus B uttiker developed Landauer's vision of quantum coherent transport as a scattering prob- lem into a theoretical framework [10, 11] and ap- plied this scattering theory of quantum transport to an impressive variety of phenomena. These ap- plications include Aharonov-Bohm oscillations [12], shot noise and current correlations [11, 13, 14], as well as edge-state transport in the integer Hall ef- fect [15] and topological insulators [16]. Frequently, B uttiker's predictions based on scattering theory provided reference points with which other theo- ries { such as the Keldysh Green-function formal- ism [17, 18, 19, 20] or master equations [21] { sought to make contact. In the present context, it is essential that scat- tering theory also provides a natural framework to study quantum coherent transport in systems un- der time-dependent driving. For adiabatic driving, B uttiker's work with Thomas and Pr^ etre [22] was instrumental in developing a description of adia- batic quantum pumping [4] in terms of scattering theory [23, 24, 25, 26] which provided a useful back- drop for later experiments [27, 28, 29, 30, 31]. Be- yond the adiabatic regime, Moskalets and B uttiker combined the scattering approach with Floquet the- ory to account for periodic driving [32]. These works describe adiabatic quantum transport as a limit of the more general problem of periodic driv- ing and ultimately triggered numerous studies on single-particle emitters and quantum capacitors (as reviewed by Moskalets and Haack in this volume[33]). The basic idea of the adiabatic quantum mo- tor [5] is easily introduced by analogy with the Archimedes screw, a device consisting of a screw inside a pipe. By turning the screw, water can be pumped against gravity. This is a classical analog of a quantum pump in which electrons are pumped be- tween reservoirs by applying periodic potentials to a central scattering region. Just as the Archimedes pump can pump water against gravity, charge can be quantum pumped against a voltage. In addi- tion, the Archimedes screw has an inverse mode of operation as a motor : Water pushed through the device will cause the screw to rotate. The adiabatic quantum motor is a quantum analog of this mode of operation in which a transport current pushed through a quantum coherent conductor induces uni- directional motion of a classical degree of freedom such as the rotations of a helical molecule. The theory of adiabatic quantum motors [5, 34] exploits the assumption that the motor degrees of freedom { be they mechanical or magnetic { are slow compared to the electronic degrees of freedom. In this adiabatic regime, the typical time scale of the mechanical dynamics is large compared to the dwell time of the electrons in the interaction region between motor and electrical degrees of freedom. In this limit, the dynamics of the two degrees of free- dom can be discussed in a mixed quantum-classical description. The motor dynamics is described in terms of a classical equation of motion, while a fully quantum-coherent description is required for the fast electronic degrees of freedom. From the point of view of the electrons, the mo- tor degrees of freedom act as acpotentials which pump charge through the conductor. Conversely, the backaction of the electronic degrees of freedom enters through adiabatic reaction forces on the mo- tor degrees of freedom [6, 7, 8, 9]. When there is just a single (Cartesian) classical degree of freedom, these reaction forces are necessarily conservative, akin to the Born-Oppenheimer force in molecular physics [35]. Motor action driven by transport cur- rents can occur when there is more than one mo- tor degree of freedom (or a single angle degree of freedom). In this case, the adiabatic reaction force need no longer be conservative when the electronic conductor is subject to a bias voltage [6, 7, 8, 9]. In next order in the adiabatic approximation, the electronic system also induces frictional and Lorentz-like forces, both of which are linear in the slow velocity of the motor degree of freedom. In- 2cluding the uctuating Langevin force which ac- companies friction yields a classical Langevin equa- tion for the motor degree of freedom. This equation can be derived systematically within the Keldysh formalism [35] and the adiabatic reaction forces ex- pressed through the scattering matrix of the coher- ent conductor [6, 7, 8]. While these developments focused on mechani- cal degrees of freedom, it was also pointed out that the scattering theory of adiabatic reaction forces extends to magnetic degrees of freedom [9]. In this case, adiabaticity requires that the precessional time scale of the magnetic moment is larger than the electronic dwell time. The eective classical de- scription for the magnetic moment takes the form of a Landau-Lifshitz-Gilbert (LLG) equation. Similar to nanoelectromechanical systems, the LLG equa- tion can be derived systematically in the adiabatic limit for a given microscopic model and the coe- cients entering the LLG equation can be expressed alternatively in terms of electronic Green functions or scattering matrices [36, 37, 38, 39, 9]. In the fol- lowing, we will apply this general theory to a mag- netic island coupled to a Kramers pair of helical edge states. This work is organized as follows. Section 2 reviews the scattering-matrix expressions for the torques entering the LLG equation. Section 3 ap- plies this theory to helical edge states coupled to a magnetic island and makes the relation to adia- batic quantum motors explicit. Section 4 denes and discusses the eciency of this device and de- rives a direct relation between charge pumping and spin transfer torque. Section 5 is devoted to con- clusions. 2. S-matrix theory of spin transfer torques and Gilbert damping 2.1. Landau-Lifshitz-Gilbert equation Consider a coherent (Landauer-B uttiker) con- ductor coupled to a magnetic moment. The latter is assumed to be suciently large to justify a classical description of its dynamics but suciently small so that we can treat it as a single macrospin. Then, its dynamics is ruled by a Landau-Lifshitz-Gilbert equation _M=M[@MU+Bel+B]: (1) Note that we use units in which Mis an angu- lar momentum and for simplicity of notation, Belas well asBdier from a conventional magnetic eld by a factor of gd, the gyromagnetic ratio of the macrospin. The rst term on the right-hand side describes the dynamics of the macrospin in the absence of coupling to the electrons. It is derived from the quantum Hamiltonian ^U=gd^M
B+D 2^M2 z; (2) where M=h^Miis the uncoupled macrospin, Bthe magnetic eld, and D> 0 the easy-plane anisotropy of the macrospin. The coupling to the electrons leads to the additional eective magnetic eld Bel. This term can be derived microscopically from the Heisenberg equation of motion of the macrospin by evaluating the commutator of ^Mwith the interac- tion Hamiltonian between macrospin and electrons in the adiabatic approximation (see, e.g., Ref. [9]). Keeping terms up to linear order in the small mag- netization \velocity" _M, we can write Bel=B0(M) (M)_M: (3) Here, the rst contribution B0can be viewed as the spin-transfer torque. The second term is a contribu- tion to Gilbert damping arising from the coupling between macrospin and electrons. In general, so derived is a tensor with symmetric and antisymmet- ric components. However, it can be seen that only the symmetric part plays a relevant role [9]. Finally, by uctuation-dissipation arguments, the Gilbert damping term is accompanied by a Langevin torque Bwith correlator hBl(t)Bk(t0)i=Dlk(tt0): (4) Its correlations are local in time as a consequence of the assumption of adiabaticity. As a result, we nd the LLG equation _M=Mh @MU+B0 _M+Bi ; (5) for the macrospin M. The spin-transfer torque, the Gilbert damping, and the correlator Dcan be expressed in terms of the scattering matrix of the coherent conductor, both in and out of equilibrium [36, 37, 38, 39, 9]. Before presenting the S-matrix expressions, a few comments are in order. First, the expression for the Gilbert damping only contains the intrinsic damp- ing originating from the coupling to the electronic degrees of freedom. Coupling to other degrees of freedom might give further contributions to Gilbert 3damping which could be included phenomenologi- cally. Second, in the study of the nanomagnet cou- pled to the helical modes we will consider the ex- pressions to lowest order in the adiabatic approxi- mation presented in Sec. 2.2. The theory can actu- ally be extended to include higher order corrections [9]. In Sec. 2.3 section, we brie y summarize the main steps of the general procedure for complete- ness. 2.2. Coecients of the LLG equation in the lowest order adiabatic approximation This section summarizes the expressions for the coecients of the LLG equation that we will use to study the problem of the nanomagnet coupled to the helical edge states. These correspond to the lowest order in the adiabatic approximation, in which we retain only terms linear in _MandeV. To this order, we can write the coecients of the LLG equation in terms of the electronic S-matrix for a static macrospin M. The coupling between macrospin and electronic degrees of freedom enters through the dependence of the electronic S-matrix S0=S0(M) on the (xed) macrospin. At this or- der, the spin-transfer torque and the Gilbert damp- ing can be expressed as [36, 37, 38, 39, 9] B0(M) =X Zd" 2ifTr ^Sy 0@S0 @M (6) and kl(M) =~X Zd" 4f0 Tr" @^Sy 0 @Mk@^S0 @Ml# s;(7) respectively. Finally, the uctuation correlator Dis expressed as [9] Dkl(M) =~X ;0Zd" 2f(1f0) Tr2 4 ^Sy 0@^S0 @Mk!y 0 ^Sy 0@^S0 @Ml!3 5 s:(8) In these expressions, =L;R denotes the reser- voirs with electron distribution function f, is a projector onto the channels of lead , and Tr traces over the lead channels. 2.3. Corrections to the adiabatic approximation of the S-matrix In order to go beyond linear response in eVand _M, we must consider the electronic S-matrix in thepresence of the time-dependent magnetization M(t) and expand it to linear order in the magnetization \velocity" _M(t). This can be done, e.g., by starting from the full Floquet scattering matrix SF ;("n;") for a periodic driving with period ![32]. The in- dicesandlabel the scattering channels of the coherent conductor and the arguments denote the energies"of the incoming electron in channel and "n="+n~!of the outgoing electron in channel . For small driving frequency !, the Floquet scatter- ing matrix can be expanded in powers of ~!, ^SF("n;") = ^S0 n(") +n~! 2@^S0 n(") @" +~!^An(") +O("2): (9) Here ^S0 n(") is the Fourier transform of the frozen scattering matrix S0(M(t)) introduced above, ^S0(M(t)) =1X n=1ein!t^S0 n("): (10) The matrix ^An("), rst introduced by Moskalets and B uttiker, is the rst adiabatic correction to the adiabatic S-matrix and can be transformed in a sim- ilar way to ^A(t;") =1X n=1ein!t^An(") = _M(t)
^A(t;"):(11) The matrix ^An(") can be straightforwardly calcu- lated from the retarded Green function of the device (see Refs. [20, 9]). We are now in a position to give expressions for the Gilbert damping to next order in the adiabatic approximation. (The spin-transfer torque and the uctuation correlator remain unchanged.) To do so, we split the Gilbert matrix into its symmetric and antisymmetric parts, = s+ a: (12) Strictly speaking, it is only the symmetric part which corresponds to Gilbert damping. The anti- symmetric part simply renormalizes the precession frequency. One nds [9] kl s(M) =~X Zd" 4f0 Tr" @^Sy 0 @Mk@^S0 @Ml# s +X Zd" 2ifTr"  @^Sy 0 @Mk^Al^Ay l@^S0 @Mk!# s(13) 4for the symmetric contribution. It can be seen, the second line is a pure nonequilibrium contribution (/eV~!). Similarly, the antisymmetric part of the Gilbert damping can be written as [9] kl a(M) =~X Zd" 2if(") Tr"  ^Sy 0@^Ak @Ml@^Ay k @Ml^S0!# a;(14) which is really a renormalization of the precession frequency as mentioned above. 3. S-matrix theory of a nanomagnet coupled to a quantum spin Hall edge We now apply the above theory to a magnetic island coupled to a quantum spin Hall edge as sketched in Fig. 1. The quantum spin Hall edge sup- ports a Kramers doublet of edge states. The mag- netization M=M?cosex+M?siney+Mzez of the magnetic island induces a Zeeman eld JM acting on the electrons along the section of length Lof the edge state which is covered by the magnet. This Zeeman eld causes backscattering between the edge modes and induces a gap
= JM?~=2 [1]. Linearizing the dispersion of the edge modes, the electronic Hamiltonian takes the form [2] ^H= (vpJMz)^z+
(x) (cos^x+ sin^y):(15) Here, thejdenote Pauli matrices in spin space and
(x) is nonzero only over the region of length Lcovered by the magnetic island. We have assumed for simplicity that the spin Hall edge conserves z. Then, a static island magnetization induces a gap whenever it has a component perpendicular to the z-direction. Indeed, ^His easily diagonalized for a spatially uniform coupling between edge modes and magnet, and the spectrum Ep=p (vpJMz)2+
2 (16) has a gap
. In the following, we assume that the easy-plane anisotropy D > 0 is suciently large so that the magnetization entering the electronic Hamiltonian can be taken in the xy-plane, i.e., Mz'0. (How- ever, we will have to keep Mzin the LLG equation when it is multiplied by the large anisotropy D.) The electronic Hamiltonian (15) is equivalent to the electronic Hamiltonian of the Thouless motor considered in Ref. [5]. Following this reference, wecan readily derive the frozen scattering matrix an- alytically [5], ^S0=1 iei 1 1iei ; (17) where we have dened the shorthands  = cos Li"p "2
2sinL; =
p "2
2sinL (18) with L(") =L ~vp "2
2: (19) Note that these expressions are exact for any Land valid for energies "both inside and outside the gap. We can now use this scattering matrix to eval- uate the various coecients in the LLG equation, employing the expressions given in Sec. 2.2. As- suming zero temperature, we nd B0=eV 2M()e; (20) for the spin transfer torque at arbitrary chemical potential. Here, we have dened the function () =
2sin2L j2
2jcos2L+2sin2L(21) withL=L() (see Fig. 2). Below, we will iden- tifywith the charge pumped between the reser- voirs during one precessional period of the mag- netization M. The vector B0points in the az- imuthal direction in the magnetization plane and indeed corresponds to a spin-transfer torque. Sim- ilarly, we can substitute Eq. (17) into Eq. (13) for the Gilbert damping and nd that the only nonzero component of the tensor is =~ 2M2(): (22) Similarly, D=~eV M2() (23) is the only nonzero component of the uctuation correlator. It is interesting to note that this yields an eective uctuation-dissipation relation D= 2Te with eective temperature Te=eV. 5With these results, we can now write the LLG equation for the nanomagnet coupled to the helical edge state, _M =DMMzez+(eV~_) 2MMe +MB; (24) where=(), we have expressed _M'M_e, and assumed zero external magnetic eld B. This com- pletes our scattering-theory derivation of the LLG equation and generalizes the result obtained in Ref. [2] on phenomenological grounds in several respects. Equation (24) applies also for nite-length magnets and chemical potentials both inside and outside the magnetization-induced gap of the edge-state spec- trum. Moreover, the identication of the _-term as a damping term necessitates the inclusion of the Langevin torque B. Indeed, Ref. [2] refers to the entire term involving eV~_as the spin-transfer torque. In contrast, our derivation produces the term involving eValready in zeroth order in mag- netization \velocity" _M, while the _term appears only to linear order. Thus, the latter term is re- ally a conrtribution to damping and related to the energy dissipated in the electron system due to the time dependence of the magnetization. 4. Eciency of the nanomagnet as a motor While the electronic Hamiltonian for the edge modes is equivalent to that of the Thouless mo- tor discussed in Ref. [5], the LLG equation for the macrospin dier from the equation of motion of the mechanical degrees of freedom discussed in Ref. [5]. In this section, we discuss the energetics and the eciency of the magnetic Thouless motor against the backdrop of its mechanical cousin. The dynamics of the macrospin is easily ob- tained from the LLG equation (24) [2]. For a large anisotropy and thus small Mz, we need to retain thez-component of Monly in combination with the large anisotropy D. Then, the steady-state value of Mzis xed by the -component of the LLG equa- tion, Mz=_ D: (25) The precessional motion of Mabout thez-axis is governed by the z-component of the LLG equation, which yields _=eV ~(26)and hence Mz=eV=(~D). It is interesting to note that the angular frequency _of the preces- sion is just given by the applied bias voltage, in- dependent of the damping strength. This should be contrasted with the mechanical Thouless motor. Here, the motor degree of freedom satises a New- ton equation of motion which is second order in time. Thus, the frequency of revolution is inversely proportional to the damping coecient. In steady state, the magnetic Thouless motor bal- ances the energy provided by the voltage source through the spin-transfer torque B0against the dis- sipation through Gilbert damping due to the intrin- sic coupling between magnetic moment and elec- tronic degrees of freedom. It is instructive to look at these contributions independently. The work per- formed by the spin-transfer torque per precessional period is given by
Wspintransfer =Z2=_ 0dtB0
_M: (27) Writing this as an integral over a closed loop of the magnetization Mand inserting the S-matrix expression (6), we nd
Wspintransfer =X Zd" 2if I dM
Tr" ^Sy 0@^S0 @M# : (28) Without applied bias, the integrand is just the gra- dient of a scalar function and the integral vanishes. Thus, we expand to linear order in the applied bias and obtain
Wspintransfer =ieV 4 X I dM
Tr" (LR)^Sy 0@^S0 @M# :(29) Comparing Eq. (29) with the familiar S-matrix ex- pression for the pumped charge [23], the right-hand side can now be identied as the bias voltage multi- plied by the charge pumped between the reservoirs during one revolution of the magnetization,
Wspintransfer =QpV: (30) With every revolution of the magnetization, a chargeQpis pumped between the reservoirs. The corresponding gain QpVin electrical energy is driv- ing the magnetic Thouless motor. This result can also be written as _Wspintransfer =QpV 2_ (31) 6for the power provided per unit time by the voltage source. The relation between spin-transfer torque and pumped charge also allows us to identify the func- tion() appearing in the LLG equation as the charge in units of epumped between the reservoirs during one precessional period of the macrospin, Qp=e: (32) This can be obtained either by deriving the pumped charge explicitly from the S-matrix expression or by evaluating Eq. (27) using the explicit expression Eq. (20). The electrical energy gain is compensated by the energy dissipated through Gilbert damping. The dissipated energy per period is given by
WGilbert =Z2=_ 0dt_MT _MT = 2M2 _: (33) Using Eq. (22), this yields the dissipated energy
WGilbert =~_ (34) per precessional period or _WGilbert =~ 2_2(35) per unit time. These expressions have a simple interpretation. Due to the nite frequency of the magnetization precession, each pumped charge ab- sorbs on average an energy ~_which is then dissi- pated in the reservoirs. Armed with these results, we can nally discuss the eciency of a magnetic Thouless motor and fol- low the framework introduced in Ref. [40] to dene an appropriate gure of merit (analogous to the ZT value of thermoelectrics). Imagine the same setup as in Fig. 1, but with an additional load coupled to the magnetization. We can now dene the e- ciency of the magnetic Thouless motor as the ratio of the power delivered to the load and the electri- cal powerIVprovided by the voltage source. In steady state, the power delivered to the load has to balance against the power provided by the elec- trons, i.e., Bel
_M. Thus, we can write the eciency as =_W IV; (36) 00.20.40.60.81ξ,ηmax0 0.5 1 1.5 2 2.5 3 µ/Δ 00.20.40.60.81ξ,ηmaxFigure 2: (Color online) The parameter (dashed lines) en- tering the coecients of the LLG equation and the maximal eciencymax(solid lines) of the motor for a xed voltage V. Upper and lower panels correspond to nanomagnets of lengthL=~v=
andL= 10 ~v=
, respectively. where _W= _Wspintorque_WGilbert = 2eV_~ 2_2: (37) The total charge current owing along the topolog- ical insulator edge averaged over the cycle is the sum of the dccurrentGVdriven by the voltage, whereGis the dcconductance of the device, and the pumping current Qp_=(2), I=GV+e 2_: (38) We can now optimize the eciency of the motor at a given bias Vas function of the frequency _of the motor revolution. Note that due to the load, the latter is no longer tied to the bias voltage eV. This problem is analogous to the problem of the optimal eciency of a thermoelectric device which leads to the denition of the important ZT value. This analogy was discussed explicitly in Ref. [40]. Applying the results of this paper to the present device yields the maximal eciency max=p1 +1p1 ++ 1; (39) with a gure of merit analogous to the ZT value dened by =e2() hG(); (40) 7where() is dened in Eq. (21) and the conduc- tance reads G() =e2 hj2
2j2 j2
2jcos2L+2sin2L(41) as obtained from the Landauer-B uttiker equation. As in thermoelectrics, the maximum eciency is realized for !1 which requires a nite pumped charge at zero conductance. Unlike thermoelectrics, the motor eciency is bounded by = 1 instead of the Carnot eciency. This re ects the fact that electrical energy can be fully converted into mag- netic energy. Specically, unit eciency is reached in the limit of a true Thouless motor with zero transmission when the Fermi energy falls into the gap and nonzero and quantized pumped charge per period. This can be realized to a good approxi- mation for a suciently long magnet, as seen from the lower panel in Fig. 2. For chemical potentials outside the gap, the conductance and the pumped charge exhibit Fabry-Perot resonances. This yields a distinct sequence of maxima and minima in the eciency. For shorter magnets, the conductance remains nonzero within the gap, leading to lower eciencies. This is shown in the upper panel of Fig. 2. Moreover, the Fabry-Perot resonances are washed out, so that there is only a feature at the gap edge where the conductance vanishes while !1=2 for arbitrary L. 5. Conclusions Implementing directional motion of a mechani- cal or magnetic degree of freedom is a fundamental problem of nanoscale systems. An attractive gen- eral mechanism relies on running quantum pumps in reverse. This is the underlying principle of adia- batic quantum motors which drive periodic motion of a classical motor degree of freedom by applying a transport current. In this paper, we emphasize that a magnetic island coupled to a quantum spin Hall edge, recently discussed by Meng et al. [2], is just such an adiabatic quantum motor. We derive the Landau-Lifshitz-Gilbert equation for the magneti- zation dynamics from a general scattering-theory approach to adiabatic quantum motors, providing a microscopic derivation of spin-transfer torque, Gilbert damping, and Langevin torque. This ap- proach does not only provide a detailed microscopic understanding of the operation of the device but also allows one to discuss its eciency. We ndthat the device naturally approaches optimal e- ciency when the chemical potential falls into the magnetization-induced gap and the conductance is exponentially suppressed. This makes this system a Thouless motor and possibly its most experimen- tally feasible variant to date. Several issues are left for future work. While we derived microscopic expressions for the Langevin torque, we have not explored its consequences for the motor dynamics. It should also be interesting to consider thermal analogs driven by a tempera- ture gradient instead of a bias voltage. Inducing the magnetization precession by a temperature gradient would realize a quantum heat engine. Conversely, forcing a magnetic precession can be used to pump heat against a temperature gradient. Setups with several magnetic islands could be engineered to ef- fect exchange of charge and energy without employ- ing a dcbattery. These devices have been explored in the literature on quantum pumps [41, 42, 43] and their eciencies could be analyzed in the thermo- electric framework of Ref. [40]. Acknowledgement We thank Gil Refael and Ari Turner for dis- cussions. This work was supported by CON- ICET, MINCyT and UBACyT (L.A.) as well as the Deutsche Forschungsgemeinschaft and the Helmholtz Virtual Institute New States of Matter and Their Excitations (F.v.O.). L.A. thanks the ICTP Trieste for hospitality and the Simons Foun- dation for support. F.v.O. thanks the KITP Santa Barbara for hospitality during the nal preparation of this manuscript. This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. 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M. and B uttiker, M., 2002, Floquet scatter- ing theory of quantum pumps, Phys. Rev. B 66, 205320. [33] Moskalets, M. and Haack G., 2015, Single-electron coherence: nite temperature versus pure dephasing, arXiv:1506.09028. [34] Fernandez-Alcazar, L.J., Bustos-Marun, R.A., Pastawski, H.M., 2015, Decoherence in current induced forces: Application to adiabatic quantum motors, Phys. Rev. B 92, 075406. [35] Pistolesi, F., Blanter, Y.M., and Martin, I., 2008, Self- consistent theory of molecular switching, Phys. Rev. B 78. [36] Tserkovnyak, Y., Brataas, A., and Bauer, G.E.W., 2002, Enhanced Gilbert Damping in Thin Ferromag- netic Films, Phys. Rev. Lett. 88, 117601. [37] Brataas, A., Tserkovnyak, Y., and Bauer, G.E.W., 2008, Scattering Theory of Gilbert Damping, Phys. Rev. Lett. 101, 037207. [38] Brataas, A., Tserkovnyak, Y., and Bauer, G.E.W., 2011, Magnetization dissipation in ferromagnets from scattering theory, Phys. Rev. B 84, 054416. [39] Hals, K.M.D., Brataas, A., Tserkovnyak, Y., 2010, Scattering theory of charge-current-induced magnetiza- tion dynamics, Europhys Lett. 90, 47002. [40] Ludovico, M.F., Battista, F., von Oppen, F., Ar- rachea, L., 2015, Adiabatic response and quan- tum thermoelectrics for acdriven quantum systems, arXiv:1506.08617. [41] Arrachea, L., Moskalets, M. and Martin-Moreno, L., 2007, Heat production and energy balance in nanoscale engines driven by time-dependent elds, Phys. Rev. B 75, 245420. [42] Juergens, S., Haupt, F., Moskalets, M., and Splettstoesser, J., 2013, Thermoelectric performance of a driven double quantum dot, Phys. Rev. B 87, 245423. [43] Moskalets, M. and B uttiker, M., 2009, Heat production and current noise for single- and double-cavity quantum capacitors, Phys. Rev. B 80, 081302. 9

2015-07-23

The precessing magnetization of a magnetic islands coupled to a quantum spin Hall edge pumps charge along the edge. Conversely, a bias voltage applied to the edge makes the magnetization precess. We point out that this device realizes an adiabatic quantum motor and discuss the efficiency of its operation based on a scattering matrix approach akin to Landauer-B"uttiker theory. Scattering theory provides a microscopic derivation of the Landau-Lifshitz-Gilbert equation for the magnetization dynamics of the device, including spin-transfer torque, Gilbert damping, and Langevin torque. We find that the device can be viewed as a Thouless motor, attaining unit efficiency when the chemical potential of the edge states falls into the magnetization-induced gap. For more general parameters, we characterize the device by means of a figure of merit analogous to the ZT value in thermoelectrics.

Nanomagnet coupled to quantum spin Hall edge: An adiabatic quantum motor

1507.06505v2

Magnetization Dissipation in the Ferromagnetic Semiconductor (Ga,Mn)As Kjetil M. D. Hals and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway We compute the Gilbert damping in (Ga,Mn)As based on the scattering theory of magnetization relaxation. The disorder scattering is included non-perturbatively. In the clean limit, spin-pumping from the localized d-electrons to the itinerant holes dominates the relaxation processes. In the diusive regime, the breathing Fermi-surface eect is balanced by the eects of interband scattering, which cause the Gilbert damping constant to saturate at around 0.005. In small samples, the system shape induces a large anisotropy in the Gilbert damping. I. INTRODUCTION The magnetization dynamics of a ferromagnet can be described phenomenologically by the Landau-Lifshitz- Gilbert (LLG) equation:1,2 1 dM dt=MHe+M"~G(M) 2M2sdM dt# :(1) Here, is the gyromagnetic ratio, Heis the eective magnetic eld (which is the functional derivative of the free energy He=F[M]=M),Mis the magnetiza- tion andMsis its magnitude. The Gilbert damping con- stant ~G(M) parameterizes the dissipative friction process that drives the magnetization towards an equilibrium state.3In the most general case, ~G(M) is a symmetric positive denite matrix that depends on the magnetiza- tion direction; however, it is often assumed to be inde- pendent of Mand proportional to the unit matrix, as- sumptions which are valid for isotropic systems. Gilbert damping is important in magnetization dynamics. It de- termines the magnitudes of the external magnetic elds4 and the current densities1that are required to reorient the magnetization direction of a ferromagnet. Therefore, a thorough understanding of its properties is essential for modeling ferromagnetic systems. The main contribution to the Gilbert damping process in metallic ferromagnets is the generation of electron- hole pairs.1,2,5,6A model that captures this process was developed by Kambersky.5In this model, the electrons are excited by a time-varying magnetization via electron- magnon coupling. If the ferromagnet is in metallic con- tact with other materials, the spin-pumping into the ad- jacent leads provides an additional contribution to the magnetization relaxation.7A general theory that cap- tures both of these eects was recently developed.8The model expresses the ~G(M) tensor in terms of the scatter- ing matrix Sof the ferromagnetic system ( mM=Ms): ~Gij(m) = 2h 4Re Tr@S @mi@Sy @mj : (2) The expression is evaluated at the Fermi energy. Instead of~G(M), one often parameterizes the damping by the dimensionless Gilbert damping parameter ~ ~G= M s. Eq. (2) allows studying both the eects of the systemshape and the disorder dependency of the magnetization damping beyond the relaxation time approximation.9 In anisotropic systems, the Gilbert damping is ex- pected to be a symmetric tensor with non-vanishing o- diagonal terms. We are interested in how this tensor structure in uences the dynamics of the precessing mag- netization in (Ga,Mn)As. Therefore, to brie y discuss this issue, let us consider a homogenous ferromagnet in which the magnetization direction m=m0+mpre- cesses with a small angle around the equilibrium direc- tionm0that points along the external magnetic eld Hext.10For clarity, we neglect the anisotropy in the free energy and choose the coordinate system such that m0= (0 0 1) and m= (mxmy0). For the lowest order of Gilbert damping, the LLG equation can be rewritten as:_m= mHext+ m(~[Hextm]), where ~[:::] is the dimensionless Gilbert damping tensor that acts on the vector Hextm. Linearizing the LLG equation re- sults in the following set of equations for mxandmy:  _mx _my = Hext (0) yy (1(0) xy) (1 +(0) xy)(0) xx! mx my : (3) Here,(0) ijare the matrix elements of ~ when the tensor is evaluated along the equilibrium magnetization direc- tionm0. For the lowest order of Gilbert damping, the eigenvalues of (3) are =i H ext Hext, and the eigenvectors describe a precessing magnetization with a characteristic life time = ( H ext)1. The eective damping coecient is:10 1 2 (0) xx+(0) yy : (4) The value of is generally anisotropic and depends on the static magnetization direction m0. The magnetiza- tion damping is accessible via ferromagnetic resonance (FMR) experiments by measuring the linear relation- ship between the FMR line width and the precession fre- quency. This linear relationship is proportional to 1 and thus depends linearly on . Therefore, an FMR ex- periment can be used to determine the eective damping coecient. In contrast, the o-diagonal terms, (0) xy and(0) yx, do not contribute to the lowest order in the damping and are dicult to probe experimentally. In this paper, we use Eq. (2) to study the anisotropy and disorder dependency of the Gilbert damping in thearXiv:1105.4148v2 [cond-mat.mtrl-sci] 2 Nov 20112 ferromagnetic semiconductor (Ga,Mn)As. Damping co- ecients of this material in the range of 0:0040:04 for annealed samples have been reported.11{14The damp- ing is anisotropically dependent on the magnetization direction.11,12,14The few previous calculations of the Gilbert damping constant in this material have indicated that0:0030:04.11,15{17These theoretical works have included the eects of disorder phenomenologically, for instance, by applying the relaxation time approxima- tion. In contrast, Eq. (2) allows for studying the disor- der eects fully and non-perturbatively for the rst time. In agreement with Ref. 15, we show that spin-pumping from the localized d-electrons to the itinerant holes dom- inates the damping process in the clean limit. In the diusive regime, the breathing Fermi-surface eect is bal- anced by eects of the interband transitions, which cause the damping to saturate. In determining the anisotropy of the Gilbert damping tensor, we nd that the shape of the sample is typically more important than the eects of the strain and the cubic symmetry in the GaAs crys- tal.18This shape anisotropy of the Gilbert damping in (Ga,Mn)As has not been reported before and provides a new direction for engineering the magnetization relax- ation. II. MODEL The kinetic-exchange eective Hamiltonian approach gives a reasonably good description of the electronic properties of (Ga,Mn)As.19The model assumes that the electronic states near the Fermi energy have the character of the host material GaAs and that the spins of the itin- erant quasiparticles interact with the localized magnetic Mn impurities (with spin 5/2) via the isotropic Heisen- berg exchange interaction. If the s-d exchange interac- tion is modeled by a mean eld, the eective Hamiltonian takes the form:19,20 H=HHoles +h(r)
s; (5) whereHHoles is the k
pKohn-Luttinger Hamiltonian de- scribing the valence band structure of GaAs and h(r)
s is a mean eld description of the s-d exchange interac- tion between the itinerant holes and the local magnetic impurities ( sis the spin operator). The exchange eld his antiparallel to the magnetization direction m. The explicit form of HHoles that is needed for realistic model- ing of the band structure of GaAs depends on the doping level of the system. Higher doping levels often require an eight-band model, but a six- or four-band model may be sucient for lower doping levels. In the four-band model, the Hamiltonian is projected onto the subspace spanned by the four 3/2 spin states at the top of the GaAs valence band. The six-band model also includes the spin-orbit split-o bands with spin 1/2. The spin- orbit splitting of the spin 3/2 and 1/2 states in GaAs is 341 meV.21We consider a system with a Fermi level of 77 meV when measured from the lowest subband. Inthis limit, the following four-band model gives a sucient description: H=1 2m ( 1+5 2 2)p22 3(p
J)2+h
J + 3 2 m(p2 xJ2 x+c:p:) +Hstrain +V(r): (6) Here, pis the momentum operator, Jiare the spin 3/2 matrices22and 1, 2and 3are the Kohn-Luttinger pa- rameters.V(r) =P iVi(rRi) is the impurity scat- tering potential, where Riis the position of the impurity iandViare the scattering strengths of the impurities23 that are randomly and uniformly distributed in the in- terval [V0=2;V0=2].Hstrain is a strain Hamiltonian and arises because the (Ga,Mn)As system is grown on top of a substrate (such as GaAs).24The two rst terms in Eq. (6) have spherical symmetry, and the term propor- tional to 3 2represents the eects of the cubic sym- metry of the GaAs crystal. Both this cubic symmetry term25and the strain Hamiltonian24are small compared to the spherical portion of the Hamiltonian. A numeri- cal calculation shows that they give a correction to the Gilbert damping on the order of 10%. However, the un- certainty of the numerical results, due to issues such as the sample-to-sample disorder uctuations, is also about 10%; therefore, we cannot conclude how these terms in- uence the anisotropy of the Gilbert damping. Instead, we demonstrate that the shape of the system is the dom- inant factor in uencing the anisotropy of the damping. Therefore, we disregard the strain Hamiltonian Hstrain and the term proportional to 3 2in our investigation of the Gilbert damping. GaAs GaAs (Ga,Mn)As yx FIG. 1: We consider a (Ga,Mn)As system attached along the [010] direction to innite ballistic GaAs leads. The scattering matrix is calculated for the (Ga,Mn)As layer and one lattice point into each of the leads. The magnetization is assumed to be homogenous. In this paper, we denote the [100] direction as the x-axis, the [010] direction as the y-axis and the [001] direction as the z-axis We consider a discrete (Ga,Mn)As system with transverse dimensions Lx2 f17;19;21gnm,Lz2 f11;15;17gnm andLy= 50 nm and connected to innite ballistic GaAs leads, as illustrated in Fig. 1. The leads are modeled as being identical to the (Ga,Mn)As system, except for the magnetization and disorder. The lattice constant is 1 nm, which is much less than the Fermi wave- lengthF10 nm. The Fermi energy is 0.077 eV when3 measured from the lowest subband edge. The Kohn- Luttinger parameters are 1= 7:0 and 2= 3= 2:5, implying that we apply the spherical approximation for the Luttinger Hamiltonian, as mentioned above.25We usejhj= 0:032 eV for the exchange-eld strength. To estimate a typical saturation value of the magnetization, we useMs= 10j jhx=a3 GaAs withx= 0:05 as the doping level andaGaAs as the lattice constant for GaAs.26 The mean free path lfor the impurity strength V0is calculated by tting the average transmission probability T=hGi=GshtoT(Ly) =l=(l+Ly),27whereGshis the Sharvin conductance and hGiis the conductance for a system of length Ly. The scattering matrix is calculated numerically us- ing a stable transfer matrix method.28The disorder ef- fects are fully and non-perturbatively included by the ensemble average hi=PNI n=1n=NI, whereNIis the number of dierent impurity congurations. All the coecients are averaged until an uncertainty hi=r h2ihi2 =NIof less than 10% is achieved. The vertex corrections are exactly included in the scattering formalism. III. RESULTS AND DISCUSSION Without disorder, the Hamiltonian describing our sys- tem is rotationally symmetric around the axis parallel toh. Let us brie y discuss how this in uences the particular form of the Gilbert damping tensor ~ .29For clarity, we choose the coordinate axis such that the ex- change eld points along the z-axis. In this case, the Hamiltonian is invariant under all rotations Rzaround the z-axis. This symmetry requires the energy dissipa- tion _E/_mT~_m(_mTis the transposed of _m) of the magnetic system to be invariant under the coordinate transformations r0=Rzr(i.e., ( _m0)T~0_m0=_mT~_m where m0=Rzmand ~0is the Gilbert damping ten- sor in the rotated coordinate system). Because ~ only depends on the direction of m, which is unchanged under the coordinate transformation Rz, ~= ~0and RT z~Rz= ~. Thus, ~andRzhave common eigenvectors ji (jxiijyi)=p 2;jzi , and the spectral decompo- sition of ~is ~=+j+ih+j+jihj +zjzihzj. Representing the damping tensor in the fjxi;jyi;jzig coordinate basis yields ~xx= ~yy= (++)=2, ~zz=z, ~yx= ~xy= 0, and+=. The last equality results from real tensor coecients. However, zzcannot be determined uniquely from the energy dis- sipation formula _E/_mT~_mbecause _mis perpendicu- lar to the z-axis. Therefore, zzhas no physical signi- cance and the energy dissipation is governed by the single parameter. For an innite system, this damping pa- rameter does not depend on the specic direction of the magnetization, i.e., it is isotropic because the symmetry of the Hamiltonian is not directly linked to the crystallo- graphic axes of the underlying crystal lattice (when the 0 0.5 1 1.5 200.2 0.4 0.6 0.8 1 φ / πθ / π 44.5 55.5 6x 10 −3 0 0.5 1 1.5 200.2 0.4 0.6 0.8 1 φ / πθ / π 4.5 55.5 6x 10 −3 a bFIG. 2: ( a) The dimensionless Gilbert damping parameter as a function of the magnetization direction for a system whereLx= 17 nm,Ly= 50 nm and Lz= 17 nm. ( b) The dimensionless Gilbert damping parameter as a function of the magnetization direction for a system where Lx= 21 nm, Ly= 50 nm and Lz= 11 nm. Here, andare the polar and azimuth angles, respectively, that describe the local magne- tization direction m= (sincos;sinsin;cos). In both plots, the mean free path is l22 nm cubic symmetry term in Eq. (6) is disregarded). For a nite system, the shape of the system induces anisotropy in the magnetization damping. This eect is illustrated in Fig. 2, which plots the eective damping in Eq. (4) as a function of the magnetization directions for dierent system shapes. When the cross-section of the conductor is deformed from a regular shape to the shape of a thin- ner system, the anisotropy of the damping changes. The magnetization damping varies from a minimum value of around 0.004 to a maximum value of 0.006, e.g., the anisotropy is around 50%. The relaxation process is largest along the axis where the ballistic leads are con- nected, i.e., the y-axis. This shape anisotropy is about four- to ve-times stronger than the anisotropy induced by the strain and the cubic symmetry terms in the Hamil- tonian (6), which give corrections of about 10 percent. For larger systems, we expect this shape eect to be-4 come less dominant. In these systems, the anisotropy of the bulk damping parameter, which is induced by the anisotropic terms in the Hamiltonian, should play a more signicant role. The determination of the system size when the strain and cubic anisotropy become comparable to the shape anisotropy eects is beyond the scope of this paper because the system size is restricted by the com- puting time. However, this question should be possible to investigate experimentally by measuring the anisotropy of the Gilbert damping as a function of the lm thickness. 0 1 2 3 4 52345678910 x 10 −3 Ly/l ααmin αmax αmean FIG. 3: The eective dimensionless Gilbert damping (4) as a function of the disorder. Here, lis the mean free path and Lyis the length of the ferromagnetic system in the transport direction.minandmaxare the minimum and maximum val- ues of the anisotropic Gilbert damping parameter and mean is the eective damping parameter averaged over all the magne- tization directions. The system dimensions are Lx= 19 nm, Ly= 50 nm and Lz= 15 nm. We next investigate how the magnetization relaxation process depends on the disorder. Ref. 15 derives an ex- pression that relates the Gilbert damping parameter to the spin- ip rate T2of the system: /T2(1 + (T2)2)1. In the low spin- ip rate regime, this expression scales withT2as/T1 2, while the damping parameter is proportional to /T2in the opposite limit . As ex- plained in Ref. 15, the low spin- ip regime is dominated by the spin-pumping process in which angular momen- tum is transferred to the itinerant particles; the trans- ferred spin is then relaxed with a rate proportional to T1 2. This process appears inside the ferromagnet itself, i.e., the spin is transferred from the magnetic system to the itinerant particles in the ferromagnet, which are then relaxed within the ferromagnet. Therefore, this relax- ation mechanism is a bulk process and should not be confused with the spin-pumping interface eect across the normal metal jferromagnet interfaces reported in Ref. 7. In (Ga, Mn)As, this bulk process corresponds to spin-pumping from the d-electrons of the magnetic Mn impurities to the itinerant spin 3/2 holes in the valence band of the host compound GaAs. The transfer of spin to the holes is then relaxed by the impurity scatteringwithin the ferromagnet. By contrast, the opposite limit is dominated by the breathing Fermi-surface mechanism. In this mechanism, the spins of the itinerant particles are not able to follow the local magnetization direction adia- batically and lag behind with a delay time of T2. In our system, which has a large spin-orbit coupling in the band structure, we expect the spin- ip rate to be proportional to the mean free path ( l/T2).30The eective dimen- sionless Gilbert damping (4) is plotted as a function of disorder in Fig. 3. The damping ( mean) partly shows the same behavior as that reported in Ref. 15. For clean systems (i.e., those with a low spin- ip rate regime), the damping increases with disorder. In such a regime, the transfer of angular momentum to the spin 3/2 holes is the dominant damping process, i.e., the bulk spin-pumping process dominates. mean starts to decrease for smaller mean free paths, implying that the main contribution to the damping comes from the breathing Fermi-surface process. Refs. 11,16,17 have reported that the Gilbert damping may start to increase as a function of disorder in dirtier samples. The interband transitions become more important with decreasing quasi-particle life times and start to dominate the intraband transitions (The intra- band transitions give rise to the breathing Fermi-surface eect). We do not observe an increasing behavior in the more diusive regime, but we nd that the damping sat- urates at a value of around 0.0046 (See Fig. 3). In this regime, we believe that the breathing Fermi-surface eect is balanced by the interband transitions. The damping does not vanish in the limit 1 =l= 0 due to scattering at the interface between the GaAs and (Ga,Mn)As lay- ers in addition to spin-pumping into the adjacent leads (an interface spin-pumping eect, as explained above). Fig. 3 shows that the shape anisotropy of the damp- ing is reduced by disorder because the dierence between the maximum ( max) and minimum ( min) values of the damping parameter decrease with disorder. We antici- pate this result because disorder increases the bulk damp- ing eect, which is expected to be isotropic for an innite system. IV. SUMMARY In this paper, we studied the magnetization damping in the ferromagnetic semiconductor (Ga,Mn)As. The Gilbert damping was calculated numerically using a recently developed scattering matrix theory of mag- netization dissipation.8We conducted a detailed non- perturbative study of the eects of disorder and an inves- tigation of the damping anisotropy induced by the shape of the sample. Our analysis showed that the damping process is mainly governed by three relaxation mechanisms. In the clean limit with little disorder, we found that the magne- tization dissipation is dominated by spin-pumping from the d-electrons to the itinerant holes. For shorter mean free paths, the breathing Fermi-surface eect starts to5 dominate, which causes the damping to decrease. In the diusive regime, the breathing Fermi-surface eect is balanced by the interband transitions and the eective damping parameter saturates at a value on the order of 0.005. For the small samples considered in this study, we found that the shape of the system was typically more important than the anisotropic terms in the Hamiltonian for the directional dependency of the damping parame- ter. This shape anisotropy has not been reported beforeand oers a new way of manipulating the magnetization damping. V. ACKNOWLEDGMENTS This work was partially supported by the European Union FP7 Grant No. 251759 \MACALO". 1For a review, see D. C. Ralph and M. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008), and reference therein. 2B. Heinrich, D. Fraitov a, and V. Kambersky, Phys. Status Solidi 23, 501 (1967); V. Kambersky, Can. J. Phys. 48, 2906 (1970); V. Korenman and R.E. Prange, Phys. Rev. B6, 2769 (1972); V.S. Lutovinov and M.Y. Reizer, Zh. Eksp. Teor. Fiz. 77, 707 (1979) [Sov. Phys. JETP 50, 355 (1979)]; V.L. Safonov and H.N. Bertram, Phys. Rev. B 61, R14893 (2000); J. Kunes and V. Kambersky, Phys. Rev. B 65, 212411 (2002); V. Kambersky Phys. Rev. B 76, 134416 (2007). 3T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 4J.A.C. Bland and B. Heinrich, Ultrathin Magnetic Struc- tures III Fundamentals of Nanomagnetism (Springer Ver- lag, Heidelberg, 2004). 5V. Kambersky, Czech. J. Phys. B 26, 1366 (1976). 6K. Gilmore, Y.U. Idzerda, and M.D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). 7Y. Tserkovnyak, A. Brataas and G.E.W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 8A. Brataas, Y. Tserkovnyak, and G.E.W. Bauer, Phys. Rev. Lett. 101, 037207 (2008); Phys. Rev. B 84, 054416 (2011). 9A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010); Y. Liu, Z. Yuan, A. A. Starikov, and P. J. Kelly, arXiv:1102.5305. 10A similar analysis is presented in J. Seib, D. Steiauf, and M. F ahnle, Physical Review B 79, 092418 (2009). 11J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J.K. Furdyna, W.A. Atkinson, and A.H. MacDonald, Phys. Rev. B 69, 085209 (2004). 12Y.H. Matsuda, A. Oiwa, K. Tanaka, and H. Munekata, Physica B 376-377 , 668 (2006). 13A. Wirthmann et al. , Appl. Phys. Lett. 92, 232106 (2008). 14Kh. Khazen et al. , Phys. Rev. B 78, 195210 (2008). 15Y. Tserkovnyak, G.A. Fiete, and B.I. Halperin, Appl. Phys. Lett. 84, 5234 (2004). 16I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064403(2009). 17I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064404 (2009). 18This shape anisotropy in the Gilbert damping should not be confused with the shape anisotropy (in the anisotropy eld) caused by surface dipoles in non-spherical systems. 19T. Jungwirth, J. Sinova, J. Ma sek, J. Ku cera, and A. H. MacDonald, Rev. Mod. Phys. 78, 809864 (2006). 20M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDon- ald, Phys. Rev. B 63, 054418 (2001). 21P.Y. Yu and M. Cardona, Fundamentals of Semicon- ductors: Physics and Materials Properties , 3rd Edition (Springer Verlag, Berlin, 2005). 22Note that in Eq. (6) the spin operator s(in the p-d ex- change term) is represented in the basis consisting of the four spin 3/2 states ( s=J=3). The factor 1 =3 is absorbed in the exchange eld h. 23In the discrete version of Eq. (6), as used in the numerical calculation, we have one impurity at each lattice site. 24A. Chernyshov, M. Overby, X. Liu, J.K. Furdyna, Y. Lyanda-Geller, and L.P. Rokhinson, Nature Physics 5, 656 (2009). 25A. Baldereschi and N.O. Lipari, Phys. Rev. B 8, 2697 (1973). 26The prefactor of 10 comes from 4 Ga atoms per unit cell times spin 5/2 per substitutional Mn, which are assumed to be fully polarized. The reduction of the net magnetization due to the interstitial Mn ions and p holes are disregarded in our estimate. 27S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, England, 1995). 28T. Usuki, M. Saito, M. Takatsu, R. A. Kiehl, and N. Yokoyama, Phys. Rev. B 52, 8244 (1995). 29D. Steiauf and M. F ahnle, Physical Review B 72, 064450 (2005). 30The spin relaxation time of holes in GaAs is on the scale of the momentum relaxation time. See D.J. Hilton and C.L. Tang, Phys. Rev. Lett. 89, 146601 (2002), and references therein.

2011-05-20

We compute the Gilbert damping in (Ga,Mn)As based on the scattering theory of magnetization relaxation. The disorder scattering is included non-perturbatively. In the clean limit, the spin-pumping from the localized d-electrons to the itinerant holes dominates the relaxation processes. In the diffusive regime, the breathing Fermi-surface effect is balanced by the effects of interband scattering, which cause the Gilbert damping constant to saturate at around 0.005. In small samples, the system shape induces a large anisotropy in the Gilbert damping.

Magnetization Dissipation in the Ferromagnetic Semiconductor (Ga,Mn)As

1105.4148v2

Real-space nonlocal Gilbert damping from exchange torque correlation applied to bulk ferromagnets and their surfaces Balázs Nagyfalusi,1,2,∗László Szunyogh,2,3,†and Krisztián Palotás1,2,‡ 1Institute for Solid State Physics and Optics, HUN-REN Wigner Research Center for Physics, Konkoly-Thege M. út 29-33, H-1121 Budapest, Hungary 2Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary 3HUN-REN-BME Condensed Matter Research Group, Budapest University of Technology and Economics, Budafoki út 8, H-1111 Budapest, Hungary (Dated: February 29, 2024) In this work we present an ab initio scheme based on linear response theory of exchange torque correlation, implemented into the real-space Korringa-Kohn-Rostoker (RS-KKR) framework to cal- culate diagonal elements of the atomic-site-dependent intrinsic Gilbert damping tensor. The method is first applied to bcc iron and fcc cobalt bulk systems. Beside reproducing earlier results from the literature for those bulk magnets, the effect of the lattice compression is also studied for Fe bulk, and significant changes for the Gilbert damping are found. Furthermore, (001)-oriented surfaces of Fe and Co are also investigated. It is found that the on-site Gilbert damping increases in the surface atomic layer and decreases in the subsurface layer, and approaches the bulk value moving further inside the magnets. Realistic atomic relaxation of the surface layers enhances the identified effects. Thefirst-neighbordampingparametersareextremelysensitivetothesurfacerelaxation. De- spite their inhomogeneity caused by the surface, the transverse Gilbert damping tensor components remain largely insensitive to the magnetization direction. I. INTRODUCTION It is highly demanded to understand and control the dynamical processes governing the manipulation of various magnetic textures, such as atomic chains1,2, magnetic skyrmions3,4or domain walls5, which can be potentially used in future magnetic recording and logic devices. These processes are often described by the phenomenological Landau-Lifshitz-Gilbert (LLG) equation6,7, ∂ ⃗ mi ∂t=−γ ⃗ mi×⃗Beff i+α mi⃗ mi×∂ ⃗ mi ∂t,(1) where ⃗ miis the magnetic moment at site i,mi=|⃗ mi| is its length, and γis the gyromagnetic ratio. The firsttermonthe rhsofEq.(1)describestheprecession of⃗ miaround the effective magnetic field ⃗Beff i, while the second term is the Gilbert damping due to the energy dissipation to the lattice. Clearly, this latter term causes the relaxation of the magnetization to its equilibrium value, which is controlled by the damping constant αand plays a crucial role in the realization of high-speed spintronic devices. The Gilbert damping constant αcan be deter- mined experimentally from the ferromagnetic reso- nance (FMR) spectroscopy where the damping pa- rameter is related to the line-width in the measured spectra8. FMR spectroscopy is a well-established method for bulk materials9,10, but especially in the low temperature measurement it is controversial be- cause the intrinsic Gilbert damping needs to be sepa- rated from various extrinsic sources of the line-width, e.g., two-magnon scattering, eddy-current damping, radiative damping, spin-pumping, or the slow relaxer mechanism11–16. The comparison of experimental measurement to theoretical calculations is also made difficult bythe sampleproperties likethe exactatomic structure.From a theoretical perspective the ultimate goal is to develop a method to calculate the Gilbert damp- ing parameters from the electronic structure of the material. In the last decades there have been sev- eral efforts to understand the damping process. The first successful method was developed by Kamberský whorelatedthedampingprocesstothespin-orbitcou- pling (SOC) in terms of the breathing Fermi surface model17, while he also proposed the spin-orbit torque correlation model18,19. Later on several other meth- ods were introduced such as the spin-pumping20and linear-response approaches11,21,22. A recent summary of these methods was published by Guimarães et al.23 Due to the increased interest in noncollinear mag- netism Fähnle et al.24suggested an inhomogeneous tensorial damping. The replacement of a scalar αby a damping matrix αmeans that the damping field in Eq.(1)isnolongerproportionaltothetimederivative of⃗ mi, it becomes a linear function of ∂ ⃗ mi/∂t. More- over, nonlocality of the damping process implies that the damping field at site iexperiences ∂ ⃗ mj/∂tfor any sitej. The LLG equation (1) is then replaced by the set of equations25, ∂ ⃗ mi ∂t=⃗ mi× −γ⃗Beff i+X jαij1 mj∂ ⃗ mj ∂t ,(2) where the damping term is unfolded to pairwise con- tributions of strength αij. The appearance of non- local damping terms was evidenced for magnetic do- main walls26,27by linking the Gilbert damping to the gradients of the magnetization. In NiFe, Co, and CoFeB thin films Li et al.28measured wave-number- dependent dissipation using perpendicular spin wave resonance, validating thus the idea of nonlocal damp- ing terms. Different analytical expressions for αijare already proposed22,25,29,30, and the nonlocal damp- ing is found for bulk materials25,31as well as its ef-arXiv:2401.09938v2 [cond-mat.mtrl-sci] 28 Feb 20242 fect on magnon properties of ferromagnets have been discussed32. Recent studies went further and, anal- ogously to the higher order spin-spin interactions in spin models, introduced multi-body contributions to the Gilbert damping33. The calculation of the Gilbert damping prop- erties of materials has so far been mostly fo- cused on 3D bulk magnets, either in chemically homogeneous11,19,23,25,34–36or heterogeneous (e.g. alloyed)11,22,31forms. There are a few studies avail- able reporting on the calculation of the Gilbert damp- ing in 2D magnetic thin films12,23,37,38, or at surfaces and interfaces of 3D magnets31,35,37. The calculation of the Gilbert damping in 1D or 0D magnets is, due to our knowledge, not reported in the literature. Fol- lowing the trend of approaching the atomic scale for functional magnetic elements in future spintronic de- vices, the microscopic understanding of energy dissi- pation through spin dynamics in magnets of reduced dimensions is inevitable and proper theoretical meth- ods have to be developed. Our present work proposes a calculation tool for thediagonalelementsofthenon-localintrinsicGilbert damping tensor covering the 3D to 0D range of mag- netic materials on an equal footing, employing a real- space embedding Green’s function technique39. For this purpose, the linear response theory of the Gilbert damping obtained by the exchange torque correlation is implemented in the real-space KKR method. As a demonstration of the new method, elemental Fe and Co magnets in their 3D bulk form and their (001)- oriented surfaces are studied in the present work. Go- ing beyond comparisons with the available literature, new aspects of the Gilbert damping in these materials are also reported. The paper is organized as follows. In Sec. II the calculation of the Gilbert damping parameters within the linear response theory of exchange torque corre- lation using the real-space KKR formalism is given. Sec. III reports our results on bulk bcc Fe and fcc Co materials and their (001)-oriented surfaces. We draw our conclusions in Sec. IV. II. METHOD A. Linear response theory within real-space KKR The multiple-scattering of electrons in a finite clus- ter consisting of NCatoms embedded into a 3D or 2D translation-invariant host medium is fully accounted for by the equation39 τC=τH I−(t−1 H−t−1 C)τH−1,(3) where τCand τHare the scattering path operator matrices of the embedded atomic cluster and the host, respectively, tCandtHare the corresponding single- site scattering matrices, all in a combined atomic site (j, k∈ {1, ..., N C}) and angular momentum ( Λ,Λ′∈ {1, ...,2(ℓmax+ 1)2}) representation: τ={τjk}= {τjk ΛΛ′}andt={tj ΛΛ′δjk}, where ℓmaxis the angularmomentum cutoff in describing the scattering events, and for simplicity we dropped the energy-dependence of the above matrices. For calculating the diagonal Cartesian elements of the nonlocal Gilbert damping tensor connecting atomic sites jandkwithin the finite magnetic atomic cluster, we use the formula derived by Ebert et al.22, αµµ jk=2 πmj sTr Tj µ˜τjk CTk µ˜τkj C , (4) where µ∈ {x, y, z}, the trace is taken in the angular-momentum space and the formula has to be evaluated at the Fermi energy ( EF). Here, mj sis the spin moment at the atomic site j, ˜τjk C,ΛΛ′= (τjk C,ΛΛ′−(τkj C,Λ′Λ)∗)/2i, and Tj µis the torque operator matrix which has to be calculated within the volume of atomic cell j,Ωj:Tj µ;ΛΛ′=R Ωjd3rZj Λ(⃗ r)×βσµBxc(⃗ r)Zj Λ′(⃗ r),wherethenotationof the energy-dependence is omitted again for simplicity. Here, βis a standard Dirac matrix entering the Dirac Hamiltonian, σµare Pauli matrices, and Bxc(⃗ r)is the exchange-correlation field in the local spin density ap- proximation (LSDA), while Zj Λ(⃗ r)are right-hand side regular solutions of the single-site Dirac equation and the superscript ×denotes complex conjugation re- stricted to the spinor spherical harmonics only22. We should emphasize that Eq. (4) applies to the diagonal (µµ) elements of the Gilbert tensor only. To calcu- late the off-diagonal tensor elements one needs to use, e.g., the more demanding Kubo-Bastin formula40,41. Note also that in noncollinear magnets the exchange field Bxc(⃗ r)is sensitive to the spin noncollinearity42 which influences the calculated torque operator ma- trix elements, however, this aspect does not concern our present study including collinear magnetic states only. Note that the nonlocal Gilbert damping is, in gen- eral, not symmetric in the atomic site indices, αµµ jk̸= αµµ kj, instead αµµ kj=mj s mksαµµ jk(5) holds true. This is relevant in the present work for the ferromagnetic surfaces. On the other hand, in ferromagnetic bulk systems αµµ jk=αµµ kjsince mj s= mk s=msfor any pair of atomic sites. In practice, the Gilbert damping formula in Eq. (4) is not directly evaluated at the Fermi energy, but a small imaginary part ( η) of the complex energy is ap- plied, which is called broadening in the following, and its physical effect is related to the scattering rate in other damping theories19,25,37,43. Taking into account the broadening η, the Gilbert damping reads αµµ jk(η) =−1 4h ˜αµµ jk(E+, E+) + ˜αµµ jk(E−, E−) −˜αµµ jk(E+, E−)−˜αµµ jk(E−, E+)i ,(6) where E+=EF+iηandE−=EF−iη, and the3 individual terms are ˜αµµ jk(E1, E2) = 2 πmj sTr Tj µ(E1, E2)τjk C(E2)Tk µ(E2, E1)τkj C(E1) (7) with E1,2∈ { E+, E−}, and the ex- plicitly energy-dependent torque opera- tor matrix elements are: Tj µ;ΛΛ′(E1, E2) =R Ωjd3rZj× Λ(⃗ r, E1)βσµBxc(⃗ r)Zj Λ′(⃗ r, E2). B. Effective damping and computational parameters Eq.(6)givesthebroadening-dependentspatiallydi- agonal elements of the site-nonlocal Gilbert damping tensor: αxx jk(η),αyy jk(η), and αzz jk(η). Since no longitu- dinal variation of the spin moments is considered, the two transversal components perpendicular to the as- sumed uniform magnetization direction are physically meaningful. Given the bulk bcc Fe and fcc Co sys- tems and their (001)-oriented surfaces with C4vsym- metry under study in the present work, in the follow- ing the scalar αrefers to the average of the xxand yyGilbert damping tensor components assuming a parallel magnetization with the surface normal z[001]- direction: αjk= (αxx jk+αyy jk)/2 =αxx jk=αyy jk. From the site-nonlocal spatial point of view in this work we present results on the on-site (" 00"), first neighbor (denoted by " 01") and second neighbor (denoted by "02") Gilbert damping parameters, and an effective, so-called total Gilbert damping ( αtot), which can be defined as the Fourier transform of αjkat⃗ q= 0. The Fourier transform of the Gilbert damping reads α⃗ q=∞X j=0α0jexp(−i⃗ q(⃗ r0−⃗ rj)) ≈X r0j≤rmaxα0jexp(−i⃗ q(⃗ r0−⃗ rj)),(8) where r0j=|⃗ r0−⃗ rj|and the effective damping is defined as αtot=α⃗ q=⃗0=∞X j=0α0j≈X r0j≤rmaxα0j.(9) Since we have a real-space implementation of the Gilbert damping, the infinite summation for both quantities is replaced by an approximative summation for neighboring atoms upto an rmaxcutoff distance measured from site "0". Moreover, note that for bulk systems the effective damping αtotis directly related to the ⃗ q= 0mode of FMR experiments. The accuracy of the calculations depends on many numerical parameters such as the number of ⃗kpoints used in the Brillouin zone integration, the choice of the angular momentum cutoff ℓmax, and the spatial cutoff rmaxused for calculating α⃗ qandαtot. Previ- ous research25showed that the Gilbert damping heav- ily depends on the broadening η, so we extended ourstudies to a wider range of η= 1meV to 1 eV. The sufficient k-point sampling was tested at the distance ofrmax= 7a0(where a0is the corresponding 2D lat- tice constant) from the reference site with the broad- ening set to 1mRy, and the number of ⃗kpoints was increased up to the point, where the 5th digit of the damping became stable. Maximally, 320400 ⃗kpoints were used for the 2D layered calculation but the re- quested accuracy was reached with 45150 and 80600 ⃗kpoints for bulk bcc Fe and fcc Co systems, respec- tively. The choice of ℓmaxwas tested through the whole η range for bcc Fe, and it was based on the comparison of damping calculations with ℓmax= 2andℓmax= 3. The maximal deviation for the on-site Gilbert damp- ing was found at around η= 5mRy, but it was still less than 10%. The first and second neighbor Gilbert damping parameters changed in a more significant way (by ≈50%) in the whole ηrange upon changing ℓmax, yet the effective total damping was practically unchanged, suggesting that farther nonlocal damping contributions compensate this effect. Since αtotis the measurable physical quantity we concluded that the lower angular momentum cutoff of ℓmax= 2is suffi- cient to be used further on. The above choice of ℓmax= 2for the angular mo- mentumcutoff, themathematicalcriterionofpositive- definite αjk(which implies α⃗ q>0for all ⃗ qvectors), and the prescribed accuracy for the effective Gilbert damping in the full considered η= 1meV to 1 eV range set rmaxto 20 a0for both bcc Fe and fcc Co. It is worth mentioning that the consideration of lattice symmetries made possible to decrease the number of atomic sites in the summations for calculating α⃗ qand αtotby an order of magnitude. III. RESULTS AND DISCUSSION Our newly implemented method was employed to study the Gilbert damping properties of Fe and Co ferromagnetsintheirbulkand(001)-orientedsurfaces. In these cases only unperturbed host atoms form the atomic cluster, and the so-called self-embedding procedure44is employed, where Eq. (3) reduces to τC=τHforthe3Dbulkmetalsand2Dlayeredmetal- vacuum interfaces. A. Bulk Fe and Co ferromagnets First we calculate and analyze the nonlocal and ef- fective dampings for bulk bcc Fe by choosing a 2D lattice constant of a0= 2.863Å. The magnitude of the magnetic moments are obtained from the self- consistent calculation. The spin and orbital moments arems= 2.168µBandmo= 0.046µB, respectively. The broadening is set to η= 68meV. The inset of Fig. 1a) shows the typical function of the nonlocal Gilbert damping α0jdepending on the normalized distance r0j/a0between atomic sites "0" and " j". In accor- dance with Ref. 25 the nonlocal Gilbert damping quickly decays to zero with the distance, and can be4 a) 5 10 15 2005 5 10 15 20−505 r0j/a0α0j[×10−4] r0j/a0α0j·(r0j/a0)2[×10−4] b) 5 10 15 20−202468 rmax/a0αtot[×10−3] FIG. 1. a) Nonlocal Gilbert damping in bulk bcc Fe as a function of distance r0jbetween atomic sites "0" and " j" shown upto a distance of 20 a0(the 2D lattice constant is a0= 2.863Å): the black squares are calculated α0jval- ues times the normalized squared-distance along the [110] crystallographic direction, and the red line is the corre- sponding fitted curve based on Eq. (10). The inset shows the nonlocal Gilbert damping α0jvalues in the given dis- tance range. b) Convergence of the effective damping pa- rameter αtot, partial sums of α0jupto rmaxbased on Eq. (9), where rmaxis varied. The broadening is chosen to be η=68 meV. well approximated with the following function: α(r)≈Asin (kr+ϕ0) r2exp(−βr).(10) To test this assumption we assorted the atomic sites lying in the [110] crystallographic direction and fit- ted Eq. (10) to the calculated data. In practice, the fit is made on the data set of α0j(r0j/a0)2, and is plotted in Fig. 1a). Although there are obvious out- liers in the beginning, the magnitude of the Gilbert damping asymptotically follows the ∝exp(−βr)/r2 distance dependence. The physical reason for this de- cay is the appearance of two scattering path operators (Green’s functions) in the exchange torque correlation formula in Eq. (4) being broadened due to the finite imaginary part of the energy argument. In our real-space implementation of the Gilbert damping, an important parameter for the effective damping calculation is the real-space cutoff rmaxin Eq. (9). Fig. 1b) shows the evolution of the ef- fective (total) damping depending on the rmaxdis- tance, within which all nonlocal damping terms α0jare summed up according to Eq. (9). An oscillation can similarly be detected as for the nonlocal damping itself in Fig. 1a), and this behavior was fitted with a similar exponentially decaying oscillating function as reported in Eq. (10) in order to determine the ex- pected total Gilbert damping αtotvalue in the asymp- totic r→ ∞limit. In the total damping case it is found that the spatial decay of the oscillation is much slower compared to the nonlocal damping case, which makes the evaluation of αtotmore cumbersome. Our detailed studies evidence that for different broaden- ingηvalues the wavelength of the oscillation stays the same but the spatial decay becomes slower as the broadening is decreased (not shown). This slower decay together with the fact that the effective (to- tal) damping value itself is also decreasing with the decreasing broadening results that below the 10meV range of ηthe amplitude of the oscillation at the dis- tance of 20 a0is much larger than its asymptotic limit. In practice, since the total damping is calculated as ther→ ∞limit of such a curve as shown in Fig. 1b), this procedure brings an increased error for αtotbelow η= 10meV, and this error could only be reduced by increasing the required number of atomic sites in the real-space summation in Eq. (10). Fig. 2 shows the dependence of the calculated on- site, first- and second-neighbor and effective total Gilbert damping parameters on the broadening η. The left column shows on-site ( α00) and total ( αtot) while the right one the first ( α01) and second ( α02) neighbor Gilbert dampings. We find very good agree- ment with the earlier reported results of Thonig et al.25, particularly that the on-site damping has the largest contribution to the total damping being in the same order of magnitude, while the first and second neighbors are smaller by an order of magnitude. The obtained dependence on ηis also similar to the one published by Thonig et al.25:α00andαtotare in- creasing with η, and α01andα02do not follow a common trend, and they are material-dependent, see, e.g., the opposite trend of α02with respect to ηfor Fe and Co. The observed negative values of some of thesite-nonlocaldampingsarestillconsistentwiththe positive-definiteness of the full (infinite) αjkmatrix, which has also been discussed in Ref. 25. The robustness of the results was tested against a small change of the lattice constant simulating the ef- fect of an external pressure for the Fe bulk. These re- sults are presented in the second row of Fig. 2, where the lattice constant of Fe is set to a0= 2.789Å. In this case the magnetic moments decrease to ms= 2.066µB andmo= 0.041µB. It can clearly be seen that the on- site, first and second neighbor Gilbert dampings be- come smaller upon the assumed 2.5% decrease of the lattice constant, but the total damping remains prac- tically unchanged in the studied ηrange. This sug- gests that the magnitudes of more distant non-local damping contributions are increased. The third row of Fig. 2 shows the selected damp- ing results for fcc Co with a 2D lattice constant of a0= 2.507Å. The spin and orbital moments are ms= 1.654µBandmo= 0.078µB, respectively. The increaseofthetotal, theon-site, andthefirst-neighbor5 10−310−210−11000246810Fe -a0= 2.863˚A η(eV)α[×10−3]α00 αtot 10−310−210−1100−202468Fe -a0= 2.863˚A η(eV)α[×10−4]α01 α02 10−310−210−11000246810Fe -a0= 2.789˚A η(eV)α[×10−3]α00 αtot 10−310−210−1100−202468Fe -a0= 2.789˚A η(eV)α[×10−4]α01 α02 10−310−210−11000246810Co -a0= 2.507˚A η(eV)α[×10−3]α00 αtot 10−310−210−1100−202468Co -a0= 2.507˚A η(eV)α[×10−4]α01 α02 FIG. 2. Left column: Local on-site ( α00, black square) and total ( αtot, red triangle) Gilbert damping as a func- tionofthebroadening ηforbccFe(001)with a0= 2.863Å, bcc Fe(001) with a0= 2.789Å, and fcc Co(001) with a0= 2.507Å. Right column: Nonlocal first nearest neigh- bor (α01, black square) and second nearest neighbor ( α02, red triangle) Gilbert damping for the same systems. dampings with increasing ηis similar to the Fe case, and the on-site term dominates αtot. An obvious difference is found for the second-neighbor damping, which behaves as an increasing function of ηfor Co unlike it is found for Fe. Concerning the calculated damping values, there is a large variety of theoretical methods and calculation parameters, as well as experimental setups used in the literature, which makes ambiguous to compare our results with others. Recently, Miranda et al.31 reported a comparison of total and on-site damping values with the available theoretical and experimen- tal literature in their Table S1. For bcc Fe bulk they reported total damping values in the range of 1.3– 4.2×10−3and for fcc Co bulk within the range of 3.2– 11×10−3, and our results fit very well within theseranges around η≈100meV for Fe and for η >100 meV for Co. Moreover, we find that our calculated on- site damping values for bcc Fe are larger ( >5×10−3) than the reported values of Miranda et al.(1.6×10−3 and 3.6 ×10−3), but for fcc Co the agreement with their reported total (3.2 ×10−3) and on-site damping (5.3×10−3) values is very good at our η= 136meV broadening value. 10−310−210−110010−510−410−310−2Fe η(eV)αtot αSOC=1 αSOC=0 10−310−210−110010−510−410−310−2Co η(eV)αtot αSOC=1 αSOC=0 FIG. 3. Effective (total) Gilbert damping for bcc Fe (left) and fcc Co (right) as a function of broadening ηon a log-log scale. The error bars are estimated from the fitting procedure of Eq. (10). The red triangles show the case with normal SOC ( αSOC=1), and the blue diamonds where SOC is switched off ( αSOC=0). Next, weinvestigatethespin-orbit-coupling-(SOC)- originated contribution to the Gilbert damping. Our methodmakesitinherentlypossibletoincludeaSOC- scaling factor in the calculations45. Fig. 3 shows the obtained total Gilbert damping as a function of the broadening ηwith SOC switched on/off for bcc Fe and fcc Co. It can be seen that the effect of SOC is not dominant at larger ηvalues, but the SOC has an important contribution at small broadening values ( η < 10−2eV), where the calculated total Gilbert damping values begin to deviate from each other with/without SOC. As discussed in Ref. 23, without SOC the damping should go toward zero for zero broadening, which is supported by our results shown in Fig. 3. B. (001)-oriented surfaces of Fe and Co ferromagnets In the following, we turn to the investigation of the Gilbertdampingparametersatthe(001)-orientedsur- faces of bcc Fe and fcc Co. Both systems are treated as a semi-infinite ferromagnet interfaced with a semi- infinite vacuum within the layered SKKR method46. In the interface region 9 atomic layers of the ferromag- net and 3 atomic layers of vacuum are taken, which is sandwiched between the two semi-infinite (ferromag- net and vacuum) regions. Two types of surface atomic geometries were calculated: (i) all atomic layers hav- ing the bulk interlayer distance, and (ii) the surface and subsurface atomic layers of the ferromagnets have6 TABLE I. Geometry relaxation at the surfaces of the fer- romagnets: change of interlayer distances relative to the bulk interlayer distance at the surfaces of bcc Fe(001) and fcc Co(001), obtained from VASP calculations. "L1" de- notes the surface atomic layer, "L2" the subsurface atomic layer, and "L3" the sub-subsurface atomic layer. All other interlayer distances are unchanged in the geometry opti- mizations. L1-L2 L2-L3 bcc Fe(001) -13.7% -7.7% fcc Co(001) -12.4% -6.4% beenrelaxedintheout-of-planedirectionusingtheVi- enna Ab-initio Simulation Package (VASP)47within LSDA48. For the latter case the obtained relaxed atomic geometries are given in Table I. Figure 4 shows the calculated layer-resolved on- site and first-neighbor Gilbert damping values (with η= 0.68eV broadening) for the bcc Fe(001) and fcc Co(001) surfaces. It can generally be stated that the surface effects are significant in the first 4 atomic lay- ers of Fe and in the first 3 atomic layers of Co. We find that the on-site damping ( α00) increases above the bulk value in the surface atomic layer (layer 1: L1), and decreases below the bulk value in the sub- surface atomic layer (L2) for both Fe and Co. This finding is interesting since the spin magnetic moments (ms, shown in the insets of Fig. 4) are also consider- ably increased compared to their bulk values in the surface atomic layer (L1), and the spin moment enters the denominator when calculating the damping in Eq. (4).α00increases again in L3 compared to its value in L2, thus it exhibits a nonmonotonic layer-dependence in the vicinity of the surface. The damping results ob- tained with the ideal bulk interlayer distances and the relaxed surface geometry ("R") are also compared in Fig. 4. It can be seen that the on-site damping is in- creasedinthesurfaceatomiclayer(L1), anddecreased in the subsurface (L2) and sub-subsurface (L3) atomic layers upon atomic relaxation ("R") for both Fe and Co. The first-neighbor dampings ( α01) are of two types for the bcc Fe(001) and three types for the fcc Co(001), see caption of Fig. 4. All damping values are approaching their corresponding bulk value mov- ing closer to the semi-infinite bulk (toward L9). In absolute terms, for both Fe and Co the maximal sur- face effect is about 10−3for the on-site damping, and 2×10−4for the first-neighbor dampings. Given the damping values, the maximal relative change is about 15% for the on-site damping, and the first-neighbor dampings can vary by more than 100% (and can even changesign)inthevicinityofthesurfaceatomiclayer. Note that Thonig and Henk35studied layer-resolved (effective) damping at the surface of fcc Co within the breathing Fermi surface model combined with a tight- binding electronic structure approach. Although they studied a different quantity compared to us, they also reported an increased damping value in the surface atomic layer, followed by an oscillatory decay toward bulk Co. So far the presented Gilbert damping results cor- respond to spin moments pointing to the crystallo- 1 3 5 7 90.81Fe 1 92.43 layermsms mR s layerα00[×10−2] α00 αR 00 1 3 5 7 90.81 1 91.71.8 layermsms mR sCo layerα00[×10−2]α00 αR 00 1 3 5 7 9−2−101Fe layerα01[×10−4] α01+αR 01+ α01−αR 01− 1 3 5 7 91234 Co layerα01[×10−4]α01+αR 01+ α01−αR 01− α01 αR 01FIG. 4. Evolution of the layer-resolved Gilbert damping from the surface atomic layer (L1) of bcc Fe(001) and fcc Co(001) toward the bulk (L9), depending also on the out- of-plane atomic relaxation "R". On-site ( α00) and first neighbor ( α01) Gilbert damping values are shown in the top two and bottom two panels, respectively. The broad- ening is η= 0.68eV. The empty symbols belong to the calculations with the ideal bulk interlayer distances, and the full symbols to the relaxed surface geometry, denoted with index "R". Note that α01is calculated for nearest neighbors of atomic sites in the neighboring upper, lower, and the same atomic layer (for fcc Co only), and they are respectively denoted by " +" (L-(L+1)), " −" (L-(L −1)), and no extra index (L-L). The insets in the top two panels show the evolution of the magnitudes of the layer-resolved spin magnetic moments ms. The horizontal dashed line in all cases denotes the corresponding bulk value. graphic [001] ( z) direction, and the transverse compo- nents of the damping αxxandαyyare equivalent due tothe C4vsymmetryofthe(001)-orientedsurfaces. In order to study the effect of a different orientation of all spin moments on the transverse components of the damping, we also performed calculations with an ef- fective field pointing along the in-plane ( x) direction: [100] for bcc Fe and [110] for fcc Co. In this case, due to symmetry breaking of the surface one expects an anisotropy in the damping, i.e., that the transverse components of the damping tensor, αyyandαzz, are not equivalent any more. According to our calcula-7 tions, however, the two transverse components of the on-site ( αyy 00andαzz 00) and nearest-neighbor ( αyy 01and αzz 01) damping tensor, at the Fe surface differed by less than 0.1 % and at the Co surface by less than 0.2 %, i.e., despite the presence of the surface the damping tensor remained highly isotropic. The change of the damping with respect to the orientation of the spin moments in zorxdirection (damping anisotropy) turned out to be very small as well: the relative dif- ference in αyy 00was 0.1 % and 0.3 %, while 0.5 % and 0.1 % in αyy 01for the Fe and the Co surfaces, respec- tively. For the farther neighbors, this difference was less by at least two orders of magnitude. IV. CONCLUSIONS We implemented an ab initio scheme of calculat- ing diagonal elements of the atomic-site-dependent Gilbert damping tensor based on linear response the- ory of exchange torque correlation into the real-space Korringa-Kohn-Rostoker (KKR) framework. To val- idate the method, damping properties of bcc Fe and fcc Co bulk ferromagnets are reproduced in good com- parison with the available literature. The lattice com- pression is also studied for Fe bulk, and important changes for the Gilbert damping are found, most pro- nounced for the site-nonlocal dampings. By investi- gating (001)-oriented surfaces of ferromagnetic Fe andCo, we point out substantial variations of the layer- resolved Gilbert damping in the vicinity of the sur- faces depending on various investigated parameters. The effect of such inhomogeneous dampings should be included into future spin dynamics simulations aim- ing at an improved accuracy, e.g., for 2D surfaces and interfaces. We anticipate that site-nonlocal damping effects become increasingly important when moving toward physical systems with even more reduced di- mensions (1D). ACKNOWLEDGMENTS The authors acknowledge discussions with Danny Thonig. Financial support of the National Research, Development, and Innovation (NRDI) Office of Hun- gary under Project Nos. FK124100 and K131938, the János Bolyai Research Scholarship of the Hungar- ian Academy of Sciences (Grant No. BO/292/21/11), the New National Excellence Program of the Min- istry for Culture and Innovation from NRDI Fund (Grant No. ÚNKP-23-5-BME-12), and the Hungarian State Eötvös Fellowship of the Tempus Public Foun- dation (Grant No. 2016-11) are gratefully acknowl- edged. Further support was provided by the Ministry of Culture and Innovation of Hungary from the NRDI Fund through the grant no. TKP2021-NVA-02. ∗nagyfalusi.balazs@ttk.bme.hu †szunyogh.laszlo@ttk.bme.hu ‡palotas.krisztian@wigner.hun-ren.hu 1B. Újfalussy, B. Lazarovits, L. Szunyogh, G. M. Stocks, and P. Weinberger, Phys. Rev. B 70, 100404(R) (2004). 2C. Etz, L. Bergqvist, A. Bergman, A. Taroni, and O. Eriksson, Journal of Physics: Condensed Matter 27, 243202 (2015). 3J. Iwasaki, M. Mochizuki, and N. 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2024-01-18

In this work we present an ab initio scheme based on linear response theory of exchange torque correlation, implemented into the real-space Korringa-Kohn-Rostoker (RS-KKR) framework to calculate diagonal elements of the atomic-site-dependent intrinsic Gilbert damping tensor. The method is first applied to bcc iron and fcc cobalt bulk systems. Beside reproducing earlier results from the literature for those bulk magnets, the effect of the lattice compression is also studied for Fe bulk, and significant changes for the Gilbert damping are found. Furthermore, (001)-oriented surfaces of Fe and Co are also investigated. It is found that the on-site Gilbert damping increases in the surface atomic layer and decreases in the subsurface layer, and approaches the bulk value moving further inside the magnets. Realistic atomic relaxation of the surface layers enhances the identified effects. The first-neighbor damping parameters are extremely sensitive to the surface relaxation. Despite their inhomogeneity caused by the surface, the transverse Gilbert damping tensor components remain largely insensitive to the magnetization direction.

Real-space nonlocal Gilbert damping from exchange torque correlation applied to bulk ferromagnets and their surfaces

2401.09938v2

Giant oscillatory Gilbert damping in superconductor/ferromagnet/superconductor junctions Authors Yunyan Yao1,2†, Ranran Cai1,2†, Tao Yu3, Yang Ma1,2, Wenyu Xing1,2, Yuan Ji1,2, Xin -Cheng Xie1,2,4,5, See -Hun Yang6*, and Wei Han1,2* Affiliations 1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China. 2Collaborative Innovation Center of Quantum Matter, Beijing 100871, China. 3Max Planck Institute for the Structure and Dynamics of Matte r, 22761 Hamburg, Germany 4CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, P. R. China 5Beijing Academy of Quantum Information Sciences, Beijing 100193, P. R. China 6IBM Research - Almaden, San Jose, California 95120, USA †These authors contributed equally to the work *Correspondence to: weihan@pku.edu.cn (W.H.); seeyang@us.ibm.com (S.H.Y.) . Abstract Interfaces between materials with differently ordered phases present unique opportunities for exotic physical properties, especially the interplay between ferromagnetism and superconductivity in the ferromagnet/superconductor heterostructures. The investig ation of zero- and π-junctions has been of particular interest for both fundamental physical science and emerging technologies. Here, we report the experimental observation of giant oscillatory Gilbert damping in the superconducting Nb/NiFe/Nb junctions wi th respect to the NiFe thickness. This observation suggests an unconventional spin pumping and relaxation via zero-energy Andreev bound states that exist only in the Nb/NiFe/Nb π-junctions, but not in the Nb/NiFe/Nb zero-junctions. Our findings could be important for further exploring the exotic physical properties of ferromagnet/superconductor heterostructures, and potential applications of ferromagnet π-junctions in quantum computing, such as half -quantum flux qubits. 2 One sentence summary: Giant oscillat ory Gilbert damping is observed in superconductor/ferromagnet/superconductor junctions with varying the ferromagnet thickness. Introduction The interplay between ferromagnetism and superconductivity has induced many exotic and exciting physical properties in ferromagnet (FM)/superconductor (SC) heterostructures (1-3). Of particular interest is the unconventional π-phase ground state SC/FM/SC junction that might be realized for cert ain FM thicknesses arising from the quantum intermixing of the wave functions between spin -singlet Cooper pairs in SC and spin -polarized electrons in FM (1, 3, 4). At the FM/SC interface, a Cooper pair moving into the FM will ha ve a finite center -of-mass momentum, resulting in the oscillation of the real part of superconducting order parameter (Re {Ψ}) with respect to the FM thickness (Fig. 1A ) (1, 5, 6). Depending on the FM thicknesses, the Cooper pair wavefunctions in the two superconductors on either side of the FM can have a phase difference from zero or π, forming so -called zero-junctions with positive Josephson coupling (Fig. 1B ) or π-junctions with the negative Josephson coupling (Fig. 1C) . The FM π-junctions can be used for quantum computing applications (7, 8), as half quantum flux qubits (9). Due to the scientific and technical importance, the research on the FM π-junctions has been active for the last tw o decades (6, 10-13). Previous experimental studies have demonstrated the switching between zero- and π-junctions in SC/FM/SC structures by varying the temperature and the FM thickness (11, 14-17). These reports mainly focus on the electrical properties of the FM zero- and π-junctions. Recently, dynamic spin injection into SCs has attracted considerable interest both the exper imentally (18-21) and theoretically (22-26). However, the spin -dependent properties in FM zero- and π-junctions have not been explored yet. The investigation of the spin -dependent properties requires the spin current probes, such as the dynamical spin pumping (27). Furthemore, for the application of the FM π- junctio ns in quantum computing technologies (9), the magnetization/spin dynamic properties are extremely important to be studi ed. Here, we report the experimental observation of giant oscillatory Gilbert damping in the superconducting Nb/NiFe/Nb junctions with respect to the NiFe thickness, which can be 3 qualitatively explained by the different spin pumping efficiency via the Andr eev bound states (ABS) of Nb/NiFe/Nb zero- and π-junctions. Using a minimal model based on the ABS, we show that an unconventional spin pumping into the zero -energy ABS penetrat ed into SCs could occur s only for the π-junctions, which can lead to the oscillatory Gilbert damping as a function of the NiFe thick ness. Results Figures 1D and 1E show the schematic s of the spin pumping, magnetization dynamics, and enhanced Gilbert damping in the SC/FM/SC zero- and π-junction s. Spin pumping refers to the spin-polarized current injection to non -magnetic materials from a FM with precessing magnetization around its ferromagnetic resonance (FMR) conditions (28, 29). In FM and its heterostructures, the Gilbert damping ( ) characterizes the magnetization dynamics , as described by the Landau -Lifshitz -Gilbert formula with an additiona l Slonczewski -torque term (30-32): 𝑑𝒎 𝑑𝑡=−𝛾𝒎×𝑯𝒆𝒇𝒇+𝛼𝒎×𝑑𝒎 𝑑𝑡+𝛾 𝑀𝑠𝑉(ℏ 4𝜋𝑔↑↓𝒎×𝑑𝒎 𝑑𝑡) (1) where 𝒎=𝑴/|𝑴| is the magnetization unit vector, 𝛾 is the gyromagnetic ratio, 𝑯𝒆𝒇𝒇 is the total effective magnetic field, 𝑀𝑠=|𝑴| is the saturation magnetization, and 𝑔↑↓ is the interface spin mixing conductance. The pumped spin current from FM into SCs can be expressed by Js = ℏ 4𝜋𝑔↑↓𝒎×𝑑𝒎 𝑑𝑡 (29). The spin pumping into the SCs give rise to an enhanced Gilbert damping constant that is proportional to the spin pumping current (αsp~ J s) (29). Fig. 1E illustrates the pumped spin current mediated by the zero -energy ABS inside the superconducting gap in π- junctions , which will be discussed later in details. While for a zero-junction, the pumped spin current is mediated by the ABS near the superconducting gap (Fig. 1D). The ABS can be formed within the FM layer and then extended into the interface of SCs with the superconducting coherent length scale (33, 34). The SC/FM/SC junctions consist of a NiFe (Ni 80Fe20) layer (thickness: ~ 5 - 20 nm) sandwiched by two Nb layers (thickness: 100 nm) grown by magnetron sputt ering (see Methods and f ig. S1). To maximize the integrity of samples for a systematic study, more than tens of samples are grown in each run via rotation mask technique in a sputtering system, which is the same as in the previous study of the oscillatory exchange coupling in mag netic multilayer 4 structures (35). The Gilbert damping and spin pumping are measured by the ferromagnetic resonance (FMR) technique (see Methods for details) . Above the TC of Nb, spin pumping in the Nb/NiFe/Nb junctions leads to the spin accumulation in Nb near the interface, which can be described by the spin -dependent chemical potentials, as illustrated in Fig. 2 A. The Gilbert damping of NiFe in the Nb/NiFe/Nb junctions is determined from the microwave frequen cy-dependent FMR spectra ( fig. S2). A typical FMR curve with the Lorentzian fitting is shown in Fig. 2 B, from which the half linewidth (ΔH) can be obtained. The Gilbert damping can be extracted from the best linear -fitting curve of ΔH vs. f (Fig. 2 C). Figure 2D shows the NiFe thickness dependence of the Gilbert damping in the Nb/NiFe/Nb junctions measured at T = 10, 15, and 20 K, respectively. Interestingly, an oscillating feature of the Gilbert damping is observed as a function of 𝑑NiFe in the region of 𝑑NiFe < ~15 nm. This oscillating behavior can be attributed to the quantum -interference effect of angular momentum transfer between the local precessing magnetic moment and conduction electrons in thin NiFe that was theoretically predicted by Mills (36), but has not been experimentally reported yet. Above TC, the continuous energy bands of Nb, similar to the normal met al in the Mills theory, overlap with both spin-up and spin -down bands of NiFe at the interface, thus allowing the conducting electrons in NiFe to flip between the spin -down and spin -up states. As illustrated in the inset of Fig. 2 D, one spin-down electron scatters with the local magnetic moment and then flips to the spin -up polarization, giving rise to the angular momentum transfer between the spin -polarized electrons and the magnetic moment. Besides the change of angular momentum, the momentum of the elect ron also changes ( ∆𝑘), due to different Fermi vectors for spin -up (𝑘𝐹↑) and spin -down ( 𝑘𝐹↓) electrons with exchange splitting (Fig. 2 A). When the NiFe layer is thin enough to become comparable with 1 ∆𝑘, quantum -interference effect of the spin -polariz ed electrons shows up, which gives rise to the oscillating spin -transfer torque to the NiFe. When the NiFe thickness is 2𝑛𝜋/[𝑘F↑−𝑘F↓] (n is an integer), the matching of the quantum levels between the spin -up and spin-down electrons in NiFe induces smaller Gilbert damping. On the other hand, when the NiFe thickness is (2n+1)𝜋/[𝑘F↑−𝑘F↓], a larger Gilbert damping is induced. Consequently, th e Gilbert damping in the Nb/NiFe/Nb structures oscillates with a period of 2𝜋/[𝑘F↑−𝑘F↓] (Supplementary Materials S1) . Experimentally, an oscillating period ( λ) of ~ 1.8 nm is identified 5 (see the red dashed arrow in Fig. 2 D). At T = 50 K, the oscillating f eature disappears since the quantum -interference effect is smeared by thermal excitations ( fig. S3) . Next, we investigate the spin pumping and spin transfer torque of the Nb/NiFe/Nb junctions in the superconducting states below TC with a superconducting gap (Fig. 3 A). TC in the Nb/NiFe/Nb junctions is obtained from typical four -probe resistance measurement as a function of the temperature. A typical temperature -dependent resistance curve measured on the Nb/NiFe (12 nm)/Nb junction is shown in Fig. 3b, indicating the TC of ~ 8.6 K. As dNiFe changes, TC of the Nb/NiFe/Nb junctions exhibits little variat ion between ~ 8.4 and ~ 8.9 K ( fig. S4). Similar to the normal states of Nb, the Gilbert damping below TC is also obtained from the be st linear -fitting result of the half linewidth vs. frequency ( fig. S5). During the FMR measurement, TC varies a little (< 1 K) ( Fig. S 6). As the temperature decreases, 𝛼 decreases abruptly from ~ 0.012 to ~ 0.0036 across the TC (fig. 3C), which indicates the decrease of spin current injected into Nb due to the formation of superconducting gap below TC. This observation is consistent with previous reports on spin pumping into SCs where the spin current is mediated by Bogoliubov quasiparticles (18, 19, 37). As the tem perature decreases far below the TC, the quasiparticle population dramatically decreases, leading to reduced spin pumping and Gilbert damping. Remarkably, the oscillating amplitude of the Gilbert damping of the Nb/NiFe/Nb junctions as a function of the NiF e thickness is dramatically enhanced as the temperature decreases into the superconducting states of Nb (Fig. 3 D). At T = 4 K, the oscillating magnitude of the Gilbert damping constant is ~ 0.005 for the first three oscillations, which is comparable to the background value of ~ 0.006. The obtained Gilbert damping values are not affected by thermal cycles, and the large oscillating feature has been confirmed on a different set of samples. Such a giant oscillation of the Gilbert damping cannot be explained by spin pumping of Bogoliubov quasiparticle - mediated spin current in SCs. Since as the temperature decreases, the population of the Bogoliubov quasiparticles monotonically and rapidly decreases with an increase of the SC gap, which would lead to lower Gilber t damping and also smaller oscillation compared to the normal states. Note that the oscillating period of the Gilbert damping at T = 4 K is the same as that at T = 10 K that is supposed to be 2𝜋/[𝑘F↑−𝑘F↓] due to the quantum interference effect . Such oscillating period of 2𝜋/[𝑘F↑−𝑘F↓] is the also same as that of the zero- and π-phase ground states transitions in FM Josephson devices, which is equal to the coherence length in NiFe film of 2𝜋/[𝑘F↑−𝑘F↓] in the 6 ballistic regime (1, 11, 17), and √ℏ𝐷𝑑𝑖𝑓𝑓 ∕𝐸𝑒𝑥 in the diffusive regime ( 𝐷𝑑𝑖𝑓𝑓 is the diffusion coefficient, and 𝐸𝑒𝑥 is the exchange energy ). The observed oscillating period of ~ 1.8 nm in our study is similar to the zero - 𝜋 oscillating period measured in the NiFe Josephson junctions in the diffusive regime reported previously (11, 17). The Gilbert damping difference (∆𝛼) between the zero- and 𝜋-junctions is extracted as a function of NiFe thickness, as shown in Fig. 4 A. We assume the larger Gi lbert damping for the 𝜋- junctions and smaller value s for the zero-junctions, which will be discuss ed later in details . The thickness -dependent Gilbert damping of the zero- and 𝜋-junctions are expected to both behave as α ~ 1 ⁄ dNiFe (29). Hence, we can treat them separately, as illustrated by the guide lines in the inse t of Fig. 4A, and ∆𝛼 is obtained by subtracting the fitted 1/d curve for the expected zero-junctions (black dashed line) . Clearly, there is a pronounced oscillating feature of ∆𝛼 for the Nb/NiFe/Nb junctions with NiFe thickness from ~ 5 nm to ~ 11 nm. When the NiFe thickness is above ~ 11 nm, the oscillating feature of the Gilbert damping is largel y suppressed compared to thinner NiFe junctions. This feature might be associated wi th the strong Josephson coupling for thin NiFe junctions and the exponential decaying of the Josephson coupling as the NiFe thickness increases (11, 17). To confirm this, the Jose phson junctions are fabricated using the shadow mask technique, and a Josephson coupling is observed from the Nb/NiFe (5 nm and 10 nm)/Nb junctions (Supplementary Materials and fig. S7). Discussion Let us discuss the physical mechanism that induces the giant oscillating Gilbert damping in the following. Apart from the spin pumping via ABS discussed above (Fig. 1 D and 1E ), the spin current in SCs can also be mediated by Bogoliubov quasiparticles (fig. S8A) (18, 19, 22, 23, 38), spin-triplet pairs (fig. S 8B) (3). Regarding Bogoliubov quasiparticles, they populate around the edge of superconducting gap at elevated temperatures close to TC (39). As shown both theoretical and experimental studies, the enhanced Gilbert damping in the SC/FM/SC heterostructures happens around TC (18, 19, 22, 23, 38). As the temperature decreases down to 0.5 TC, the Bogoliubov quasiparticles are mostly frozen out, for which the spin pumping is forbidden that will no longer contribute to the enhanced Gilbert damping. Hence, the Bogoliubov quasi particles are very unlikely to account for our experimental results. Regarding the spin-triplet pairs, it has been shown in previous studies that the spin -triplet current under FMR conditions and spin triplet 7 correlations would be different for zero- and 𝜋-junctions (4, 38, 40, 41), which might result in different Gilbert damping theoretically. However, in our study, there are not spin sinks adjacent to the Nb layers , thus not allowing the spin -triplet Cooper pairs to be relaxed in the Nb. This is diffe rent from previous report on the Pt/SC/FM/SC/Pt heterostructures (20), where the Pt is used as the spin sink. Experimentally, as the temperature below TC, the Gilbert damping exhibits a monotonic decrease for the Nb/NiFe/Nb heterostructures (Fig. 3 C), which is different from the enhanced Gilbert damping due to spin -triplet pairs (20). Furthermore, no Josephson current in the Nb/NiFe/Nb heterostructures is observed i n Nb/NiFe (30 nm)/Nb junction ( fig. S7), which indicates the absence of long -range spin -triplet Josephson coupling. Both these experimental results indicate that the contribution from the spin -triplet pairs is not significant to the enhanced Gilbert damping in the superconducting Nb/NiFe/Nb junctions . To our best understanding, the most reasonable mechanism is the spin pumping via the ABS, which can qualitatively describe our experimental observation. Previous studies have demonstrated that the energy of ABS inside the superconducting gap depends on the superconducting -phase (42, 43). For the FMR measurement under open -circuit ed conditions , the inversion symmetry of the current -phase (𝜑) relationships is preserved (43-45). For 𝜋-junctions, there is a 𝜋-phase shift in the current -phase relationship curves compared to zero-junctions , i.e., the properties of 𝜑 = 0 of a 𝜋-junction is the same as those of 𝜑 = 𝜋 of a zero-junction. Since this 𝜋-phase shift is already taken into account by the FM exchange field , the ABS energy of the 𝜋- junctions can be obtained at 𝜑 = 0 in the ground states , which is similar to that of 𝜑 = 𝜋 of zero- junctions . For 𝜋-junctions, ABS is located around the zero-energy inside of the superconducting gap (Fig. 1D). The ABS could penetrate into the superconducting Nb films with scale of superconducting coherent length (~ 30 nm), which is evanescent to dissipate the spin angular moment um (25, 26, 44). As shown in Fig. 4 B, the transfer efficiency of spin angular momentum via the zero -energy ABS can lead to an enhanced Gilbert damping . Whileas, for zero-junctions, the distribution of the ABS is near the edge of the superconducting gap (Fig. 1C) , thus, the spin pumping effic iency is suppressed due to the reduced population of the ABS at low temperatures (Fig. 4 C). Furthermore , the oscil latory energy levels of the ABS between the zero- and 𝜋-junctions is also consistent with the density of states (DOS) oscillating in supercon ductors between the zero- and 𝜋-junctions (1, 6, 8 38, 44, 46). In consequence, as the NiFe thickness increas es, the oscillatory spin pumping efficiency via ABS at the FM/SC interface (or DOS in SC s) gives rise to the oscillatory Gilbert damping. We have proposed a simplified model for the case of ideal transparency of electrons (Supplementary Materials S2 and f ig. S9 ). For the less transparency cases, i.e., in diffusive regime , (42, 43), the energy level s of the ABS in 𝜋-junctions locates away from zero -energy, but they are still much smaller than those of the ABS in zero-junctions. Actually, the similar oscillating behaviors of ABS (or DOS) can be preserved in the diffusive regime (6, 46). Hence, an oscillating spin pumping efficiency would also be expected in the diffusive regime, which could lead to the oscillating Gilbert damping observed in our experiment . To fully understand the experimental observation of the oscillatory Gilbert damping and the detailed spin relaxation process in the diffusive regime, further theoretical studies are needed. Furthe rmore, the control samples of bilayer Nb/NiFe heterostructur es do not exhibit the large oscillatory feature for the Gilbert damping as the NiFe thickness varies at T = 4 K (fig. S10), which further presents the important role of phase difference across NiFe in the large the oscillatory Gilbert damping observed in t he trilayer Nb/NiFe/Nb heterostructures . In conclusion, giant oscillatory Gilbert damping is observed in the superconducting Nb/NiFe/Nb junctions with respect to the NiFe thickness. To our best knowledge, neither the Bogoliubov quasiparticles, nor the spin -triplet pairs are relevant to this observation. The most possible explanation for such giant oscillatory Gilbert damping could be related to the different ABS energy levels and the DOS at the NiFe/SC interface in zero- and π- junctions. To full y understand these results, further theoretical studies are needed. Looking forward, our experimental results might pave the way for controlling the magnetization dynamics by the superconducting phase in a FM Josephson junction in the SQUID setup, and could be important potential applications of ferromagnet π-junctions in quantum computing, such as half -quantum flux qubits. Materials and Methods Materials growth The SC/FM/SC heterostructures consisting of Nb (100 nm) and Ni 80Fe20 (NiFe ; ~ 5 - 20 nm) were grown on thermally oxidized Si substrates in a d.c. magnetron sputtering system with a base pressure of ∼1× 10−8 torr. To systematically vary the NiFe thickness that is crucial for the quantum -9 size effect, we adopted the rotating multi -platter technique that allows us to grow dozens of Nb/NiFe/Nb samples in each run (35). The thickness of the Nb layer is fixed to be ~100 nm that is much larger than the spin diffusion length of Nb (20, 47). After the growth, a thin Al 2O3 layer (~ 10 nm) was deposited in situ as a capping layer to avoid sample degradation against air/water exposure. The crystalline properties of Nb/NiFe/Nb heterostructures were chara cterized by X -ray diffraction (fig. S1A ) and high -resolution cross -section al tra nsmission electron microscopy (f ig. S1B) using a 200 -kV JEOL 2010F field -emission microscope. The NiFe thickness is determined by the growth rate that is calibrated by TEM measurement, where the uncertainty of the NiFe thickness is obtained to be smaller than ~ 0.8 nm (f ig. S1B). The resistivity of the NiFe layers (thickness: 5 - 20 nm) is ranging from 60 to 35 μΩ ·cm, which corresponds to the mean free path between 2.3 and 3.9 nm. Ferromagnetic resonance measurement. The spin pumping of Nb/NiFe/Nb heterostructures was characterized via FMR using the coplanar wave guide technique connected with a vector network analyzer (VNA; Agilent E5071C) in the variable temperature insert of a Physical Properties Measurement System (PPMS; Quantum Design) (19). The FMR spectra were characterized by measuring the amplitudes of forward complex transmission coefficients (S 21) as the in -plane magnetic field decreases from 4000 to 0 Oe under the microwave power of 1 mW. The typical FMR results measured on the Nb/NiFe (12 nm)/Nb het erostructures are shown in the fig. S2A (T = 10 K) and fig. S 5A (T = 4 K). Weaker FMR signals are observed in the superconducting states compared to the normal states. The half linewidth ( ∆𝐻) can be obtained by the Lorentz fitting of the magnetic field -dependent FMR sign al following the relationship ( figs. S2B and S4 B): 𝑆21∝𝑆0(∆𝐻)2 (∆𝐻)2+(𝑯−𝑯𝒓𝒆𝒔)2 (3) where 𝑆0 is the coefficient for the transmitted microwave power, 𝑯 is the external in -plane magnetic field, and 𝑯𝒓𝒆𝒔 is the resonance magnetic field. The Gilbert damping constant (α) can be obtained from the slope of the best linear -fitting results of the ∆𝐻 vs. the microwave frequency ( f) (48-51): 10 ∆𝐻=∆𝐻0+(2𝜋𝛼 𝛾)𝑓 (4) where ∆𝐻0 is the zero -frequency line broadening that is related to the inhomogeneous properties, and 𝛾 is the gyromagnetic ratio. From the best linearly fits of the ∆𝐻 vs. f results measured on the typical Nb/Py ( 12 nm)/Nb sample (red lines in f igs. S2 C and S 5C), 𝛼 is determined to be 0.012 and 0.0054 at T = 10 and 4 K, respectively. A larger zero -frequency line broadening ∆𝐻0 is observed for the superconducting state compared to the normal state of Nb/Py/Nb heterostructures, which could be attributed to Meissn er screening effect and the formation of trapped magnetic fluxes in Nb (51). The thickness dependent ∆𝐻0 is shown in f ig. S1 1C, and no obviously oscillatory behaviors are observed . The effective magnetization and the gyromagn etic ratio can be fitted via the in -plane Kittel formula (51): 𝑓𝑟𝑒𝑠=𝛾 2𝜋√(𝐻𝑟𝑒𝑠+ℎ)(𝐻𝑟𝑒𝑠+ℎ+4𝜋𝑀𝑒𝑓𝑓), (5) where 𝑓𝑟𝑒𝑠 and 𝐻𝑟𝑒𝑠 are the resonant microwave frequency and magnetic field respectively , 4𝜋𝑀𝑒𝑓𝑓 is the effective saturated magnetization , and ℎ is the shifted magnetic field induced by superconducting proximity effect. The thickness -dependent gyromagnetic ratio and eff ective magnetization can be found in fig. S1 1A and fig. S1 1B. Both parameters do not exhibit any oscillatory features as the Gilbert damping does (Fig . 4A), which demonstrates that the oscillatory Gilbert damping is not caused by any unintentional experimental error . Superconducting transition temperature measurement. The superconducting transition temperature ( TC) of the Nb/NiFe/Nb heterostructures was determined via the zero -resistance temperature measured by four -probe method in a P PMS using standard a.c. lock -in technique at a low frequency of 7 Hz. The TC of Nb (100 nm)/NiFe/Nb (100 nm) heterostructures exhibits little variation as a f unction of the NiFe thickness ( fig. S 4). It is noticed that the FMR measurement can affect the TC a little (< 1 K), as shown in fig. S6. 11 Supplementary Materials Supplementary Materials and Methods fig. S1 . The crystalline properties of the Nb/NiFe/Nb heterostructures. fig. S2. Gilbert dampin g measurement of Nb/NiFe/Nb heterostructures at T = 10 K. fig. S3. NiFe thickness dependence of Gilbert damping at T = 50 K. fig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures. fig. S5. Measurement of the Gilbert damping of Nb/ NiFe/Nb heterostructures at T = 4 K. fig. S6. 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Y.Y., R.C., Y.M., W.X., Y.J., X.C.X., and W.H. acknowledge the financial support from National Basic Research Programs of China (No. 2019YFA0308401), National Natural Science Foundation of China (No. 11974025 ), Beijing Natural Science Foundation (No. 1192009), and the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB28000000). T.Y. is financially supported by DFG Emmy Noether program (SE 2558/2 -1). Author contributions: W.H. conceived and supervised the project. Y.Y. and R.C. performed the ferromagnetic resonance measurements. Y.Y. and Y.M. performed X -ray diffraction measurements. T.Y. performed the theor etical calculations. S.H.Y. synthesized the Nb/NiFe/Nb heterostructures. Y.Y. and W.H. wrote the manuscript with the contribution from all authors. All the authors discussed the results. Competing interests: The authors declare no competing interests. Data Availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. 16 Fig. 1. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC heterostructures. (A) The oscillatory real part of the superconducting order parameter (Re {Ψ}, green curve ) penetrated into FM leads to the zero-state and 𝜋-state. (B) The symmetric order parameter in the zero-junction s. (C) The anti -symmetric order parameter in the 𝜋-junction s. (D-E) Spin pumping via t he ABS in SCs in the zero- and 𝜋-junction s. M and 𝛼FM are the magnetization and Gilbert damping of the FM layer itself, and 𝛼sp is the enhanced Gilbert damping, which arises from the spin dissipation in SC layers during the spin pumping process. 17 Fig. 2. Oscillatory Gilbert damping of the Nb/NiFe/Nb heterostructures above TC. (A) The illustration of spin pumping into the normal states of Nb layers and the electronic band structure of NiFe with different spin-up and spin -down Fermi vectors (𝑘F↑ and 𝑘F↓) due to the exchange splitting (2 𝐸𝑒𝑥). The spin pumping gives rise to the spin accumulation in the Nb layers, indicated by the spin -split chemical potential ( 𝜇↑ and 𝜇↓). (B) A typical FMR curve measured with f = 12 GHz (black circles) and the Lorentzian fitting curve (red line) measured on Nb/Py (12 nm)/Nb . ΔH is the half line width at the half maximum of FMR signal. (C) The determination of the Gilbert 18 damping from ΔH vs. f. The red line represents the best linear -fitting curve. (D) The oscillatory Gilbert damping as a function of NiFe thickness ( dNiFe) measured at T = 10, 15, and 20 K, respectively. The experimental oscillating period ( 𝜆) is marked by the red dashed arrow. The inset: Illustration of the quantum -interference effect o f the angular momentum transfer between the local magnetic moment and the spin -polarized electrons . When the NiFe thickness decreases to scale of 1 ∆𝑘, the quantum -interference effect starts to be significant in the angular momentum transfer and spin pumping into the Nb layers . 19 Fig. 3. Giant oscillatory Gilbert damping in Nb/NiFe/Nb heterostructures below TC. (A) The illustration of electronic band structures of Nb in the normal and superconducting states. (B) The determination of TC via zero -resistance temperature measured on the typical Nb/NiFe (12 nm)/Nb heterostructures. (C) Temperature dependence of Gilbert d amping of the typical Nb/NiFe (12 nm)/Nb heterostructures. (D) The oscillatory Gilbert damping as a function of the NiFe thickness in the Nb/NiFe/Nb heterostructures measured at T = 10, 7, 5, and 4 K, respectively. The oscillating feature below TC (T = 4 and 5 K) is dramatically enhanced compared to that above TC (T = 10 K). 20 Fig. 4. Physical mechanism of the giant oscillatory Gilbert dam ping in Nb/NiFe/Nb junctions. (A) The NiFe thickness dependence of the Gilbert damping difference ( ∆α) between the Nb/NiFe/Nb π- and zero-junctions at T = 4 K . Inset: The Gilbert damping of zero- and π-junctions. The solid balls represent the experimental data, the blue and black dash lines are the guide lines for π- and zero-junctions, respectively. For bo th guide lines, the damping is expected to behave as α ~ 1 ⁄ dNiFe. (B-C) Illustration of the spin pumping via the ABS and the enhanced Gilbert damping for Nb/NiFe/Nb π- and zero-junctions, respectively. The red thick -arrows indicate the pumping and relaxation of the spin current in SCs . 21 Supplementary Materials for Giant oscillatory Gilbert damping in superconductor/ferromagnet/superconductor junctions Authors Yunyan Yao1,2†, Ranran Cai1,2†, Tao Yu3, Yang Ma1,2, Wenyu Xing1,2, Yuan Ji1,2, Xin -Cheng Xie1,2,4,5, See -Hun Yang6*, and Wei Han1,2* This SM file includes : ⚫ Supplementary Materials and Methods ⚫ fig. S1. The crystalline properties of the Nb/NiFe/Nb heterostructures. ⚫ fig. S2. Gilbert damping measurement of Nb/NiFe/Nb heterostructures at T = 10 K. ⚫ fig. S3. NiFe thickness dependence of Gilbert damping at T = 50 K . ⚫ fig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures. ⚫ fig. S5. Measurement of the Gilbert damping of Nb/NiFe/Nb heterostructures at T = 4 K. ⚫ fig. S6. The effect of FMR measurement on the TC of Nb/NiFe/N b heterostructures. ⚫ fig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions. ⚫ fig. S8. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC heterostructures due to Bogoliubov quasiparticles and equal spin -triplet Cooper pai rs. ⚫ fig. S9. Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at T = 4 K. ⚫ fig. S10. Gilbert damping of control sample of bilayer Nb/NiFe junctions. ⚫ fig. S11. Thickness dependence of gyromagnetic ratio, effective magnetization an d inhomogeneous half -linewidth . 22 Supplementary Materials and Methods Section 1: Model of oscillating Gilbert damping above TC. The oscillatory Gilbert damping in normal metal (NM)/ferromagnet (FM)/NM heterostructures arising from quantum interference effect is analyzed based on previous theory by Mills (36). Within the linear response theory, the enhanced Gilbert damping is related to the dynamical spin susceptibility ( 𝜒−+(𝛺)) of conduction electrons in a FM, 𝛼sp=𝐽2𝑀𝑠𝑉 2𝑁2ℏ3𝛾Λ2 (S1) where Λ2=Im(𝑑𝜒−+(𝛺) 𝑑𝛺| Ω=0). Using one dimensional model, we obtain Λ2=1 𝜋2∫𝑑𝑥𝑑 𝑥′Im[𝐺↑(𝑥,𝑥′,𝜖F)]Im[𝐺↓(𝑥,𝑥′,𝜖F)] FM (S2) where 𝐺𝜎(𝑥,𝑥′,𝜖F) is the Green’s function for conduction electrons with 𝜎-spin at the Fermi energy ( 𝜖F). In a FM, 𝐺𝜎(𝑥,𝑥′,𝜖F) is related to the exchange energy. [−ℏ2 2𝑚𝑑 𝑑𝑥2−𝜖±𝐸ex]𝐺𝜎(𝑥,𝑥′,𝜖F)=𝛿(𝑥−𝑥′) (S3) For the FM film with a thickness dFM in − dFM/2 < x < dFM/2, the Green’s function satisfies the relation 𝐺𝜎(𝑥,𝑥′,𝜖F)=𝐺𝜎(𝑥′,𝑥,𝜖F)=𝐺𝜎(−𝑥,−𝑥′,𝜖F) (S4) Hence, the imaginary part of the Green’s function could be expressed by Im[𝐺𝜎(𝑥,𝑥′,𝜖F)]=−π{𝑁F𝜎cos [𝑘F𝜎(𝑥−𝑥′)]+𝑁F𝜎′cos [𝑘F𝜎(𝑥+𝑥′)]} (S5) where 𝑘F𝜎=√2𝑚 ℏ2(𝜖F∓𝐸ex) is the Fermi wave -vector in the FM, 𝑁F𝜎 and 𝑁F𝜎′ are equivalent to the density of states and the modulation amplitude of the local density of states, respectively. For the same position of x, the local density of states is equal to 𝑁𝜎(𝑥,𝜖F)=𝑁F𝜎+𝑁F𝜎′cos [2𝑘F𝜎𝑥] (S6) Since 𝐸ex is much smaller compared to 𝜖F, the spatial modulation of the local density of states is negligible. The combination of equations (S2) and (S5) leads to Λ2=∫ 𝑑𝑥𝑑 𝑥′𝑑FM 2⁄ −𝑑FM 2⁄{𝑁F↑cos[𝑘F↑(𝑥−𝑥′)]}∗{𝑁F↓cos[𝑘F↓(𝑥−𝑥′)]} (S7) 23 =2𝑁F↑𝑁F↓{1 (𝑘F↑−𝑘F↓)2sin2[𝑘F↑−𝑘F↓ 2𝑑FM]+1 (𝑘F↑+𝑘F↓)2sin2[𝑘F↑+𝑘F↓ 2𝑑FM]} Clearly, the enhanced Gilbert damping is expected to oscillate as a function of the FM thickness with two periods of 2𝜋/[𝑘F↑−𝑘F↓] and 2𝜋/[𝑘F↑+𝑘F↓]. For real FM materials, such as NiFe with (𝑘F↑+𝑘F↓)≫(𝑘F↑−𝑘F↓), the second term in the equation (S7) could be negligible, leaving only one oscillating period of 2𝜋/[𝑘F↑−𝑘F↓]. When the FM thickness is equal to 2𝑛𝜋/[𝑘F↑−𝑘F↓], a lower Gilbert damping is obtained. On the other hands with FM thickness of (2𝑛+1)𝜋/[𝑘F↑−𝑘F↓], a larger Gilbert damping is obtained. Section 2: Calculation of the enhanced Gilbert damping in Nb/NiFe/Nb by spin pumping via Andreev bound states (ABS). As the reciprocal process of the spin transfer torque, conventional spin pumping is achieved by the magnetization torques provided by the driven quasiparticle carriers (29, 52-54), which are the electrons in the normal metals. In SC/FM heterostructures, however, the quasiparticle carriers can be either Bogoliubov quasiparticles or ABS (42), which lie above and within the superconducting gaps, respectively. Therefore, it is desirable to formulate and estim ate the contribution to the spin pumping via the ABS (55), when the temperature is much smaller than the superconducting critical temperature. Without loss of generality, we start the analysis from a left -propagating electron of energy 𝜀 and spin 𝜎 = {↑, ↓} = {+, −,}. When the Zeeman splitting J is much smaller than the Fermi energy EF, it has momentum 𝑘𝜎=𝑘𝐹+(𝜀+𝜎𝐽)/(ℏ𝜈𝐹), (S8) where 𝜈𝐹 is the Fermi velocity of the electron. When one electron goes from the FM to the SCs, it is reflected as a hole by the Andreev reflection at the right FM/SC interface; this hole has a phase shift χ=−arccos (𝜀/Δ) with respect to the electron (56), where Δ is the superconducting gap . Similarly, when a hole goes from the metal to the superconductor at the left FM/SC interface, an electron can be reflected. With a proper energy, the Andreev reflections can form a closed path, as a result of which the ABS forms. This requires that the phase accumulated in the reflections satisfies the Sommerfeld quantization condition, i.e. in the ballistic regime, 24 𝜀𝐿 ℏ𝜈𝐹+𝜎𝐽𝑑NiFe ℏ𝜈𝐹−arccos (𝜀 ∆)=𝑛𝜋+𝜑 2, (S9) where 𝜑 is the phase difference between the two superconductors , dNiFe is the thickness of the FM layer and n is an integer. Since ℏ𝜈𝐹/∆ ≥ 100 nm with 𝜈𝐹 = 2.2 × 105 m/s and ∆ = 1 meV at T = 4 K in our experiment (17, 57, 58), dNiFe < 19 nm << ℏ𝜈𝐹/∆ such that the first term in Eq. (S9) can be safely disregarded. For the FMR measurements with open -circuited configuration, the junctions always stay in the ground states (43-45). For 𝜋-junctions, there is a 𝜋-phase shift in the current - phase relationship curves compared to zero-junctions , i.e., the properties of 𝜑 = 0 of a 𝜋-junction is the same as those of 𝜑 = 𝜋 of a zero-junction. Since this 𝜋-phase shift is already taken into account by the FM exchange field , the ABS energy of the 𝜋-junctions can be obtained at 𝜑 = 0 in the ground states , which is similar to that of 𝜑 = 𝜋 of zero-junctions . Hence, the energy of the ABS can be described by ε0=±∆cos (𝐽𝑑NiFe ℏ𝜈𝐹) for ideal case with perfect transparency of electrons/holes. In reality, the interfacial scattering and transport conditions (ballistic or diffusive regimes) of FM could affect th e energy of the ABS. Following previous studies (42, 59), a transmission coefficient ( D) could be introduced to describe this issue , which is close to unity in the ballistic regime but can also be large in the diffusive regime with an ideal transparency at the interface (43, 60, 61). In this work, we focus on the ideal cases with perfect transparency of electrons/holes . The energy of the ABS oscillates from the ed ge of the superconducting gap to the zero -energy with respect to the FM thickness (fig. S9A). The pumped spin current reads (29, 52-54), 𝐉𝑠(𝑡)=ℏ 4𝜋𝑔eff↑↓𝒎×𝑑𝒎 𝑑𝑡, (S10) where m is the magnetization unit vector, and we define the effective mixing spin conductivity 𝑔eff↑↓ at the finite temperature via the zero -temperature one 𝑔↑↓ by (53, 55) 𝑔eff↑↓=𝑛0∫𝑑𝜀𝑑𝑓(𝜀) 𝑑𝜀Re[𝑔↑↓(𝜀)]. (S11) Here, n0 is the number of the conduction channel that roughly corresponds the conduction electron density at the interface and 𝑓(𝜀)=1/{exp [𝜀/(𝑘𝐵𝑇)]+1} is the Fermi -Dirac distribution of electron at the temperature T. Importantly, in the ballistic limit Re[𝑔↑↓(𝜀)]=1 when 𝜀=𝜀0; it 25 has width ∆𝜀 depending on the FM thickness dNiFe in the ballistic regime or the mean free path 𝑙𝑚 in the diffusive regime . By the uncertainty principle, ∆𝜀∆𝑡=2𝜋ℏ, where ∆𝑡=𝑙𝑚/𝜈𝐹 is the propagation time of the electron in the junction, leading to ∆𝜀 ~ 2𝜋ℏ𝜈𝐹/𝑙𝑚. By further considering the degeneracy due to spin (× 2) and the existence of two interfaces (× 2), we thus can estimate 𝑔eff↑↓ ~ 8𝜋𝑛0ℏ𝜈𝐹 𝑙𝑚𝑑𝑓(𝜀0) 𝑑𝜀. (S12) The pumped spin current carries the angular momentum away from the precessing magnetization and hence cause an enhanced Gilbert damping, which is described by 𝛿𝛼=2𝛾ℏ2𝜈𝐹 𝑀𝑠𝑙𝑚𝑑NiFe𝑑𝑓(𝜀0) 𝑑𝜀 , (S13) where 𝛾 is electron gyromagnetic ratio and 𝑀𝑠 is the saturated magnetization of the ferromagnet. We are now ready to estimate the contribution of ABS to the Gilbert damping at T = 4 K with varying transmission coefficient. We take 𝑛0 =0.5 × 1016 m−2 following Ref. 44 , 𝑙𝑚~ 3 nm, 𝜈𝐹= 2.2 × 105 m/s, J = 400 meV and 𝜇0𝑀𝑠≈1 𝑇 from previous experimental results (17). With superconducting gaps ∆ ≈ 1 meV at T = 4 K for Nb (57, 58), Fig.S8A plots the normalized energy of ABS by the superconducting gap at T = 4 K as a function of dNiFe for the ideal transparency case . The oscillation of the Gilbert damping can be resolved by using the FM exchange field -induced phase shift of 𝐽𝑑NiFe ℏ𝜈𝐹 (fig. S9B) . For simplicity, we have disregarded the possible thickness dependence of the superconducting gaps and magnetizations. To be noted, our theoretical estimation is based on a simplified model that assumes D = 1. For the diffusive regime or the case of non -perfect transparency of electrons at the interface (42, 43), similar oscillating behaviors of ABS (or DOS) in the SCs can also be preserved. For example, the oscillating ABS (or DOS) in the SCs have been shown to exist in the diffusive regime theoretically (6, 46), and indeed, the zero to 𝜋 transitions have been experimentally observed in both the ballistic and diffusiv e regimes from the supercurrent measurements (11, 17). To fully understand the experimental observation of the oscillatory Gilbert damping in the diffusive regime, further theoreti cal studies are needed. 26 Section 3: Measurement of the Josephson coupling in Nb/NiFe/Nb. The Nb/NiFe/Nb Josephson devices are fabricated using the shadow mask techniques during the films growth. As shown in figs. S7A and S7B, the Josephson devices have a junction area ( A) of ~ 80 μm × 80 μm, and the other areas are electrically isolated by a 100 nm AlO x layer. The Josephson current is measured by standard a.c. lock -in technique. The normalized differential resistances (dV/dI) measured on the Nb/NiFe (5 nm)/Nb junction at various temperatures are shown in fig. S7C. The critical current ( Ic) is defined as poi nt where the differential resistance increases above the value for the zero -bias current. The normal resistance (R n) is determined to be the saturated value of the normal states of the Josephson coupling measurement. The measured area-resistance product (R nA) of ~ 5×10−10 Ω𝑚2 is higher than that reported in metallic Josephson junction (17, 61), and comparable to that of FM Josephson junction with a thin tunnel barrier (62). This behavior indicates that there is more likely a thin NiFeO x layer (indicated by Fig. S8B) in the junction formed during the AlO x growth step in the presence of oxygen gas. As the temperature increases, I c and the characteristic voltage (I cRn) decrease (figs. S7C and S7D). Clear Josephson currents are observed on the Nb/NiFe (5 nm)/Nb junction and Nb/NiFe (10 nm)/Nb junction (figs. S7E and S78F). And the estimated SC gap energy is ~ 0.9 meV at T = 2 K (1, 43), which is comparable to the value of ~1.36 meV at T = 0 K estimated from TC of ~ 8.5 K (fig. S4). On the other hand, no Josephson current could be observed in the Nb/NiFe (30 nm)/Nb junction (figs. S7E and S7F). The absence of Jo sephson current in Nb/NiFe (30 nm)/Nb junction indicates that there is no long -range spin -triplet Josephson coupling in the Nb/NiFe/Nb heterostructures in our experiment. 27 fig. S1. The crystalline properties of the Nb/NiFe/Nb heterostructures. (A) The θ -2θ X -ray diffraction results measured on the typical Nb/NiFe (12 nm)/Nb sample, where Nb (110) and NiFe (111) peaks are observed. ( B) High -resolution transmission electron micrographs mea sured on the typical Nb/NiFe (12 nm)/Nb sample. The dashed lines show the interfaces between Nb and NiFe layers. The red bars indicate the deviation of NiFe at the interface. 28 fig. S2. Gilbert damping measurement of Nb/NiFe /Nb heterostructures at T = 10 K. (A) The typical FMR spectra as a function of magnetic field with microwave frequency ( f) of 10, 12, 14, 16, and 18 GHz, respectively. (B) The typical FMR spectrum measured with f = 12 GHz (black circles) and the Lorentz fi tting curve (red line). ΔH is the half linewidth of the FMR signal. (C) The determination of the Gilbert damping from ΔH vs. f. The red line indicates the best linear - fitting curve. These results are obtained on the typical Nb/Py (12 nm)/Nb sample. 29 fig. S3. NiFe thickness dependence of Gilbert damping at T = 50 K . 30 fig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures. The TC is determined from the zero -resistance temperature via four -probe resistance measurement. 31 fig. S5. Measurement of the Gilbert damping of Nb/NiFe/Nb heterostructures at T = 4 K. (A) The typical FMR spectra as a function of magnetic field with microwave frequency ( f) of 10, 12, 14, 16, and 18 GHz, respectively. (B) The typical FMR spectrum measured with f = 12 GHz (black circles) and the Lorentz fitting curve (red line). ΔH is the half linewidth of the FMR signal. (C) The determination of the Gilbert damping from ΔH vs. f. The red line indicates the best linear - fitting curve. These results are obtained on the typical Nb/Py (12 nm)/Nb sample. 32 fig. S6. The effect of FMR measurement on the TC of Nb/NiFe/Nb heterostructures. The four - probe resistances vs. temperature are probed from the typical Nb/NiFe (12 nm)/Nb sample with/without the presence of the in -plane magnetic field and microwave power. 33 fig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions. (A) The optical image of a typical Nb/NiFe/Nb Josephson device and schematic of the electrical measurement geometry. (B) The cross -section of the Josephson devices with a junction area of ~ 80 μm × 80 μm. At the junction, a thin oxide layer of NiFeO x is mostly liked formed on the top surface of NiFe during the growth of A lOx in the presence of oxygen. ( C) The normalized differential resistance (dV/dI) as a function of the bias current measured on the Nb/NiFe (5 nm)/Nb junction from T = 2 to 6 K. (D) The temperature dependence of the characteristic voltage (I cRn) of the Nb/NiFe (5 nm)/Nb Josephson junction . (E) The normalized differential resistance as a function of the bias current of the Nb/NiFe/Nb junctions ( dNiFe = 5, 10 and 30 nm) at T = 2 K. ( F) The NiFe thickness dependence of the characteristic voltages a t T = 2 K. 34 fig. S8. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC heterostructures due to Bogoliubov quasiparticles (A) and equal spin -triplet Cooper pairs (B). The dark and light balls represent the electron -like and hole-like quasiparticles respectively. The red and blue arrows indicate the spin up and spin down respectively. 35 fig. S9. Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at T = 4 K. (A) The normalized energy of ABS by the superconducting gap at T = 4 K as a function of dNiFe for the ideal transparency case. ( B) The enhanced Gilbert damping via ABS as a function of dNiFe. 36 fig. S10. Gilbert damping of control sample of bilayer Nb/Ni Fe junctions. (A) Gilbert damping of bilayer Nb/NiFe junctions at T = 4, 5, and 10 K. (B) Comparison of the Gilbert damping of bilayer Nb/NiFe and trilayer Nb/NiFe/Nb junctions at T = 4 K. 37 fig. S11. Thickness dependence of gyromagnetic ratio (and g factor) (A), effective magnetization (B) and inhomogeneous half -linewidth (C) . The blue, black, green and red dotted -lines represent to temperature of T = 10, 7, 5 and 4 K, respectively.

2021-11-05

Interfaces between materials with differently ordered phases present unique opportunities for exotic physical properties, especially the interplay between ferromagnetism and superconductivity in the ferromagnet/superconductor heterostructures. The investigation of zero- and pi-junctions has been of particular interest for both fundamental physical science and emerging technologies. Here, we report the experimental observation of giant oscillatory Gilbert damping in the superconducting Nb/NiFe/Nb junctions with respect to the NiFe thickness. This observation suggests an unconventional spin pumping and relaxation via zero-energy Andreev bound states that exist only in the Nb/NiFe/Nb pi-junctions, but not in the Nb/NiFe/Nb zero-junctions. Our findings could be important for further exploring the exotic physical properties of ferromagnet/superconductor heterostructures, and potential applications of ferromagnet pi-junctions in quantum computing, such as half-quantum flux qubits.

Giant oscillatory Gilbert damping in superconductor/ferromagnet/superconductor junctions

2111.03233v1

The interplay of large two-magnon ferromagnetic resonance linewidths and low Gilbert damping in Heusler thin lms W. K. Peria,1T. A. Peterson,1A. P. McFadden,2T. Qu,3C. Liu,1C. J. Palmstrm,2;4and P. A. Crowell1 1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455 2Department of Electrical & Computer Engineering, University of California, Santa Barbara, California 93106 3Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455 4Department of Materials, University of California, Santa Barbara, California 93106 We report on broadband ferromagnetic resonance linewidth measurements performed on epitaxial Heusler thin lms. A large and anisotropic two-magnon scattering linewidth broadening is observed for measurements with the magnetization lying in the lm plane, while linewidth measurements with the magnetization saturated perpendicular to the sample plane reveal low Gilbert damping constants of (1:50:1)103, (1:80:2)103, and<8104for Co 2MnSi/MgO, Co 2MnAl/MgO, and Co2FeAl/MgO, respectively. The in-plane measurements are t to a model combining Gilbert and two-magnon scattering contributions to the linewidth, revealing a characteristic disorder lengthscale of 10-100 nm. I. INTRODUCTION The theoretical understanding of the damping mech- anism believed to govern longitudinal magnetization re- laxation in metallic ferromagnets, originally due to Kam- bersk y [1, 2], has in recent years resulted in quantita- tive damping estimates for realistic transition metal band structures [3{5]. Although of great interest where engi- neering of damping is desired [6], these calculations re- main largely uncompared to experimental data. Kam- bersk y damping may be characterized by the so-called Gilbert damping constant in the Landau-Lifshitz- Gilbert macrospin torque equation of motion, and for- mally describes how the spin-orbit interaction in itinerant electron systems results in damping of magnetization dy- namics [2]. Schoen et al. [7] have reported that is mini- mized for Co-Fe alloy compositions at which the density- of-states at the Fermi level is minimized, in reasonable agreement with Kambersk y model predictions [8]. Fur- thermore, half-metallic, or near half-metallic ferromag- nets such as full-Heusler compounds have been predicted to demonstrate an ultralow Kambersk y (103) due to their spin-resolved band structure near the Fermi level [9]. Finally, anisotropy of the Kambersk y damping in sin- gle crystals has been predicted, which is more robust for Fermi surfaces with single-band character [5, 10]. The Gilbert damping constant is often reported through measurements of the ferromagnetic resonance (FMR) linewidth
H, which may be expressed as a sum of individual contributions
H=2f +
H0+
HTMS; (1) where the rst term is the Gilbert damping linewidth (fis the FMR frequency, is the gyromagnetic ratio),
H0is a frequency-independent inhomogeneous broad- ening, and
HTMS represents an extrinsic two-magnon scattering (TMS) linewidth contribution [11, 12] that is, in general, a nonlinear function of frequency. In recentyears it has been realized that TMS linewidths are per- vasive for the conventional in-plane geometry of thin lm FMR measurements, requiring either the perpendicular- to-plane FMR geometry [13] (for which TMS processes are suppressed) or suciently broadband measurements [14] to extract the bare Gilbert . For instance, recent FMR linewidth studies on Heusler compounds have re- ported distinct TMS linewidths [15, 16], which challenged simple inference of the Gilbert . In this article, we present FMR linewidth measure- ments for epitaxial Heusler thin lms for all principal ori- entations of the magnetization with respect to the sym- metry axes. For the in-plane conguration, large and anisotropic TMS-dominated linewidths are observed. In the perpendicular-to-plane conguration, for which the TMS process is inactive [11], the Gilbert and inhomo- geneous broadening are measured. We nd evidence of a low (103) Gilbertin these Heusler thin lms, accom- panied by a large and anisotropic TMS contribution to the linewdith for in-plane magnetization. We conclude by discussing the interplay of low Gilbert and large TMS, and we emphasize the nature by which the TMS may conceal the presence of anisotropic Kambersk y . II. SAMPLES The Heusler alloy lms used for these measurements were grown by molecular beam epitaxy (MBE) by co- evaporation of elemental sources in ultrahigh vacuum (UHV). The MgO(001) substrates were annealed at 700C in UHV followed by growth of a 20 nm thick MgO buer layer by e-beam evaporation at a substrate temper- ature of 630C. The 10 nm thick Co 2MnAl and Co 2MnSi lms were grown on the MgO buer layers at room tem- perature and then annealed at 600C for 15 minutes in situ in order to improve crystalline order and surface morphology. The 24 nm thick Co 2FeAl sample was grown using the same MgO substrate and buer layer prepa-arXiv:1909.02738v2 [cond-mat.mtrl-sci] 9 Apr 20202 ration, but at a substrate temperature of 250C with no post-growth anneal. Re ection high energy electron diraction (RHEED) was monitored during and after growth of all samples and conrmed the expected epitax- ial relationship of MgO(001) h110ijjHeusler(001)h100i. X-ray diraction (XRD) demonstrated the existence of a single phase of (001)-oriented Heusler, along with the presence of the (002) re ection, conrming at least B2 ordering in all cases. In addition, for the Co 2MnSi lm only, the (111) re ection was observed, indicating L2 1 ordering [see Fig. 1(a)]. All of the lms were capped with several nm of e-beam evaporated AlOx for pas- sivation prior to atmospheric exposure. The eective magnetization for the 24 nm thick Co 2FeAl lm was determined from anomalous Hall eect saturation eld to be 1200 emu/cm3, which is consistent with measure- ments of Ref. [17] for L2 1or B2-ordered lms, along with 990 emu/cm3and 930 emu/cm3for the Co 2MnSi and Co 2MnAl lms, respectively. Hereafter, we will refer to the Co 2MnSi(10 nm)/MgO as the \CMS" lm, the Co2MnAl(10 nm)/MgO lm as the \CMA" lm, and the Co2FeAl(24 nm)/MgO lm as the \CFA" lm. III. EXPERIMENT Broadband FMR linewidth measurements were per- formed at room temperature with a coplanar waveguide (CPW) transmission setup, similar to that discussed in detail in Refs. [18, 19], placed between the pole faces of an electromagnet. A cleaved piece of the sample (2 mm1 mm) was placed face-down over the center- line of the CPW. A rectifying diode was used to detect the transmitted microwave power, and a 100 Hz mag- netic eld modulation was used for lock-in detection of the transmitted power, resulting in a signal /d=dH (whereis the lm dynamic magnetic susceptibility). The excitation frequency could be varied from 0-50 GHz, and a microwave power near 0 dBm was typically used. It was veried that all measurements discussed in this arti- cle were in the small precession cone angle, linear regime. The orientation of the applied magnetic eld could be rotated to arbitrary angle in the lm plane (IP), or ap- plied perpendicular to the lm plane (PP). We empha- size again that TMS contributions are suppressed in the PP conguration [12]. The resonance elds were t as a function of applied frequency in order to extract various magnetic properties of the lms. The magnetic free energy per unit volume used to gen- erate the resonance conditions for these samples is given by FM=M
H+K1sin2cos2+ 2M2 effcos2;(2) where His the applied eld, andare the azimuthal and polar angles of the magnetization, respectively, K1 is a rst order in-plane cubic anisotropy constant, and 4Meffis the PP saturation eld, which includes the usual demagnetization energy and a rst order uniaxial -150-75075150-6-30361 0 GHz20 GHz30 GHzdχ/dH (arb. u.)H - HFMR (Oe)40 GHzCMS 090180270360101102103104 <111> <202>Intensity (arb. u.)φ (°)CMS -5000 5 00-101F ield (Oe)M/MSH || 〈110〉H || 〈100〉H || 〈110〉CFA (d)(b)( c)(a)C FAFIG. 1. (a) Wide-angle x-ray diraction -scans ofh202i (blue) andh111i(red) peaks for the CMS lm. (b) Typical derivative susceptibility lineshapes for these samples at dif- ferent microwave excitation frequencies. The ts are shown as solid lines. (c) In-plane hysteresis loops for CFA obtained with a vibrating-sample magnetometer (VSM). (d) Atomic force microscopy (AFM) image of surface topography for CFA. RMS roughness is 0.2 nm. anisotropy due to interfacial eects. The parameters ob- tained by tting to Eq. 2 are shown in Table I. The uncer- tainty in these parameters was estimated by measuring a range of dierent sample pieces, and using the standard deviation of the values as the error bar. The long-range inhomogeneity characteristic of epitaxial samples makes this a more accurate estimate of the uncertainty than the tting error. The magnetic-eld-swept FMR line- shapes were t to the derivative of Lorentzian functions [19] in order to extract the full-width at half-maximum linewidths
H[magnetic eld units, Fig. 1(b)], which are the focus of this article. The maximum resonant fre- quency was determined by the maximum magnetic eld that could be applied for both IP and PP electromagnet congurations, which was 10.6 kOe and 29 kOe, respec- tively. For the IP measurement, the angle of the applied eld in the plane of the lm was varied to determine the in-plane magnetocrystalline anisotropy of our samples, which was fourfold-symmetric for the three lms char- acterized in this article. The anisotropy was conrmed using vibrating-sample magnetometry (VSM) measure- ments, an example of which is shown in Fig. 1(c), which shows IP easy and hard axis hysteresis loops for the CFA lm. For the PP measurement, alignment was ver- ied to within0.1to ensure magnetization saturation just above the PP anisotropy eld, thus minimizing eld-3 TABLE I. Summary of the magnetic properties extracted from the dependence of the resonance eld on applied frequency for both eld in-plane ( jj) and eld perpendicular-to-plane ( ?) congurations, along with the Gilbert and inhomogeneous broadening from the perpendicular-to-plane conguration. 2 K1=Msand 4Meffare the in-plane and perpendicular-to-plane anisotropy elds, respectively (see Eq. 2), and gis the Land e g-factor. Sample 2 K1=Ms(Oe) 4Mjj eff(kOe) 4 M? eff(kOe) gjjg?001(103)
H0(Oe) CMS 280 12.3 13.3 2.04 2.04 1 :50:1 91 CMA 35 11.3 11.7 2.06 2.08 1 :80:2 123 CFA 230 15.1 15.5 2.06 2.07 <0.8 1006 CFA 500C anneal N/A N/A 15.1 N/A 2.07 1 :10:1 451 01 02 03 04 05 00306090120α 001 = 1.1×10-3CFA 500 °C annealCFAC MAα001 < 8×10-4α 001 = 1.8×10-3ΔH (Oe)F requency (GHz)α001 = 1.5×10-3CMS FIG. 2. Linewidths as a function of frequency with the eld applied perpendicular to plane, for which two-magnon scat- tering is inactive. The black squares are data for the CMS lm, the red circles are for the CMA lm, and the blue trian- gles are for the CFA lm. In addition, linewidths are shown for a CFA lm that was annealed at 500Cex situ (magenta diamonds). Corresponding linear ts are shown along with the extracted Gilbert damping factor . The blue dashed lines indicate an upper bound of 001= 8104and a lower bound of001= 0 for CFA. dragging contributions to the linewidth. IV. RESULTS AND ANALYSIS A. Perpendicular-to-plane linewidths First we discuss the results of the PP measurement. As stated in Sec. III, the TMS extrinsic broadening mecha- nism is suppressed when the magnetization is normal to the plane of the lm. We can thus t our data to Eq. 1 with
HTMS = 0, greatly simplifying the extraction of the Gilbert damping constant and the inhomogeneous broadening
H0. Prior knowledge of
H0is particu- larly important for constraining the analysis of the IP measurements, as we shall discuss. 3035404550C MS〈100〉C MS〈110〉ΔH (Oe)CMS2 0 GHz2 800300032003400H FMR(Oe)- 450 4 5200400600C FA〈100〉ΔH (Oe)A ngle (°)CFA2 0 GHzCFA〈110〉2 600280030003200H FMR(Oe)50100150200(a)( c)C MA〈100〉C MA〈110〉ΔH (Oe)CMA1 5 GHz(b)2 00020502100H FMR(Oe)FIG. 3. Azimuthal angular dependence of the linewidths (left ordinate, blue circles) and resonance elds (right ordinate, black squares) for (a) CMS, (b) CMA, and (c) CFA. The excitation frequency was 20 GHz for CMS, 15 GHz for CMA, and 20 GHz for CFA. The solid lines are sinusoidal ts. The dependence of
Hon frequency for the CMS, CMA, and CFA lms in the PP conguration is summarized in Fig. 2, in which ts to Eq. 1 are shown with
HTMS set to zero. For the CMS lm,001= (1:50:1)103and
H0= 9 Oe, while for the CMA lm 001= (1:80:2)1034 and
H0= 12 Oe. Co 2MnSi 2=3Al1=3/MgO and Co2MnSi 1=3Al2=3/MgO lms (both 10 nm thick) were also measured, with Gilbert damping values of 001= (1:80:2)103and001= (1:50:1)103, re- spectively (not shown). For CFA, we obtained a damp- ing value of 001= 3104with an upper bound of 001<8104and
H0= 100 Oe. These t param- eters are also contained in Table I. The source of the large inhomogeneous broadening for the CFA lm is un- clear: AFM measurements [Fig. 1(d)] along with XRD in- dicate that the lm is both crystalline and smooth. Note that the range of frequencies shown in Fig. 2 are largely governed by considerations involving the Kittel equation [20]: measurements below 10 GHz were not used due to the increasing in uence of slight misalignment on
H (through eld-dragging) for resonant elds just above the saturation value. A piece of the CFA sample was annealed at 500Cex situ , which reduced the inhomoge- noeus broadening to 45 Oe (still a relatively large value) and increased the Gilbert damping to 001= 1:1103 (similar behavior in CFA was seen in Ref. [21]). The constraint of 001<8104is among the lowest of re- ported Gilbert damping constants for metallic ferromag- nets, but the 104range is not unexpected based on Kambersk y model calculations performed for similar full-Heusler compounds [9] or other recent experimental reports [22, 23]. It should be noted that Schoen et al. [7] have recently reported = 5104for Co 25Fe75thin lms, where spin pumping and radiative damping con- tributions were subtracted from the raw measurement. Spin pumping contributions to the intrinsic damping are not signicant in our lms, as no heavy-metal seed layers have been used and the lms have thicknesses of 10 nm or greater. For the radiative damping contribution [13] in the geometry of our CPW and sample, we calculate contributions rad<1104, which is below the uncer- tainty in our damping t parameter. B. In-plane linewidths With the intrinsic damping and inhomogeneous broad- ening characterized by the PP measurement, we turn our attention to the IP linewidth measurements, for which TMS contributions are present. For hard-axis measure- ments, frequencies <5 GHz were not used due to the in uence of slight magnetic eld misalignment on the linewidths. For easy-axis measurements, the lower limit is determined by the zero-eld FMR frequency. Fig- ure 3 shows the dependences of the resonance elds and linewidths on the angle of the in-plane eld. An im- portant observation seen in Fig. 3 is that the linewidth extrema are commensurate with those of the resonance elds and therefore the magnetocrystalline anisotropy en- ergy. This rules out eld-dragging and mosaicity contri- butions to the linewidth, which can occur when the reso- nance eld depends strongly on angle [24]. We note that similar IP angular dependence of the FMR linewidth, 01 02 03 04 05 002004006008000204060801000 60120180240300C FA(c)ΔH (Oe)F requency (GHz)〈100〉ΔH (Oe)(a)C MSC MA[ 001]〈110〉(b)ΔH (Oe)FIG. 4. Linewidths along all three principal directions for CMS (a), CMA (b), and CFA (c). Heusler crystalline axes are labeled byh100i(black),h110i(red), and [001] (blue). In all three cases,h110iis the in-plane easy axis and h100iis the in-plane hard axis. The corresponding ts are shown as the solid curves, where the in-plane linewidths are t using Eq. 3 and the out-of-plane linewidths are t to the Gilbert damping model. The t parameters are given in Table II. which was attributed to an anisotropic TMS mechanism caused by a rectangular array of mist dislocations, has been reported by Kurebayashi et al. [25] and Woltersdorf and Heinrich [14] for epitaxial Fe/GaAs(001) ultrathin lms. To further study the anisotropy of the IP
Hin our lms, we have measured
Hat the angles correspond- ing to the extrema of HFMR (and
H) in Fig. 3 over a range of frequencies. These data are shown in Fig. 4, along with the PP ([001]) measurements for each sample. A distinguishing feature of the data shown in Fig. 4 is the signicant deviation between IP and PP linewidths in all but one case (CMS h100i). Large and nonlinear frequency dependence of the IP linewidths is strongly suggestive of an active TMS linewidth broadening mechanism. In the presence of TMS, careful analysis is required to separate5 0°90°1 80°2 70°01x10501 02 03 04 00.00.51.01.52 x1045x1041 1.411.82 4 GHz32 GHz1 6 GHz |q2M| (cm-1)ξ -1ξ = 100 nm10-41 0-35 ×10-3ΔHTMS/H'2 (Oe-1) (×10-4)f FMR (GHz)α = 10-2( q || M)(q ^ M)(b) |q| (cm-1)q ^ Mωm (GHz)q || MD egeneracyH = 1 kOe(a) FIG. 5. (a) Two-magnon scattering linewidth contribution for values of Gilbert damping = 102;5103;103;and 104. The inset shows magnon dispersions for an applied eld ofH= 1 kOe. (b) Contours of the degenerate mode wavenumber q2Min the lm plane as a function of wavevector angle relative to the magnetization for fFMR = 16, 24, and 32 GHz. The dashed circle indicates the wavenumber of a defect with size = 100 nm. the Gilbert damping from the TMS linewidth contribu- tions. We therefore describe the TMS mechanism in more detail in the following section in order to analyze the IP linewidths in Fig. 4 and extract the Gilbert damping. C. Two-magnon scattering model The TMS mechanism leads to a characteristic nonlin- ear frequency dependence of
H[11, 12]. In Fig. 4, the IP
His not a linear function of frequency, but possesses the \knee" behavior characteristic of the fre- quency dependence of linewidths dominated by the TMS mechanism. We have t our data to the TMS model described by McMichael and Krivosik [12], in which theTMS linewidth
HTMS is given by [26, 27]
HTMS = 22H02 df=dHjfFMRZ 0qCq()(!!q)d2q;(3) where 0qis the defect-mediated interaction term be- tween magnons at wavevector 0 and q,Cq() = (1 + (q)2)3=2is the correlation function of the magnetic sys- tem with correlation length , andH0is the magnitude of the characteristic inhomogeneity (units of magnetic eld). The -function in Eq. 3 selects only the magnon scattering channels that conserve energy. In the limit of zero intrinsic damping, it is identical to the Dirac delta function, but for nite it is replaced by a Lorentzian function of width != (2!= )d!=dH . The magnon dispersion relation determining !qis the usual Damon- Eshbach thin lm result [26, 28] with the addition of mag- netocrystalline anisotropy stiness eld terms extracted from the dependence of the resonance eld on the applied frequency for the IP conguration. The lm thickness daects the states available for two-magnon scattering through the dispersion relation, namely, the linear term which gives rise to negative group velocity for small q (/qd). The IP FMR linewidth data shown in Fig. 4 were t to Eq. 1 (with Eq. 3 used to evaluate
HTMS) with,, andH0as tting parameters (shown in Table II). The correlation length remains approximately con- stant for dierent in-plane directions, while the strength H0is larger for theh100idirections in the CMA and CFA samples and the h110idirections in the CMS sample. Some degree of uncertainty results from this tting proce- dure, because for linewidth data collected over a limited frequency range, andare not completely decoupled as tting parameters. In absolute terms, however, the largest systematic errors come from the exchange sti- ness, which is not well-known. The error bars given in Table II were calculated by varying the exchange sti- ness over the range 400 meV A2to 800 meV A2, and recording the change in the t parameters. This range of values was chosen based on previous Brillouin light scat- tering measurements of the exchange stiness in similar Heusler compounds [29, 30]. In addition, we note that in Eq. 1
H0is taken to be isotropic, with the value given by the PP linewidth measurements shown in Fig. 2. Although certain realizations of inhomogeneity may result in an anisotropic
H0(see Ref. [14] for a good discussion), doing so here would only serve to create an additional tting parameter. D. Eect of low intrinsic damping The eect of low intrinsic damping on the two-magnon linewidth can be seen in Fig. 5(a). As decreases, with all other parameters xed,
HTMS steadily increases and becomes increasingly nonlinear (and eventually non- monotonic) with frequency. In particular, a \knee" in the frequency dependence becomes more pronounced for6 TABLE II. Summary of the tting parameters used to t the in-plane data of Fig. 4 (black squares and red circles) to Eqs. 1 and 3. CFA refers to the unannealed Co 2FeAl sample. Sample (Field Direction) (103)(nm)H0(Oe) CMSh110i 1:60:2 4025 5530 CMSh100i 1:50:1 4025 3015 CMAh110i 3:10:2 7020 305 CMAh100i 4:70:4 5510 905 CFAh110i 2:00:3 2010 17560 CFAh100i N/A N/A N/A low damping (see e.g. Fig. 5(a) curve for = 104). The physics giving rise to the knee behavior is illustrated in Fig. 5(b). The TMS process scatters magnons from zero to non-zero wavevector at small q. There is assuemd to be sucient disorder to allow for the momentum q to be transferred to the magnon system. There will al- ways be, however, a length scale below which the disor- der decreases, so that the lm becomes eectively more uniform at large wavevectors. The corresponding FMR frequencies are those for which the contours of constant frequency (the gure eights in Fig. 5) in q-space have ex- trema atq1. The TMS rate is also determined by the interplay of the magnon density of states, the eec- tive area in q-space occupied by the modes that conserve energy, and the Gilbert damping. The knee behavior is more pronounced for low due to the increased weight of the van Hove singularity coming from the tips of the gure eights, in the integrand of Eq. 3. Although a larger window of energies, set by the width of , is available for larger, this smears out the singularity in the magnon density of states, removing the sharp knee in the TMS linewidth as a function of frequency. The PP measure- ment conrms that all of these epitaxial Heusler lms lie within the range <2103. Ferromagnetic lms with ultraloware therefore increasingly prone to large TMS linewidths (particularly for metals with large Ms). The TMS linewidths will also constitute a larger fraction of the total linewidth due to a smaller contribution from the Gilbert damping. In practice, this is why experimental reports [7, 22, 23] of ultralow have almost all utilized the PP geometry. E. Discussion The results of the IP linewidth ts to Eqs. 1 and 3 are summarized in Table II. In the case of CMS, the high- frequency slopes in Fig. 4(a) approach the same value along each direction, as would be expected when the fre- quency is large enough for the TMS wavevector to exceed the inverse of any defect correlation length. In this limit, is isotropic (within error limits). Next, we discuss the CMA IP data shown in Fig. 4(b) and Table II. It is clear from this gure that a good tcan be obtained along both h100iandh110idirections. In Table II it can be seen that the value of the defect cor- relation length is approximately the same along both directions. However, the values of we obtain from t- ting to Eqs. 1 and 3 do not agree well with the PP value of001= 1:8103(Fig. 2). Anisotropic values of  have been both predicted [5, 10] and observed [31], and an anisotropic is possibly the explanation of our best- t results. The in-plane h100iand [001] directions are equivalent in the bulk, so the anisotropy would neces- sarily be due to an interface anisotropy energy [31] or perhaps a tetragonal distortion due to strain [32]. Finally, we discuss the CFA linewidths shown in Fig. 4(c) and Table II. This sample has by far the largest two- magnon scattering contribution, which is likely related to the anomalously large inhomogeneous broadening and low intrinsic damping [see Fig. 5(a)] observed in the PP measurement. A good t of the data was obtained when the eld was applied along the h110idirection. Notably, the IPh110ibest t value of 2 :1103is nearly a factor of 3 larger than the 001upper bound on the same sample (Table I), strongly suggesting an anisotropic Gilbert . A striking anisotropy in the IP linewidth was revealed upon rotating the magnetization to the h100iorientation. For theh100icase, which yielded the largest TMS linewidths measured in this family of lms, we were not able to t the data to Eq. 3 using a set of physically reasonable in- put parameters. We believe that this is related to the consideration that higher order terms in the inhomoge- neous magnetic energy (see Ref. [26]) need to be taken into account. Another reason why this may be the case is that the model of McMichael and Krivosik [12] assumes the inhomogeneities to be grain-like, whereas the samples are epitaxial [see Fig. 1(a)]. Atomic force microscopy im- ages of these samples [Fig. 1(d)] imply that grains, if they exist, are much larger than the defect correlation lengths listed in Table II, which are of order 10's of nm. We also note that there does not appear to be a correlation be- tween the strength of two-magnon scattering H0and the cubic anisotropy eld 2 K1=Ms, which would be expected for grain-induced two-magnon scattering. V. SUMMARY AND CONCLUSION We conclude by discussing the successes and limita- tions of the McMichael and Krivosik [12] model in an- alyzing our epitaxial Heusler lm FMR linewidth data. We have shown that two-magnon scattering is the ex- trinsic linewidth-broadening mechanism in our samples. Any model which takes this as its starting point will predict much of the qualitative behavior we observe, such as the knee in the frequency dependence and the large linewidths IP for low lms. The TMS model used in this article (for the purpose of separating TMS and Gilbert linewidth contributions) is, however, only as accurate as its representation of the inhomogeneous magnetic eld and the underlying assumption for the7 functional form of Cq(). Grain-like defects are as- sumed, which essentially give a random magnetocrys- talline anisotropy eld. We did not, however, explicitly observe grains in our samples with AFM, at least below lengthscales of10m [Fig. 1(d)]. Mist dislocations, a much more likely candidate in our opinion, would cause an eective inhomogeneous magnetic eld which could have a more complicated spatial prole and therefore lead to anisotropic two-magnon scattering (see Ref. [14]). The perturbative nature of the model also brings its own limitations, and we believe that the CFA h100idata, for which we cannot obtain a satisfactory t, are exemplary of a breakdown in the model for strong TMS. Future work should go into methods of treating the two-magnon scattering dierently based on the type of crystalline de- fects present, which will in turn allow for a more reli- able extraction of the Gilbert damping and facilitate the observation of anisotropic Gilbert damping, enabling quantitative comparison to rst-principles calculations. Regardless of the limitations of the model, we empha- size three critical observations drawn from the linewidth measurements presented in this article. First, in all cases we observe large and anisotropic TMS linewidth contri- butions, which imply inhomogeneity correlation length- scales of order tens-to-hundreds of nanometers. The mi- croscopic origin of these inhomogeneities is the subjectof ongoing work, but are likely caused by arrays of mist dislocations [14]. The relatively large lengthscale of these defects may cause them to be easily overlooked in epi- taxial lm characterization techniques such as XRD and cross-sectional HAADF-STEM, but they still strongly in- uence magnetization dynamics. These defects and their in uence on the FMR linewidth through TMS complicate direct observation of Kambersk y's model for anisotropic and (in the case of Heusler compounds) ultralow intrinsic damping in metallic ferromagnets. Second, we observed low intrinsic damping through our PP measurement, which was<2103for all of our samples. Finally, we have presented the mechanism by which FMR linewidths in ultralow damping lms are particularly likely to be en- hanced by TMS, the anisotropy of which may dominate any underlying anisotropic Kambersk y damping. This work was supported by NSF under DMR-1708287 and by SMART, a center funded by nCORE, a Semi- conductor Research Corporation program sponsored by NIST. The sample growth was supported by the DOE under de-sc0014388 and the development of the growth process by the Vannevar Bush Faculty Fellowship (ONR N00014-15-1-2845). Parts of this work were carried out in the Characterization Facility, University of Minnesota, which receives partial support from NSF through the MRSEC program. [1] V. Kambersk y, On the Landau-Lifshitz relaxation in fer- romagnetic metals, Canadian Journal of Physics 48, 2906 (1970). [2] V. Kambersk y, On ferromagnetic resonance damping in metals, Czechoslovak Journal of Physics 26, 1366 (1976). [3] D. Steiauf and M. F ahnle, Damping of spin dynamics in nanostructures: An ab initio study, Physical Review B 72, 064450 (2005). [4] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Identica- tion of the Dominant Precession-Damping Mechanism in Fe, Co, and Ni by First-Principles Calculations, Physical Review Letters 99, 027204 (2007). [5] K. Gilmore, M. D. Stiles, J. Seib, D. Steiauf, and M. 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Ren, R.-R. Hao, H. Zhong, Y. Kang, S.-S. Kang, Y.-F. Qin, S.-Y. Yu, G.-B. Han, S.-S. Yan, and L.-M. Mei, Broad-Band FMR Linewidth of Co 2MnSi Thin Films with Low Damping Factor: The Role of Two- Magnon Scattering, Chinese Physics Letters 33, 047601 (2016). [17] Y. Cui, J. Lu, S. Sch afer, B. Khodadadi, T. Mewes, M. Osofsky, and S. A. Wolf, Magnetic damping and spin polarization of highly ordered B2 Co 2FeAl thin lms, Journal of Applied Physics 116, 073902 (2014).8 [18] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, Fer- romagnetic resonance linewidth in metallic thin lms: Comparison of measurement methods, Journal of Ap- plied Physics 99, 093909 (2006). [19] E. Montoya, T. McKinnon, A. Zamani, E. Girt, and B. Heinrich, Broadband ferromagnetic resonance system and methods for ultrathin magnetic lms, Journal of Magnetism and Magnetic Materials 356, 12 (2014). [20] C. Kittel, On the Theory of Ferromagnetic Resonance Absorption, Physical Review 73, 155 (1948). [21] A. Kumar, F. Pan, S. Husain, S. Akansel, R. Bru- cas, L. Bergqvist, S. Chaudhary, and P. Svedlindh, Temperature-dependent Gilbert damping of Co 2FeAl thin lms with dierent degree of atomic order, Physi- cal Review B 96, 224425 (2017). [22] M. Oogane, A. P. McFadden, K. Fukuda, M. Tsunoda, Y. Ando, and C. J. Palmstrm, Low magnetic damp- ing and large negative anisotropic magnetoresistance in half-metallic Co 2xMn 1+xSi Heusler alloy lms grown by molecular beam epitaxy, Applied Physics Letters 112, 262407 (2018). [23] C. Guillemard, S. Petit-Watelot, L. Pasquier, D. Pierre, J. Ghanbaja, J.-C. Rojas-S anchez, A. Bataille, J. Rault, P. Le F evre, F. Bertran, and S. Andrieu, Ultralow Mag- netic Damping in Co 2Mn-Based Heusler Compounds: Promising Materials for Spintronics, Physical Review Ap- plied 11, 064009 (2019). [24] K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. 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2019-09-06

We report on broadband ferromagnetic resonance linewidth measurements performed on epitaxial Heusler thin films. A large and anisotropic two-magnon scattering linewidth broadening is observed for measurements with the magnetization lying in the film plane, while linewidth measurements with the magnetization saturated perpendicular to the sample plane reveal low Gilbert damping constants of $(1.5\pm0.1)\times 10^{-3}$, $(1.8\pm0.2)\times 10^{-3}$, and $<8\times 10^{-4}$ for Co$_2$MnSi/MgO, Co$_2$MnAl/MgO, and Co$_2$FeAl/MgO, respectively. The in-plane measurements are fit to a model combining Gilbert and two-magnon scattering contributions to the linewidth, revealing a characteristic disorder lengthscale of 10-100 nm.

The interplay of large two-magnon ferromagnetic resonance linewidths and low Gilbert damping in Heusler thin films

1909.02738v2

arXiv:1908.06862v1 [math-ph] 19 Aug 2019SPECTRAL DETERMINANT FOR THE DAMPED WAVE EQUATION ON AN INTERVAL PEDRO FREITAS Departamento de Matem´ atica, Instituto Superior T´ ecnico , Universidade de Lisboa, Av. Rovisco Pais 1, P-1049-001 Lisboa, Portugal andGrupo de F´ ısica Matem´ atica, Faculdade de Ciˆ encias, Universidade de Lisboa, Campo Gran de, Edif´ ıcio C6, P-1749-016 Lisboa, Portugal JIˇR´I LIPOVSK ´Y Department of Physics, Faculty of Science, University of Hr adec Kr´ alov´ e, Rokitansk´ eho 62, 50003 Hradec Kr´ alov´ e, Czechia Abstract. We evaluate the spectral determinant for the damped wave equat ion on an interval of length Twith Dirichlet boundary conditions, proving that it does not depend on the damping. This is achieved by analysing the square of th e damped wave operator using the general result by Burghelea, Friedlander , and Kappeler on the determinant for a differential operator with matrix coefficients . PACS: 46.40.Ff, 03.65.Ge 1.Introduction We consider the simple mathematical model of wave propagationon a damped string fixed at both ends given by ∂2v(t,x) ∂t2+2a(x)∂v(t,x) ∂t=∂2v(t,x) ∂x2, (1.1) with the space variable xon an interval [0 ,T],v(0) =v(T) = 0 and a(x)∈C([0,T]). Despite its apparent simplicity, the problem is nontrivial and interest ing and has re- ceived much attention over the last two decades – see, for instanc e, [GH11, FL17, CFN+91, CZ94, BF09]. The operator associated with (1.1) is non-selfadjoint and the asym ptotical location of its eigenvalues was determined to first order in [CFN+91, CZ94], wh ere it was shown that eigenvalues λconverge to the vertical line Re λ=−/a\}bracketle{ta/a\}bracketri}htas their imaginary part goes to±∞, where/a\}bracketle{ta/a\}bracketri}htdenotes the average of the damping function. The general asymptotic behaviour was analysed in [BF09], where further spectr al invariants were determined. Comingfromothersourcesintheliterature, thenotionofdetermin antofamatrixhas been generalised to operators. In this analogy, we would like to obta in a regularisation corresponding to the product of eigenvalues of the given operato r. If the considered E-mail addresses :psfreitas@fc.ul.pt, jiri.lipovsky@uhk.cz . 12 DETERMINANT FOR THE DAMPED WAVE EQUATION operator Shas eigenvalues {λj}∞ j=1, in agreement with [RS71] (see also [GY60, MP49]) we define the generalised zeta function associated with the operat orSby ζS(s) =∞/summationdisplay j=1λ−s j, for complex sin a half-plane such that the above Dirichlet series converges. The spectral determinant may then be defined by the formula DetS= e−ζ′ S(0), (1.2) where the prime denotes the derivative with respect to the variable s. Note that the series defining the zeta function will not, in general, be convergent fors= 0. We use the definition of ζSfor the real part of slarge enough and understand the formula in the sense of the analytic continuation of the generalized zeta func tion to the complex plane. The spectral determinant was computed for the Sturm-Liouville op erator in [LS77], where an elegant expression using the solution of a corresponding C auchy problem was presented. This was extended to the case of quantum graphs in [AC D+00, Fri06]. We point out that spectral determinants have several applications e .g. in quantum field theory [Dun08]. To the best of our knowledge the spectral determinant for the da mped wave equation had not been studied previously, so in this note we bring together th ese two topics and evaluate this object. From a mathematical perspecive there is also what we believe to be the interesting feature of applying the concept of the determin ant of an operator to a non-selfadjoint operator. Furthermore, and as we will see, the determinant does not, in fact, depend on the damping. This may be expected form a formal analysis, and our purpose is to give a rigorous justification of this fact. This note is structured as follows. In the next section, we ellaborat e on the math- ematical description of the model and state the main result. In Sec tion 3 we then address the problem for the case without damping, as this already d isplays some of the important features which we will need to consider later, namely, the fact that the asso- ciated zeta function will depend on the branch cut which is chosen fo r the logarithm. In Section 4 we recall the general result of Burghelea, Friedlander , and Kappeler. In Section 5 we apply this result to the square of our operator, since a direct application is not possible. Finally, we find the sought determinant for the dampe d wave equation in Section 6. 2.Basic setting and formulation of the main result Equation (1.1) may be written in a different form, namely, ∂ ∂t/parenleftbigg v0(t,x) v1(t,x)/parenrightbigg =/parenleftbigg0 1 ∂2 ∂x2−2a(x)/parenrightbigg/parenleftbigg v0(t,x) v1(t,x)/parenrightbigg which will prove to be more convenient for our purposes. Using the a nsatzv0(t,x) = eλtu0(x),v1(t,x) = eλtu1(x), we can translate the initial value problem into the follow- ing spectral problem H/parenleftbigg u0(x) u1(x)/parenrightbigg =λ/parenleftbigg u0(x) u1(x)/parenrightbiggDETERMINANT FOR THE DAMPED WAVE EQUATION 3 whereHdenotes the matrix operator H=/parenleftbigg0 1 ∂2 ∂x2−2a(x)/parenrightbigg . The domain of this operator consists of functions u(x) =/parenleftbigg u0(x) u1(x)/parenrightbigg with components in the Sobolev spaces uj(x)∈W2,2([0,T]),j= 0,1 satisfying the Dirichlet boundary conditions uj(0) =uj(T) = 0, j= 0,1. Our main result is the following. Theorem 2.1. Assumea(x)∈C([0,T]), and let εbe a positive number such that there are no eigenvalues with phase on the interval [π−ε,π), Then the spectral determinant of the operator Hdoes not depend on the damping and equals ±2T, where the plus and minus signs correspond to whether we define λ−s j= e−slogλjin such a way that the branch cut of the logarithm is λ=tei(π−ε), orλ=tei(2π−ε),t∈[0,∞), respectively. Remark 2.2. Note that a value of εas above always exists, since on any compact set there are only a finite number of eigenvalues. Remark 2.3. The case of the damped wave equation where a potential is adde d to the right-hand side of (1.1)may be treated in a similar fashion and the corresponding determinant also turns out to be independent of the damping t erm. We discuss this situation in Remark 6.2. 3.The case of a(x) = 0 We begin by considering the case without damping, and denote the co rresponding operator by H0. It is a simple exercise that its eigenvalues are of the form λj=ijπ T, j∈Z\{0}. To obtain the spectral determinant in this instance, we start fro m the zeta function resulting from the definition (1.2). However, one must pro ceed carefully here, as the result depends on the definition of λ−s jand, in particular, on which branch of the logarithm we use when defining i−sand (−i)−s. First, we consider that the logarithm has the cut in the negative rea l axis, i.e. the eigenvalues of H0in the upper half-plane are λj=jπ Teiπ 2,j∈Nand the eigenvalues in the lower half-plane are λj=jπ Te−iπ 2,j∈N. The generalized zeta function for this operator is ζH0(s) =∞/summationdisplay j=1/bracketleftBigg/parenleftbiggjπ Teiπ 2/parenrightbigg−s +/parenleftbiggjπ Te−iπ 2/parenrightbigg−s/bracketrightBigg =∞/summationdisplay j=1/parenleftBig e−iπ 2s+eiπ 2s/parenrightBig/parenleftbiggjπ T/parenrightbigg−s = 2eslogT πcos/parenleftBigπs 2/parenrightBig ζR(s), whereζR(s) =∞/summationdisplay j=1j−sis the Riemann zeta function. We obtain −ζ′ H0(0) =−2logT πζR(0)−2ζ′ R(0) = logT π+log(2π) = log(2 T),4 DETERMINANT FOR THE DAMPED WAVE EQUATION where we have used ζR(0) =−1 2andζ′ R(0) =−1 2log(2π). Hence the spectral determi- nant for the operator H0is given by DetH0= e−ζ′ H0(0)= 2T . Now we are going to compute the determinant in the case where we ch oose the cut to be the positive real axis. The eigenvalues are λj=jπ Teiπ 2,j∈N, for the upper half- plane and λj=jπ Te3iπ 2,j∈Nfor the lower half-plane. The generalized zeta function is now ζH0(s) =∞/summationdisplay j=1/bracketleftBigg/parenleftbiggjπ Teiπ 2/parenrightbigg−s +/parenleftbiggjπ Te3iπ 2/parenrightbigg−s/bracketrightBigg = e−iπs∞/summationdisplay j=1/parenleftBig eiπ 2s+e−iπ 2s/parenrightBig/parenleftbiggjπ T/parenrightbigg−s = 2e−iπseslogT πcos/parenleftBigπs 2/parenrightBig ζR(s). Hence we have −ζ′ H0(0) =−2logT πζR(0)+2iπζR(0)−2ζ′ R(0) = logT π−iπ+log(2π) =−iπ+log(2T). The spectral determinant for the operator H0is DetH0= e−ζ′ H0(0)=−2T . 4.A general result The starting point for finding the determinant for the damped wave equation is a general result by Burghelea, Friedlander and Kappeler [BFK95]. Th is result gives a formula for the determinant of a more general matrix-valued oper ator on an interval. For convenience, we state this result here, together with the nec essary definitions. Definition 4.1. Let us for n∈Ndefine the operator A=2n/summationdisplay k=0ak(x)(−i)kdk dxk, where akarer×rmatrices, in general smoothly dependent on x∈[0,T]. We assume that the leading term a2nis nonsingular and that there exist an angle θso thatspeca2n∩ {ρeiθ,0≤ρ <∞}=∅. We assume the following boundary conditions at the end poin ts of the interval. αj/summationdisplay k=0bjku(k)(T) = 0,βj/summationdisplay k=0cjku(k)(0) = 0,1≤j≤n. Here,bjkandcjkare for each j,kconstant r×rmatrices and bjαj=cjβj=I(Idenotes ther×ridentity matrix). The integer numbers αjandβjsatistfy 0≤α1< α2<···< αn<2n−1, 0≤β1< β2<···< βn<2n−1. Moreover, we define |α|=/summationtextn j=1αj,|β|=/summationtextn j=1βj. We define the 2n×2nmatrices B= (Bjk)andC= (Cjk), whose entries are r×rmatrices. Here 1≤j≤2n,DETERMINANT FOR THE DAMPED WAVE EQUATION 5 0≤k≤2n−1and Bjk:=/braceleftbigg bjkfor 1≤j≤nand 0≤k≤αj 0 otherwise, Cjk:=/braceleftbigg bj−n,kforn+1≤j≤2nand 0≤k≤αj−n 0 otherwise. We define a 2n×2nmatrixY(x) = (ykℓ(x)),0≤k,ℓ≤2n−1whose entries are r×rmatrices ykℓ(x)defined by ykℓ(x) :=dkyℓ(x) dxk, whereyℓ(x)is the solution of the Cauchy problem Ayℓ(x) = 0with the initial conditions ykℓ(0) =δkℓI. We are interested in the value of the matrix Yat the point T. Finally, we introduce gα:=1 2/parenleftbigg|α| n−n+1 2/parenrightbigg , hα:= det wα1 1... wα1n......... wαn 1... wαnn , wherewk= exp/parenleftbig2k−n−1 2nπi/parenrightbig . Similarly, we define gβandhβ. We denote by γj,j= 1,...,rthe eigenvalues of the matrix a2nand define (deta2n)gα θ:=r/productdisplay j=1|γj|gαexp(igαarg(γj)) withθ−2π <argγj< θ. Theorem 4.2. (Burghelea, Friedlander and Kappeler) The spectral determinant for the operator Ais DetA=Kθexp/parenleftbiggi 2/integraldisplayT 0Tr(a2n(x)−1a2n−1(x))dx/parenrightbigg det(BY(T)−C), where Kθ= [(−1)|β|(2n)nh−1 αh−1 β]r(deta2n(0))gβ θ(deta2n(T))gα θ. 5.The square of the operator H OurpurposeistouseTheorem4.2toobtainthespectraldetermina ntfortheoperator H. However, a direct application of the theorem is not possible since th e highest order derivative is only present in one of the entries of the matrix which give s the operator H. This would contradict the assumption that the matrix a2nis nonsingular. In order to overcome this difficuly, we shall consider the operator A=H◦H=H2for which it is then possible to apply Theorem 4.2. However, we have to be careful when computing the spectral det erminant of Hfrom the spectral determinant of A, as the determinant of the composition of two operators does not necessarily have to equal the product of their determina nts. This is known in the literature as the multiplicative anomaly and, as has been shown in [BS03, Woj01], even the determinant of the square of an operator is not always th e square of the determinant.6 DETERMINANT FOR THE DAMPED WAVE EQUATION A direct calculation yields A=/parenleftbigg0 1 ∂2 ∂x2−2a(x)/parenrightbigg2 =/parenleftbigg∂2 ∂x2 −2a(x) −2a(x)∂2 ∂x2∂2 ∂x2+4a2(x)/parenrightbigg =/parenleftbigg −1 0 2a(x)−1/parenrightbigg/parenleftbigg −i∂ ∂x/parenrightbigg2 +/parenleftbigg 0−2a(x) 0 4a2(x)/parenrightbigg/parenleftbigg −i∂ ∂x/parenrightbigg0 . Hence we have n= 1, a2(x) =/parenleftbigg −1 0 2a(x)−1/parenrightbigg , a1=/parenleftbigg 0 0 0 0/parenrightbigg , a0=/parenleftbigg 0−2a(x) 0 4a2(x)/parenrightbigg . We deal with matrices 2 ×2, sor= 2, and have α1=β1= 0 and hence |α|=|β|= 0. The boundary conditions according to Definition 4.1 are b10u(T)+b11u′(T) = 0, c10u(0)+c11u′(0) = 0. We have the Dirichlet boundary conditions u(T) =u(0) =0, and thus the matrices bjkandcjkmust be chosen as b10=c10=I , b 11=c11= 0, whereIis the 2×2 identity matrix. The matrices BandCare 2×2 matrices with the entries being 2 ×2 matrices. Their rows are indexed by 1 and 2, their columns by 0 and 1. According to De finition 4.1 we have B10=b10=I, B 11=B20=B21= 0, C20=c10=I , C 11=C10=C21= 0. Hence we have B=/parenleftbigg I0 0 0/parenrightbigg , C=/parenleftbigg 0 0 I0/parenrightbigg . The matrix a2n=a2clearly has eigenvalues γ1=γ2=−1. Moreover, we have gα=gβ=1 2/parenleftbigg0 2−1+1 2/parenrightbigg =−1 4, hα=hβ=w0 1= 1. and (deta2n)gα θ=2/productdisplay j=11−1/4ei(−1/4)(−π)=i To sum up, Kθ= [(−1)0(2·1)11−2]2i2=−4. Now, we are going to compute the matrix Y(x). By its definition, we have Y(x) =/parenleftbigg y00(x)y01(x) y10(x)y11(x)/parenrightbigg =/parenleftbigg y0(x)y1(x) y′ 0(x)y′ 1(x)/parenrightbigg , where the entries of this matrix are 2 ×2 matrices with the following boundary condi- tions atx= 0 y0(0) =I, y′ 0(0) = 0, y1(0) = 0, y′ 1(0) =I . (5.1)DETERMINANT FOR THE DAMPED WAVE EQUATION 7 For the matrix in the formula in Theorem 4.2 we have det(BY(T)−C) = det/bracketleftbigg/parenleftbigg I0 0 0/parenrightbigg/parenleftbigg y0(T)y1(T) y′ 0(T)y′ 1(T)/parenrightbigg −/parenleftbigg 0 0 I0/parenrightbigg/bracketrightbigg = det/parenleftbigg y0(T)y1(T) −I0/parenrightbigg = dety1(T). Now, we will find the solutions of the Cauchy problem, i.e. the equation H2/parenleftbigg u0(x) u1(x)/parenrightbigg = 0 with the boundary conditions (5.1). We obtain u′′ 0(x)−2a(x)u1(x) = 0, (5.2) −2a(x)u′′ 0(x)+u′′ 1(x)+4a2(x)u1(x) = 0. (5.3) Multiplying (5.2) by 2 a(x) and adding the result to (5.3) we obtain u′′ 1(x) = 0. (5.4) The boundary conditions are y(k) l(0) =δklI. Hence to obtain the matrix y1(x), we have to find two vectors/parenleftbigg u01(x) u11(x)/parenrightbigg and/parenleftbigg u02(x) u12(x)/parenrightbigg , for which y1(0) =/parenleftbigg u01(0)u02(0) u11(0)u12(0)/parenrightbigg = 0, y′ 1(0) =/parenleftbigg u′ 01(0)u′ 02(0) u′ 11(0)u′ 12(0)/parenrightbigg =I.(5.5) The general form of the solution of (5.4) is u1(x) =ax+b, and from the first condition in (5.5) we have b= 0. The second condition in (5.5) yields u11(x) = 0, u12(x) =x. Hence in the first case we have from (5.2) u′′ 01(x) = 0, hence we have u01(x) =cx+d. By the first condition in (5.5) we have d= 0, by the second one u01(x) =x. In the second case ( u12(x) =x) we obtain from (5.2) u′′ 02(x) = 2a(x)x. Hence with the use of the second condition in (5.5) we have u′ 02(x) =/integraltextx 02a(s)sdsand with the use of the first condition in (5.5) we have u02(x) =/integraltextx 0/integraltextq 02a(s)sdsdq. To sum up, y1(x) =/parenleftbigg x/integraltextx 0/integraltextq 02a(s)sdsdq 0 x/parenrightbigg and hence det y1(T) =T2. We obtained DetA=−4T2. 6.The determinant of H Let us consider the operator Hin the general case. In each horizontal strip |Imλ|< Kthere are finitely many eigenvalues. Hence there exists a positive εso that there are no eigenvalues whose arguments are in the intervals (0 ,ε], [π−ε,π), (π,π+ε], and [2π−ε,2π). We will choose the cut of the logarithm as the half-line λ=tei(π−ε), t∈[0,∞). We denote the eigenvalues of Hin the upper half-plane by ˜ µj, and their complex conjugates by ¯˜µj, the positive real eigenvalues by ˜ νj,j∈I1(hereI1is a finite8 DETERMINANT FOR THE DAMPED WAVE EQUATION index set) and the negative real eigenvalues as −˜ωj,j∈I2(hereI2is a finite index set). It can be easily proven that there are no zero eigenvalues of H. For the operator A, we denote the eigenvalues as µ2 j, ¯µ2 j,ν2 jandω2 j. We consider the phases of all these eigenvalues in the interval [0 ,2π−2ε), the cut of the logarithm for the operator Ais the half-line λ=te−2iε. Hence one can easily obtain log ˜µj=1 2logµ2 j,log¯˜µj=1 2log ¯µ2 j−πi, log˜νj=1 2logν2 j,log(−˜ωj) =1 2logω2 j−πi. The zeta functions for these operators are ζA(s) =∞/summationdisplay j=1[(µ2 j)−s+(¯µ2 j)−s]+/summationdisplay j∈I1(ν2 j)−s+/summationdisplay j∈I2((−ωj)2)−s =∞/summationdisplay j=1(e−slogµ2 j+e−slog ¯µ2 j)+/summationdisplay j∈I1e−slogν2 j+/summationdisplay j∈I2e−slogω2 j. ζH(s) =∞/summationdisplay j=1[˜µ−s j+¯˜µj−s]+/summationdisplay j∈I1˜ν−s j+/summationdisplay j∈I2(−˜ωj)−s =∞/summationdisplay j=1(e−slog ˜µj+e−slog¯˜µj)+/summationdisplay j∈I1e−slog ˜νj+/summationdisplay j∈I2e−slog(−˜ωj) =∞/summationdisplay j=1(e−1 2slogµ2 j+e−1 2slog ¯µ2 jeπis)+/summationdisplay j∈I1e−1 2slogν2 j+/summationdisplay j∈I2e−1 2slogω2 jeπis. Hence we have ζH(s)−ζA/parenleftBigs 2/parenrightBig = (eπis−1)/parenleftBigg∞/summationdisplay j=1e−1 2slog ¯µ2 j+/summationdisplay j∈I2e−1 2slogω2 j/parenrightBigg = 2ieπis 2sinπs 2/parenleftBigg∞/summationdisplay j=1e−1 2slog ¯µ2 j+/summationdisplay j∈I2e−1 2slogω2 j/parenrightBigg . For the derivatives at zero we obtain, using the fact that the sine w ill vanish in that case, ζ′ H(0)−1 2ζ′ A(0) =iπlim s→0∞/summationdisplay j=1e−1 2slog ¯µ2 j+iπcardI2 =iπcardI2+iπlim s→0∞/summationdisplay j=1e−iπs 2eslogT πe−slogj +iπlim s→0∞/summationdisplay j=1/parenleftBig e−1 2slog ¯µ2 j−e−iπs 2eslogT πe−slogj/parenrightBig . Wewillprove thatthelasttermiszero. Wewillusetheasymptotics (s eee.g. [BF09]) ¯µ2 j=−j2π2 T2/parenleftbigg 1+O/parenleftbigg1 j/parenrightbigg/parenrightbiggDETERMINANT FOR THE DAMPED WAVE EQUATION 9 and by our assumption ¯ µ2 jis not close to zero. We have lim s→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay j=1(e−1 2slog ¯µ2 j−e−iπs 2eslogT πe−slogj)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle= lim s→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay j=1e−iπs 2eslogT πe−slogj ×(e−1 2slog(1+O(1 /j))−1)/vextendsingle/vextendsingle/vextendsingle ≤lim s→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle∞/summationdisplay j=1e−slogj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|eslogT π||e−sC−1| = lim s→0|ζR(s)||eslogT π||e−sC−1| = 0. whereCis a real constant and ζR(0) =−1 2. Hence we have ζ′ H(0)−1 2ζ′ A(0) =iπcardI2+iπζR(0) =iπcardI2−iπ 2. (6.1) Then the determinant is equal to DetH= e−ζ′ H(0)= e−1 2ζ′ A(0)eiπ 2e−iπcardI2=∓i√ DetA=∓i√ −4T2=±2T . The number of negative real eigenvalues card I2is always even, so this term does not influence the result. The aim of the rest of our analysis is to find t he sign of the determinant for the case of the cut we chose. First, we will follow an approach different from that which was used in Section 3 to find the determinant for the operator without damping. The zeta function for the operator A0is ζA0(s) = 2∞/summationdisplay j=1/parenleftbiggjπ T/parenrightbigg−2s (−1)−s= 2/parenleftbiggT π/parenrightbigg2s e−slog(−1)∞/summationdisplay j=1j−2s= 2e−iπs/parenleftbiggT π/parenrightbigg2s ζR(2s). Then we have −ζ′ A0(0) = 2πiζR(0)−4logT πζR(0)−4ζ′ R(0) =−πi+2log(2 T). Using equation (6.1) we obtain DetH0= e−ζH′ 0(0)= e−1 2ζA′ 0(0)eiπ 2= e−iπ 2eiπ 2elog(2T)= 2T . Now, we are going to generalize this approach to the operator with d amping. Our aim is to find the imaginary part of ζ′ H0(0). We denote the absolute value of µ2 jbyrj and its phase by π−ϕj. We have µ2 j=rjei(π−ϕj),¯µ2 j=rjei(π+ϕj). We obtain (µ2 j)−s+(¯µ2 j)−s=r−s je−iπs(eiϕjs+e−iϕjs) = 2e−iπsr−s jcos(ϕjs) and since r−s jcos(ϕjs),ν2 jandω2 jare real Imd ds/braceleftBigg∞/summationdisplay j=1/bracketleftbig (µ2 j)−s+(¯µ2 j)−s/bracketrightbig +/summationdisplay j∈I1(ν2 j)−s+/summationdisplay j∈I1(ω2 j)−s/bracerightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle s=0=−2πlim s→0∞/summationdisplay j=0r−s jcos(ϕjs) =−πlim s→0∞/summationdisplay j=1/bracketleftbig (µ2 j)−s+(¯µ2 j)−s/bracketrightbig10 DETERMINANT FOR THE DAMPED WAVE EQUATION We will use the asymptotic behaviour of the eigenvalues of the opera torH(see [BF09]) µj=jπ Ti−/a\}bracketle{ta/a\}bracketri}ht−i/a\}bracketle{ta2/a\}bracketri}htT 2πj+O/parenleftbigg1 j2/parenrightbigg (here/a\}bracketle{t·/a\}bracketri}htdenotestheaverageofthefunctionontheinterval)whichleadsto thefollowing behaviour of the eigenvalues of the operator A µ2 j=−j2π2 T2−2jπ/a\}bracketle{ta/a\}bracketri}ht Ti+O(1). Hence we obtain (µ2 j)−s= (−1)−sj−2sT2s π2s/parenleftbigg 1−2T/a\}bracketle{ta/a\}bracketri}hts πji+O(j−2)/parenrightbigg . Since ¯µ2 jis the complex conjugate of µ2 j, we have ∞/summationdisplay j=1/bracketleftbig (µ2 j)−s+(¯µ2 j)−s/bracketrightbig = 2T2s π2s(−1)−s∞/summationdisplay j=0j−2s+∞/summationdisplay j=1O(j−2s−2). Thesecondseriesisabsolutelyconvergent downtoRe s= 0,andhencewecanexchange the sum and the limit. Moreover, the term under the sum goes to zer o ass→0, as it is multiplied by s. The first term can be written as 2e−iπsT2s π2sζR(2s), hence its limit is equal to −1. We conclude that −Imζ′ A(0) =−π. Using equation (6.1) similarly to the case of no damping leads to the pos itive sign for the determinant with the cut of the logarithm taken just above the negative real axis. This proves Theorem 2.1. Remark 6.1. It is possible prove in the similar manner that if one moves th e cut so that it passes finitely many eigenvalues of H, the determinant does not change. Remark 6.2. If one considers the dampedwave equation with a potential te rm, namely, ∂2v(t,x) ∂t2+2a(x)∂v(t,x) ∂t=∂2v(t,x) ∂x2+b(x)v(t,x), then it is possible to use a similar approach to the above. If t he operator on the right- hand side has only negative eigenvalues, the negative eigen values of the damped wave equation always appear in pairs, as in the case without the po tential. The only difference is in the solution of the Cauchy problem. We define y(x)satisfying y′′(x)+b(x)y(x) = 0, y(0) = 0,y′(0) = 1. The result for the determinant is DetH= 2y(T)for the cut of the logarithm just above the negative real axis, and −2y(T)for the opposite case. If the operator on the right-hand side also has positive eigenvalu es (but no zero eigenvalue), the phase (−1)cardI2appears multiplying the determinant, changing the sign acc ording to whether the number of negative eigenvalues of the damped w ave equation is odd or even.DETERMINANT FOR THE DAMPED WAVE EQUATION 11 Acknowledgements P.F. was partially supported by the Funda¸ c˜ ao para a Ciˆ encia e a T ecnologia, Portu- gal, through project PTDC/MAT-CAL/4334/2014. J.L. was suppo rted by the project “International mobilities for research activities of the University o f Hradec Kr´ alov´ e” CZ.02.2.69/0.0/0.0/16 027/0008487. J.L. thanks the University of Lisbon for its hos- pitality during his stay in Lisbon. References [ACD+00] E. Akkermans, A. Comtet, J. Desbois, G. Montambaux, C . Texier, Spectral determinant on quantum graphs, Annals of Physics 284(2000), 10–51. DOI: 10.1006/aphy.2000.6056. [BF09] D. Borisov, P. Freitas, Eigenvalue asymptotics, inverse pro blems and a trace for- mula for the linear damped wave equation, J. Diff. Eq. 247(2009), 3028–3039. DOI: 10.1016/j.jde.2009.07.029. [BFK95] D. Burghelea, L. Friedlander, T. Kappeler, On the determin ant of elliptic boundary value problems on a line segment, Proc. Amer. Math. Soc. 123(1995), 3027–3038. DOI 10.2307/2160656. [BS03] C. B¨ ar, S. Schopka, The Dirac Determinant of Spherical Sp ace Forms, in Hildebrandt S., Karcher H. (eds), Geometric Analysis and Nonlinear Part ial Differential Equations . Springer, Berlin, Heidelberg (2003). DOI: 10.1007/978-3-642-55 627-23. [CFN+91] G. Chen, S.A. Fulling, F.J. Narcowich, S. Sun, Exponential d ecay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math. 51(1991), 266–301. DOI: 10.1137/0151015. [CZ94] S. Cox, E. Zuazua, The rate at which energy decays in a damp ed string, Comm. Partial Differential Equations 19(1994) 213–243. DOI: 10.1080/03605309408821015. [Dun08] G. V. Dunne, Functional determinants in quantum field theo ry,J. Phys. A: Math. Theor. 41(2008) 304006. DOI: 10.1088/1751-8113/41/30/304006 [FL17] P. Freitas, J. Lipovsk´ y, Eigenvalue asymptotics for the da mped wave equation on metric graphs,J. Diff. Eq. 263(2017), 2780–2811. DOI: 10.1016/j.jde.2017.04.012. [Fri06] L. Friedlander, Determinant of the Schr¨ odinger operator on a metric graph. Quantum graphs and their applications, 151–160, Contemp. Math. 415, Amer. Math. Soc., Provi- dence, RI, 2006. DOI: 10.1090/conm/415/07866 [GY60] I.M. Gelfand and A.M. Yaglom, Integration in functional space s and its applications in quantum physics, J. Math. Phys. 1(1960), 48–69. DOI: 10.1063/1.1703636. [GH11] F. Gesztesy, H. Holden, The damped string problem revisited ,J. Diff. Eq. 251(2011) 1086–1127. DOI: 10.1016/j.jde.2011.04.025. [LS77] S. Levit, U. Smilansky, A Theorem on Infinite Products of Eige nvalues of Sturm-Liouville Type Operators, Proc. Am. Math. Soc. 65(1977), 299–302. DOI: 10.2307/2041911. [MP49] S. Minakshisundaram and ˚A. Pleijel, Some properties of the eigenfunctions of the Laplace- operator on Riemannian manifolds, Can. J. Math. 1(1949) 242–256. DOI: 10.4153/CJM- 1949-021-5. [RS71] D.B. Ray and I.M. Singer, R-torsion and the Laplacian on Rieman nian manifolds, Adv. Math.7(1971), 145–210. DOI 10.1016/0001-8708(71)90045-4. [Woj01] K. P. Wojciechowski, Heat equation and spectral geometr y: Introduction for beginners, in H. Ocampo, S. Paycha, A. Reyes, Geometric Methods for Quantu m Field Theory , World Scientific, pp. 238-292 (2001). DOI: 10.1142/9789812810571 0004.

2019-08-19

We evaluate the spectral determinant for the damped wave equation on an interval of length $T$ with Dirichlet boundary conditions, proving that it does not depend on the damping. This is achieved by analysing the square of the damped wave operator using the general result by Burghelea, Friedlander, and Kappeler on the determinant for a differential operator with matrix coefficients.

Spectral determinant for the damped wave equation on an interval

1908.06862v1

arXiv:2304.11522v1 [math.AP] 23 Apr 2023Decay rates for a variable-coefficient wave equation with nonlinear time-dependent damping Menglan Liao∗ College of Science, Hohai University, Nanjing, 210098, China Abstract: In this paper, a class of variable-coefficient wave equations equipped with time- dependent damping and the nonlinear source is considered. W e show that the total energy of the system decays to zero with an explicit and precise deca y rate estimate under different assumptions on the feedback with the help of the method of wei ghted energy integral. Keywords: Variable coefficient; Time-dependent damping; Energy estim ates; Nonlinear source. MSC(2020): 35B35; 35L70; 35A01. 1 Introduction In this paper, we are concerned with the following initial-b oundary value problem for a class of variable-coefficient wave equations with time-dependent damping utt−Au+γ(t)g(ut) =f(u)x∈Ω, t>0, u(x,t) = 0 x∈∂Ω, t>0, u(x,0) =u0(x), ut(x,0) =u1(x)x∈Ω,(1.1) where Ω ⊂R3, for simplicity, is a bounded domain with C∞boundary. In particular, present results can be extended to bounded domain in Rnby adopting the Sobolev embeddings, and by adjusting some parameters imposed. Here γ(t)g(ut) is a time-dependent damping term, and f∈C1(R) such that |f′(s)| ≤C(|s|p−1+ 1) with 1 ≤p <6. To simplify computations, we choosef(u) :=|u|p−1uwith 1≤p<6 in this paper. The second-order differential operator A is defined by Au= div(A(x)∇u) =∞/summationdisplay i,j=1∂ ∂xi/parenleftBig aij(x)∂u ∂xj/parenrightBig , whereA(x) = (ai,j(x)) is symmetric and positive definite matrices with ai,j(x)∈C∞(¯Ω) and satisfies the uniform ellipticity conditions n/summationdisplay i,j=1ai,j(x)ξiξj≥ωn/summationdisplay i=1ξ2 i, x∈¯Ω, ω>0. (1.2) It is easily verified that the bilinear form a(·,·) :H1 0(Ω)×H1 0(Ω)→Rdefined by a(u,v) =n/summationdisplay i,j=1/integraldisplay Ωai,j(x)∂u ∂xj∂u ∂xidx=/integraldisplay ΩA∇u·∇vdx ∗Corresponding author: Menglan Liao Email addresses: liaoml@hhu.edu.cn2 is symmetric and continuous. Further, it follows from ( 1.2) that a(u,v)≥ω/ba∇dbl∇u/ba∇dbl2 2. (1.3) The problem of asymptotic stability of solutions of dissipa tive wave systems equipped with time-dependent nonlinear damping forces has been given a lo t of attention. P. Pucci and J. Serrin [11] investigated the following nonlinear damped wave system w ith Dirichlet data utt−∆u+Q(t,x,u,ut)+f(x,u) = 0, (1.4) where the function Qrepresents a nonlinear damping satisfying ( Q(t,x,u,v),v)≥0,fis a restoring force (that is (f(x,u),u)≥0). Based on the a priori existence of a suitable auxiliary function, they proved the natural energy Eu(t) associated with solutions of the system satisfies limt→+∞Eu(t) = 0. Problem ( 1.4) is well-known in a variety of models in mathematical physic s, forinstance, elasticvibrationsinadissipativemedium,t hetelegraphicequation, andthedamped Klein-Gordon equation. P. Pucci and J. Serrin [ 13] studied again problem ( 1.4) not only for potential energies which arise from restoring forces, but a lso for the effect of amplifying forces . They pointed out that global asymptotic stability can no lon ger be expected, and should be replaced by local stability. However, whether an explicit a nd precise decay rate estimate of the total energy of the system can be obtained was unknown. P. Martinez [ 10] considered the following time-dependent dissipative systems utt−∆u+ρ(t,ut) = 0 with Dirichlet boundary, where ρ:R+×R→Ris a continuous function differentiable on R+×(−∞,0). By generalizing the method introduced to study autonomo us wave equation damped with a boundary nonlinear velocity feedback ρ(ut) in [9], P. Martinez obtained that the total energy of the system decays to zero with an explicit and precise decay rate estimate under sharp assumptions on the feedback. M. Daoulatli [ 1] studied problem ( 1.1) withoutf(u), and they illustrated that if given suitable conditions on th e nonlinear terms, and the damping is modeled by a continuous monotone function without any gro wth restrictions imposed at the origin and infinity, the decay rate of the energy functional w as obtained by solving a nonlinear non-autonomous ODE. The Cauchy problem for second order hyp erbolic evolution equations with restoring force in a Hilbert space, under the effect of non linear time-dependent damping has also been studied, the interested readers can refer to pa pers [5–8]. It is certainly beyond the scope of the present paper to give a comprehensive review for dissipative wave systems equipped with time-dependent non linear damping forces. However, the literature on energy decay estimates is rare for the weak solution of the variable coefficient wave equations when it is concerned with the interaction bet ween time-dependent damping and source term. Inspired by the paper [ 10], in this paper, the primary goal is to establish the energy decay rates when the time-dependent damping satisfies differe nt assumptions. Because of the the interaction between time-dependent damping and source term, some new difficulties need to be solved. The outline of the paper is as follows. In Section 2, we shall give some assumptions, main results and several remarks. In Section 3, we prove that the local (in time) existence of the weak solution can be extended globally. Sections 4and5are used to prove the energy decay rates. 2 Preliminaries and main results Throughout the paper, denote by Mthe optimal embedding constant for the embedding theoremH1 0(Ω)֒→Lp+1(Ω).The symbol Cis a generic positive constant, which may bedifferent3 in various positions. Ci(i= 0,1,2,···,33) represent some positive constants. We impose the following assumptions: (H1)γ∈W1,∞ loc(R+) is bounded and nonnegative on R+= [0,+∞). (H2)gis continuous and monotone increasing feedback with g(0) = 0. In addition, there exist positive constants b1andb2such that b1|s|m+1≤g(s)s≤b2|s|m+1,wherem≥1 and|s|>1. (H3)pm+1 m<6. (H4)u0(x)∈H1 0(Ω), u1(x)∈L2(Ω). Definition 2.1 (Weak solution) .A functionu(t) :=u(t,x) is said to a weak solution of problem (1.1) on [0,T] ifu∈C([0,T];H1 0(Ω)) withut∈C([0,T];L2(Ω))∩Lm+1(Ω×(0,T)). In addition, for allt∈[0,T], (ut(t),φ(t))−(ut(0),φ(0)) +/integraldisplayt 0a(u(s),φ(s))ds−/integraldisplayt 0/integraldisplay Ωut(s)φt(s)dxds +/integraldisplayt 0/integraldisplay Ωγ(t)g(ut(s))φt(s)dxds=/integraldisplayt 0/integraldisplay Ωf(u(s))φ(s)dxds(2.1) forφ∈ {φ:φ∈C([0,T];H1 0(Ω))∩Lm+1(Ω×(0,T)) withφt∈C([0,T];L2(Ω))}. Theorem 2.2 (Local existence) .Let(H1)−(H4)hold, then there exists a local (in time) weak solutions u(t)to problem (1.1)forT >0depending on the initial quadratic energy E(0). Moreover, the following identity holds E(t)+/integraldisplayt 0/integraldisplay Ωγ(t)g(ut)utdxds=E(0)+/integraldisplayt 0/integraldisplay Ωf(u(s))ut(s)dxds (2.2) with the quadratic energy is defined by E(t) =1 2/ba∇dblut(t)/ba∇dbl2 2+1 2a(u(t),u(t)). (2.3) The following result illustrates that when the damping domi nates the source, then the solu- tion is global in time. Theorem 2.3 (Global existence I) .In addition to (H1)−(H4), ifu0(x)∈Lp+1(Ω)andm≥p, then the weak solution of problem (1.1)is global in time. Theorem 2.2can be proved directly by employing the theory of monotone op erators and nonlinear semigroups combined with energy methods. Theore m2.3also follows from a standard continuation argument from ODE theory. The interested read ers can follow from line to line as shown in [ 2](see also a recent paper [ 3]) with slight differences to achieve it. The proof of Theorem 2.2and Theorem 2.3is not the point in question, so we omit it. Foru∈Lp+1(Ω), define the total energy E(t) by E(t) :=E(t)−1 p+1/ba∇dblu/ba∇dblp+1=1 2/ba∇dblut(t)/ba∇dbl2 2+1 2a(u(t),u(t))−1 p+1/ba∇dblu/ba∇dblp+1 p+1.(2.4) Then (2.2) can be transferred as E(t)+/integraldisplayt 0/integraldisplay Ωγ(t)g(ut)utdxds=E(0). (2.5) Further, we obtain the total energy E(t) is monotone decreasing in time, and E′(t) =−/integraldisplay Ωγ(t)g(ut)utdx. (2.6)4 Theorem 2.4 (Global existence II) .Let1<p≤5and(H1)−(H4)hold. Assume 0<E(0)</parenleftBig1 2−1 p+1/parenrightBig/parenleftBigωp+1 2 Mp+1/parenrightBig2 p−1, a(u0,u0)</parenleftBigωp+1 2 Mp+1/parenrightBig2 p−1, then the weak solution of problem (1.1)is global. We need to impose additional conditions to discuss the energ y decay rates. (H5)γ:R+→R+a nonincreasing function of class C1and/integraltext∞ 0γ(t)dt=∞. (H6) There exists a strictly increasing and odd function h∈C1[1,1] such that |h(s)s| ≤g(s)s≤ |h−1(s)s|,where|s| ≤1, whereh−1is the inverse function of h. Theorem 2.5 (Energy decay rates I) .In addition to all conditions of Theorem 2.4,(H5)and (H6), we assume 1≤m≤5. Then for all t≥0 (1)ifh(s)is linear, E(t)≤E(0)e1−C/integraltextt 0γ(s)ds. (2.7) (2)ifh(s)has polynomial growth, E(t)≤CE(0)/parenleftBigg 1 1+/integraltextt 0γ(s)ds/parenrightBigg2 m−1 withm>1. (2.8) Whenh(s) does not necessarily have polynomial growth, the energy de cay rates still can be obtained if we replace ( H5) by the following (H′ 5)γ(t)≥γ0>0, whereγ0is constant. Theorem 2.6 (Energy decay rates II) .In addition to all conditions of Theorem 2.4,(H′ 5)and (H6), we assume 1≤m≤5. Then for all t≥1 E(t)≤CE(0)/parenleftbigg H−1/parenleftBig1 t/parenrightBig/parenrightbigg2 . (2.9) HereH(s) :=h(s)s. Remark 2.7.In the proof of Theorem 2.5, we only discuss the energy decay rates for m≤5, that is, the nonlinear damping is subcritical andcritical. The restraint results from the embedding theoremH1 0(Ω)֒→Lm+1(Ω). A recent paper [ 4] investigated the energy decay estimates for the automous wave equation with supercritical nonlinear damping in the absence of the driving source. Inspired by the paper [ 4], we reasonably conjecture that if there exists κ(t) satisfying some suitable conditions, then E(t)≤CE(0)/parenleftBigg 1 1+/integraltextt 0κ(t)γ(s)ds/parenrightBigg2 m−1 , form>5, that is the supercritical nonlinear damping.5 Remark2.8.Inthis paper, wediscussthe variable-coefficient wave equat ion with nonlineartime- dependent damping and nonlinear source for standard growth conditions. However, the energy decay rates for wave systems with nonstandard growth condit ion can be of equal importance. By following this paper, it is possible to discuss the energy decay rates to the initial-boundary value problem utt−∆u+γ(t)|ut|m(x)−2ut=|u|p(x)−2uin Ω×(0,T), u(x,t) = 0 on ∂Ω×(0,T), u(x,0) =u0(x), ut(x,0) =u1(x) in Ω . Remark 2.9.The rest field u(t,x) = 0 will be called asymptotically stable in the mean , or simply asymptotically stable , if and only if lim t→∞E(t) = 0 for all solutions u(t) :=u(t,x) of problem ( 1.1). This concept was proposed first by P. Pucci and J. Serrin [ 12]. Obviously, based on Theorem 2.3or Theorem 2.4, by Lemma 3.2, we obtain lim t→∞E(t) = 0.Hence, the rest field u(x,t) = 0 is asymptotically stable. 3 Proof of Theorem 2.4 Let us introduce a function Fas follows F(s) =1 2s−Mp+1 (p+1)ωp+1 2sp+1 2. (3.1) By a direct computation, the function Fsatisfies that 1.F(0) = 0; 2. lims→+∞F(s) =−∞; 3.Fis strictly increasing in (0 ,s1), and is strictly decreasing in ( s1,+∞); 4.Fhas a maximum at s1with the maximum value F1. Here s1=/parenleftBigωp+1 2 Mp+1/parenrightBig2 p−1,F1=/parenleftBig1 2−1 p+1/parenrightBig s1. Lemma 3.1. Ifu(t)is a solution for problem (1.1)andE(0)<F1, a(u0,u0)<s1,then there exists a positive constant s2satisfying 0<s2<s1such that a(u(t),u(t))≤s2for allt≥0. (3.2) Proof.Using (1.3) and (2.1), the embedding theorem H1 0(Ω)֒→Lp+1(Ω), we obtain E(t)≥1 2a(u(t),u(t))−1 p+1/ba∇dblu/ba∇dblp+1 p+1≥1 2a(u(t),u(t))−Mp+1 p+1/ba∇dbl∇u/ba∇dblp+1 2 ≥1 2a(u(t),u(t))−Mp+1 (p+1)ωp+1 2[a(u(t),u(t))]p+1 2:=F(a(u(t),u(t))).(3.3)6 SinceE(0)<F1, there exists a s2< s1such that F(s2) =E(0). It follows from ( 3.3) thatF(a(u0,u0))≤E(0) =F(s2), which implies a(u0,u0)≤s2due to the given condition a(u0,u0)< s1. To complete the proof of ( 3.2), we suppose by contradiction that for some t0>0,a(u(t0),u(t0))>s2.The continuity of a(u(t),u(t)) illustrates that we may choose t1such thats1>a(u(t1),u(t1))>s2, then we have E(0) =F(s2)<F(a(u(t1),u(t1)))≤E(t1).This is a contradiction for E(t) is nonincreasing. Lemma 3.2. Under all the conditions of Lemma 3.1,for allt≥0,the following holds 0≤E(t)≤C0E(t)≤C0E(0). (3.4) Proof.Similar to ( 3.3), and then using ( 3.2) and (2.4), then 1 p+1/ba∇dblu/ba∇dblp+1 p+1≤Mp+1 p+1/ba∇dbl∇u/ba∇dblp+1 2≤Mp+1 (p+1)ωp+1 2[a(u(t),u(t))]p+1 2 =Mp+1 (p+1)ωp+1 2[a(u(t),u(t))]p−1 2a(u(t),u(t)) ≤Mp+1 (p+1)ωp+1 2sp−1 2 2/parenleftBig 2E(t)+2 p+1/ba∇dblu/ba∇dblp+1 p+1/parenrightBig , which indicates /ba∇dblu/ba∇dblp+1 p+1≤ ME(t)≤ ME(0), (3.5) by recalling E(t) is monotone decreasing. Here M:=2Mp+1 ωp+1 2sp−1 2 2 1−2Mp+1 (p+1)ωp+1 2sp−1 2 2<2Mp+1 ωp+1 2sp−1 2 1 1−2Mp+1 (p+1)ωp+1 2sp−1 2 1=2(p+1) p−1. Combining ( 3.5) with (2.4), we directly obtain E(t) =E(t)+1 p+1/ba∇dblu/ba∇dblp+1≤C0E(t)≤C0E(0), which yields ( 3.4). It follows from Lemma 3.2that the weak solution u(t) of problem ( 1.1) exists globally, that isT=∞. Remark 3.3.By the well-known potential well theory, we can also prove th e global existence. In fact,F1is equal to the potential well depth(the mountain pass level )ddefined by d:= inf u∈H1 0(Ω)\{0}sup λ≥0J(λu),whereJ(u) :=1 2a(u,u)−1 p+1/ba∇dblu/ba∇dblp+1 p+1. 4 Proof of Theorem 2.5 The proof of Theorem 2.5relies on the following crucial lemma. We refer to [ 9] for the detailed proof.7 Lemma 4.1 ([9]).LetE:R+→R+be a non-increasing function and ψ:R+→R+be a strictly increasing function of class C1such that ψ(0) = 0andψ(t)→+∞ast→+∞. Assume that there exist σ≥0,σ′≥0,c≥0andω>0such that : /integraldisplay+∞ tE1+σ(s)ψ′(s)ds≤1 ωEσ(0)E(t)+c (1+ψ(s))σ′Eq(0)E(s) thenEhas the following decay property : (1)ifσ=c= 0,thenE(t)≤E(0)e1−ωψ(t)for allt≥0; (2)ifσ>0,then there exists C >0such thatE(t)≤CE(0)(1+ψ(t))−1+σ′ σfor allt≥0. Inwhatfollows, letusproveTheorem 2.5. Itdosemakesensetomultiply( 1.1)byEβ(t)φ′(t)u(t) and to integrate over Ω ×[S,T]. Hereφ:R+→R+is a concave nondecreasing function of class C2, such that φ′is bounded, and β≥0 is constant. Then we obtain /integraldisplayT SEβ(t)φ′(t)/integraldisplay Ωu(t)[utt(t)−Au(t)+γ(t)g(ut(t))]dxdt=/integraldisplayT SEβ(t)φ′(t)/ba∇dblu(t)/ba∇dblp+1 p+1dt.(4.1) Integrating by parts yields /integraldisplayT SEβ(t)φ′(t)/integraldisplay Ωu(t)utt(t)dxdt=Eβ(t)φ′(t)/integraldisplay Ωu(t)ut(t)dx/vextendsingle/vextendsingle/vextendsingleT S−/integraldisplayT SEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2 2 −/integraldisplayT S[βEβ−1(t)E′(t)φ′(t)+Eβ(t)φ′′(t)]/integraldisplay Ωu(t)ut(t)dxdt.(4.2) Substituting ( 4.2) into (4.1), one obtains Eβ(t)φ′(t)/integraldisplay Ωu(t)ut(t)dx/vextendsingle/vextendsingle/vextendsingleT S−/integraldisplayT SEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2 2dt −/integraldisplayT S[βEβ−1(t)E′(t)φ′(t)+Eβ(t)φ′′(t)]/integraldisplay Ωu(t)ut(t)dxdt +/integraldisplayT SEβ(t)φ′(t)a(u(t),u(t))dt +/integraldisplayT SEβ(t)φ′(t)γ(t)/integraldisplay Ωu(t)g(ut(t))dxdt=/integraldisplayT SEβ(t)φ′(t)/ba∇dblu(t)/ba∇dblp+1 p+1dt.(4.3) It follows from ( 2.4) that (4.3) can be written as 2/integraldisplayT SEβ+1(t)φ′(t)dt=−Eβ(t)φ′(t)/integraldisplay Ωu(t)ut(t)dx/vextendsingle/vextendsingle/vextendsingleT S+2/integraldisplayT SEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2 2dt +/integraldisplayT S[βEβ−1(t)E′(t)φ′(t)+Eβ(t)φ′′(t)]/integraldisplay Ωu(t)ut(t)dxdt +p−1 p+1/integraldisplayT SEβ(t)φ′(t)/ba∇dblu(t)/ba∇dblp+1 p+1dt−/integraldisplayT SEβ(t)φ′(t)γ(t)/integraldisplay Ωu(t)g(ut(t))dxdt =J1+J2+J3+J4+J5.(4.4)8 In what follows, let us estimate every term on the right hand s ide above. Using the embedding theorem H1 0(Ω)֒→L2(Ω), (1.3) and (3.4) yields /ba∇dblu(t)/ba∇dbl2≤M/ba∇dbl∇u(t)/ba∇dbl2≤M/bracketleftBiga(u(t),u(t)) ω/bracketrightBig1 2≤C1E1 2(t). (4.5) Applying Cauchy’s inequality, the boundedness of φ′(t)(we can denote by θ), (4.5) and (3.4), we arrive at |J1|=/vextendsingle/vextendsingle/vextendsingle−Eβ(t)φ′(t)/integraldisplay Ωu(t)ut(t)dx/vextendsingle/vextendsingle/vextendsingleT S/vextendsingle/vextendsingle/vextendsingle ≤θEβ(S)/bracketleftbig /ba∇dblut(T)/ba∇dbl2/ba∇dblu(T)/ba∇dbl2+/ba∇dblut(S)/ba∇dbl2/ba∇dblu(S)/ba∇dbl2/bracketrightbig ≤C2Eβ+1(S),(4.6) moreover, |J3|=/vextendsingle/vextendsingle/vextendsingle/integraldisplayT S[βEβ−1(t)E′(t)φ′(t)+Eβ(t)φ′′(t)]/integraldisplay Ωu(t)ut(t)dxdt/vextendsingle/vextendsingle/vextendsingle ≤ −θβ/integraldisplayT SEβ(t)E′(t)dt+C3/integraldisplayT SEβ+1(t)(−φ′′(t))dt ≤θβ β+1Eβ+1(S)+C4Eβ+1(S) =C5Eβ+1(S).(4.7) It follows from ( 3.5) that |J4|=p−1 p+1/integraldisplayT SEβ(t)φ′(t)/ba∇dblu(t)/ba∇dblp+1 p+1dt≤p−1 p+1M/integraldisplayT SEβ+1(t)φ′(t)dt. (4.8) Using Young’s inequality with ε1>0, (4.5), one has /integraldisplay |ut(t)|≤1u(t)g(ut(t))dx≤ε1 2/ba∇dblu(t)/ba∇dbl2 2+1 2ε1/integraldisplay |ut(t)|≤1|g(ut(t))|2dx ≤C6ε1E(t)+1 2ε1/integraldisplay |ut(t)|≤1|g(ut(t))|2dx. It follows from Young’s inequality with ε1>0, the embedding theorem H1 0(Ω)֒→Lm+1(Ω), (1.3) and (3.4) that /integraldisplay |ut(t)|>1u(t)g(ut(t))dx≤ε2 m+1/ba∇dblu(t)/ba∇dblm+1 m+1+mε−1 m 2 m+1/integraldisplay |ut(t)|>1|g(ut(t))|m+1 mdx ≤Mm+1ε2 m+1/ba∇dbl∇u(t)/ba∇dblm+1 2+mε−1 m 2 m+1/integraldisplay |ut(t)|>1|g(ut(t))|m+1 mdx ≤Mm+1ε2 m+1/bracketleftBiga(u(t),u(t)) ω/bracketrightBigm+1 2+mε−1 m 2 m+1/integraldisplay |ut(t)|>1|g(ut(t))|m+1 mdx ≤C7ε2E(t)+mε−1 m 2 m+1/integraldisplay |ut(t)|>1|g(ut(t))|m+1 mdx.9 Therefore, by recalling the boundedness of γ(t)(we can denote by c1), then |J5|=/vextendsingle/vextendsingle/vextendsingle−/integraldisplayT SEβ(t)φ′(t)γ(t)/integraldisplay Ωu(t)g(ut(t))dxdt/vextendsingle/vextendsingle/vextendsingle ≤c1/integraldisplayT SEβ(t)φ′(t)/vextendsingle/vextendsingle/vextendsingle/integraldisplay |ut(t)|≤1u(t)g(ut(t))dx+/integraldisplay |ut(t)|>1u(t)g(ut(t))dx/vextendsingle/vextendsingle/vextendsingledt ≤c1/integraldisplayT SEβ(t)φ′(t)/bracketleftBig C6ε1E(t)+1 2ε1/integraldisplay |ut(t)|≤1|g(ut(t))|2dx +C7ε2E(t)+mε−1 m 2 m+1/integraldisplay |ut(t)|>1|g(ut(t))|m+1 mdx/bracketrightBig dt ≤C8(ε1+ε2)/integraldisplayT SEβ+1φ′(t)dt+c1 2ε1/integraldisplayT SEβ(t)φ′(t)/integraldisplay |ut(t)|≤1|g(ut(t))|2dx +c1mε−1 m 2 m+1/integraldisplayT SEβ(t)φ′(t)/integraldisplay |ut(t)|>1|g(ut(t))|m+1 mdxdt.(4.9) Inserting ( 4.6)-(4.9) into (4.4) indicates /bracketleftBig 2−p−1 p+1M−C8(ε1+ε2)/bracketrightBig/integraldisplayT SEβ+1(t)φ′(t)dt ≤C9Eβ+1(S)+2/integraldisplayT SEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2 2dt +c1 2ε1/integraldisplayT SEβ(t)φ′(t)/integraldisplay |ut(t)|≤1|g(ut(t))|2dxdt +c1mε−1 m 2 m+1/integraldisplayT SEβ(t)φ′(t)/integraldisplay |ut(t)|>1|g(ut(t))|m+1 mdxdt.(4.10) Note that 2 −p−1 p+1M>0, one gets 2−p−1 p+1M−C8(ε1+ε2)>0 for sufficiently small positive constants ε1andε2. 4.1 Case I: h(s)is linear Let usφ(t) =/integraltextt 0γ(s)dsin this section. From ( H6), there exists two positive constant b3, b4 such that b3s2≤g(s)s≤b4s2,where|s| ≤1. (4.11)10 Using (4.11), (H2) and (2.6) yields 2/integraldisplayT SEβ(t)γ(t)/ba∇dblut(t)/ba∇dbl2 2dt ≤2/integraldisplayT SEβ(t)γ(t)/bracketleftBig/integraldisplay |ut(t)|≤1|ut(t)|2dx+/integraldisplay |ut(t)|>1|ut(t)|2dx/bracketrightBig dt ≤2 b3/integraldisplayT SEβ(t)/integraldisplay Ωγ(t)g(ut(t))ut(t)dxdt+2/integraldisplayT SEβ(t)γ(t)/integraldisplay |ut(t)|>1|ut(t)|m+1dxdt ≤2 b3/integraldisplayT SEβ(t)/integraldisplay Ωγ(t)g(ut(t))ut(t)dxdt+2 b1/integraldisplayT SEβ(t)γ(t)/integraldisplay Ωg(ut(t))ut(t)dxdt ≤ −/bracketleftBig2 b3+2 b1/bracketrightBig/integraldisplayT SEβ(t)E′(t)dt≤C10Eβ+1(S).(4.12) By using ( 4.11) and (2.6), we gets c1 2ε1/integraldisplayT SEβ(t)γ(t)/integraldisplay |ut(t)|≤1|g(ut(t))|2dx ≤c1b4 2ε1/integraldisplayT SEβ(t)γ(t)/integraldisplay |ut(t)|≤1g(ut(t))ut(t)dx ≤ −c1b4 2ε1/integraldisplayT SEβ(t)E′(t)dt≤C11Eβ+1(S).(4.13) It follows from ( H2) and (2.6) that c1mε−1 m 2 m+1/integraldisplayT SEβ(t)γ(t)/integraldisplay |ut(t)|>1|g(ut(t))|m+1 mdxdt =c1mε−1 m 2 m+1/integraldisplayT SEβ(t)γ(t)/integraldisplay |ut(t)|>1|g(ut(t))|1 m|g(ut(t))|dxdt ≤c1mε−1 m 2b1 m 2 (m+1)/integraldisplayT SEβ(t)γ(t)/integraldisplay |ut(t)|>1g(ut(t))ut(t)dxdt≤C12Eβ+1(S).(4.14) Inserting ( 4.12)-(4.14) into (4.10), we easily deduce /integraldisplayT SEβ+1(t)γ(t)dt≤1 C13Eβ(0)E(S). (4.15) WhenTgoes to + ∞, chooseβ= 0,ψ(t) =/integraltextt 0γ(s)dsin Lemma 4.1, then we derive ( 2.7). 4.2 Case II: h(s)has polynomial growth In this subsection, we still choose φ(t) =/integraltextt 0γ(s)ds. To simplicity, denote h(s) =b5|s|m−1s withm>1,|s| ≤1, b5>0. From ( H6), there exists two positive constant b6, b7such that b6sm+1≤g(s)s≤b7|s|m+1 m,where|s| ≤1. (4.16)11 It follows from H¨ older’s inequality, ( 4.16), Young’s inequality with ε3>0, (H2) and (2.6) yields 2/integraldisplayT SEβ(t)γ(t)/ba∇dblut(t)/ba∇dbl2 2dt ≤2/integraldisplayT SEβ(t)γ(t)/bracketleftBig/integraldisplay |ut(t)|≤1|ut(t)|2dx+/integraldisplay |ut(t)|>1|ut(t)|2dx/bracketrightBig dt ≤2|Ω|m−1 m+1/integraldisplayT SEβ(t)γ(t)/parenleftBig/integraldisplay |ut(t)|≤1|ut(t)|m+1dx/parenrightBig2 m+1dt+2 b1/integraldisplayT SEβ(t)γ(t)/integraldisplay Ωg(ut(t))ut(t)dxdt ≤2|Ω|m−1 m+11 b2 m+1 6/integraldisplayT SEβ(t)γm−1 m+1(t)/parenleftBig/integraldisplay Ωγ(t)ut(t)g(ut(t))dx/parenrightBig2 m+1dt−2 b1/integraldisplayT SEβ(t)E′(t)dxdt ≤C14ε3/integraldisplayT SEβ(m+1) m−1(t)γ(t)dt+C15ε−m−1 2 3/integraldisplayT S(−E′(t))dt+C16Eβ+1(S) ≤C14ε3/integraldisplayT SEβ(m+1) m−1(t)γ(t)dt+C17E(S)+C16Eβ+1(S). (4.17) Similar to ( 4.17), using the second inequality in ( 4.16), we obtain c1 2ε1/integraldisplayT SEβ(t)γ(t)/integraldisplay |ut(t)|≤1|g(ut(t))|2dx ≤c1 2ε1|Ω|m−1 m+1/integraldisplayT SEβ(t)γ(t)/parenleftBig/integraldisplay |ut(t)|≤1|g(ut(t))|m+1dx/parenrightBig2 m+1dt ≤/integraldisplayT SEβ(t)γm−1 m+1(t)·c1 2ε1|Ω|m−1 m+1b2m m+1 7/parenleftBig/integraldisplay Ωγ(t)ut(t)g(ut(t))dx/parenrightBig2 m+1dt ≤C18ε4/integraldisplayT SEβ(m+1) m−1(t)γ(t)dt+C19ε−m−1 2 4/integraldisplayT S(−E′(t))dt ≤C18ε4/integraldisplayT SEβ(m+1) m−1(t)γ(t)dt+C20E(S).(4.18) Moreover, ( 4.14) still holds in this case. Inserting ( 4.14), (4.17) and (4.18) into (4.10), and choosing β=m−1 2, we easily deduce /bracketleftBig 2−p−1 p+1M−C8(ε1+ε2)−C21(ε3+ε4)/bracketrightBig/integraldisplayT SEβ+1(t)γ(t)dt≤C22Eβ+1(0)E(S).(4.19) We also have 2−p−1 p+1M−C8(ε1+ε2)−C21(ε3+ε4)>0 for sufficiently small positive constants ε1,ε2,ε3andε4. Therefore, we get/integraldisplayT SEβ+1(t)γ(t)dt≤1 C23Eβ(0)E(S). (4.20) WhenTgoes to + ∞, note that β=m−1 2,ψ(t) =/integraltextt 0γ(s)dsin Lemma 4.1then we derive ( 2.8).125 Proof of Theorem 2.6 To prove Theorem 2.6, we need to the following lemma included in [ 9]. Lemma 5.1. The function φ: [1,+∞)→[1,∞)defined by φ(t) :=˜ψ−1(t) (5.1) is a strictly increasing concave function of class C2and satisfies lim t→+∞φ(t) = +∞,lim t→+∞φ′(t) = 0, and/integraltext∞ 1φ′(t)|h−1(φ′(t))|2dtconverges. Here ˜ψ(t) := 1+/integraldisplayt 11 h(1/s)ds, (5.2) and the function hsatisfies (H6). In this section, define χ(t) :=1 φ(t)=1 ˜ψ−1(t). (5.3) Now we are in a position to estimate ( 4.10). Fort≥1, set Ω1={x∈Ω :|ut(t)| ≤χ(t)}; Ω2={x∈Ω :χ(t)<|ut(t)| ≤1}; Ω3={x∈Ω :|ut(t)|>1}. Using Lemma 5.1, (H6), (H′ 5) and (2.6), we gets /integraldisplayT SEβ(t)/integraldisplay Ω2φ′(t)|ut(t)|2dxdt =/integraldisplayT SEβ(t)/integraldisplay Ω2h(χ(t))|ut(t)|2dxdt≤/integraldisplayT SEβ(t)/integraldisplay Ω2h(ut(t))|ut(t)|2dxdt ≤/integraldisplayT SEβ(t)1 γ(t)/integraldisplay Ω2γ(t)g(ut(t))ut(t)dxdt≤1 γ0Eβ+1(S).(5.4) It follows from ( 2.6), (5.3) and Lemma 5.1that /integraldisplayT SEβ(t)φ′(t)/integraldisplay Ω1|ut(t)|2dxdt≤ |Ω|/integraldisplayT SEβ(t)φ′(t)χ2(t)dt ≤ |Ω|Eβ(S)/integraldisplayT Sφ′(t)χ2(t)dt≤ |Ω|Eβ(S) φ(S).(5.5)13 Recalling ( 4.17), and using ( 5.4) and (5.5), one gets 2/integraldisplayT SEβ(t)φ′(t)/ba∇dblut(t)/ba∇dbl2 2dt ≤2/integraldisplayT SEβ(t)φ′(t)/integraldisplay |ut(t)|≤1|ut(t)|2dxdt+2 b1/integraldisplayT SEβ(t)φ′(t) γ(t)/integraldisplay Ω3γ(t)g(ut(t))ut(t)dxdt ≤2/integraldisplayT SEβ(t)φ′(t)/integraldisplay Ω1∪Ω2|ut(t)|2dxdt+C24Eβ+1(S) ≤C25Eβ+1(S)+|Ω|Eβ(S) φ(S).(5.6) Next we continue to estimate the remaining terms in ( 4.10). Fort≥1 andφ′(t)≤1, set Ω1={x∈Ω :|ut(t)| ≤φ′(t)}; Ω2={x∈Ω :φ′(t)<|ut(t)| ≤1}; Ω3={x∈Ω :|ut(t)|>1}. Fort≥1 andφ′(t)>1, set Ω4={x∈Ω :|ut(t)| ≤1<φ′(t)}; Ω5={x∈Ω : 1<|ut(t)| ≤φ′(t)}; Ω6={x∈Ω :|ut(t)|>φ′(t)>1}. Hence {x∈Ω :|ut(t)| ≤1}= Ω1∪Ω2(or Ω4),{x∈Ω :|ut(t)|>1}= Ω3(or Ω5∪Ω6). Similar to ( 4.14), c1mε−1 m 2 m+1/integraldisplayT SEβ(t)φ′(t)/integraldisplay Ωi|g(ut(t))|m+1 mdxdt≤C26Eβ+1(S),i=3,5,6. Thus c1mε−1 m 2 m+1/integraldisplayT SEβ(t)φ′(t)/integraldisplay |ut(t)|>1|g(ut(t))|m+1 mdxdt≤C27Eβ+1(S). (5.7) Using (H6) and (H′ 5), then c1 2ε1/integraldisplayT SEβ(t)φ′(t)/integraldisplay Ω2|g(ut(t))|2dxdt ≤c1 2ε1/integraldisplayT SEβ(t)/integraldisplay Ω2|ut(t)||g(ut(t))|2dxdt ≤c1 2ε1/integraldisplayT SEβ(t)/integraldisplay Ω2ut(t)g(ut(t))|h−1(ut(t))|dxdt ≤c1 2ε1h−1(1)/integraldisplayT SEβ(t)1 γ(t)/integraldisplay Ω2γ(t)ut(t)g(ut(t))dxdt≤C28Eβ+1(S),(5.8)14 and c1 2ε1/integraldisplayT SEβ(t)φ′(t)/integraldisplay Ωi|g(ut(t))|2dxdt ≤c1 2ε1/integraldisplayT SEβ(t)φ′(t)/integraldisplay Ωi|h−1(ut(t))|2dxdt ≤c1 2ε1/integraldisplayT SEβ(t)φ′(t)/integraldisplay Ωi|h−1(φ′(t))|2dxdt ≤C29Eβ(S)/integraldisplayT Sφ′(t)|h−1(φ′(t))|2dt,i=1,4.(5.9) Substubiting ( 5.6)-(5.9) into (4.10), we get /bracketleftBig 2−p−1 p+1M−C8(ε1+ε2)/bracketrightBig/integraldisplayT SEβ+1(t)φ′(t)dt ≤C30Eβ+1(S)+|Ω|Eβ(S) φ(S)+C31Eβ(S)/integraldisplayT Sφ′(t)|h−1(φ′(t))|2dt.(5.10) Since/integraltext∞ 1φ′(t)|h−1(φ′(t))|2dtconverges in Lemma 5.1, then /integraldisplayT SEβ+1(t)φ′(t)dt≤1 C32Eβ(0)E(S) +C33 φ(S)Eβ(0)E(S). (5.11) WhenTgoes to + ∞, chooseβ= 1, and choose ψ(t) =φ(t)−1 in Lemma 4.1, then we derive E(t)≤CE(0) φ2(t). (5.12) Let us choose s0such thath(1 s0)≤1, then by ( 5.2) andH(s) :=h(s)s, fors≥s0 ˜ψ(s)≤1+(s−1)1 h/parenleftBig 1 s/parenrightBig≤s1 h/parenleftBig 1 s/parenrightBig=1 H/parenleftBig 1 s/parenrightBig. Hence, by ( 5.1), fors≥s0 s≤φ 1 H/parenleftBig 1 s/parenrightBig =φ(t) witht=1 H/parenleftBig 1 s/parenrightBig. Further, 1 φ(t)≤1 s=H−1/parenleftBig1 t/parenrightBig . which together with ( 5.12) implies ( 2.9). Acknowledgements ThisworkissupportedbytheFundamentalResearchFundsfor CentralUniversities(B230201033).15Competing Interests The authors declare that they have no competing interests. Data Availability Data sharing is not applicable to this article as no new data w ere created or analyzed in this study. References [1] M. Daoulatli, Rates of decay for the wave systems with time-dependent damp ing, Discrete Contin. Dyn. Syst., 31(2011) 407–443. [2] Y.Q. Guo, M.A. Rammaha, et al., Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping , J. Differential Equations, 257(2014), 3778–3812. [3] T.G. Ha, Global solutions and blow-up for the wave equation with vari able coefficients: I. Interior supercritical source. Appl. Math. Optim., 84(2021), 767–803. [4] A. Haraux, L. Tebou, Energy decay estimates for the wave equation with supercrit ical non- linear damping , arXiv preprint arXiv:2204.11494. [5] J.R. Luo, T.J. Xiao, Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping , Evol. Equ. Control Theory, 9(2020), 359–373. [6] J.R. Luo, T.J. Xiao, Decay rates for semilinear wave equations with vanishing da mping and Neumann boundary conditions , Math. Methods Appl. Sci., 44(2021), 303–314. [7] J.R.Luo, T.J.Xiao, Optimal energy decay rates for abstract second order evolut ion equations with non-autonomous damping , ESAIM: COCV, 27(2021) 59. [8] J.R. Luo, T.J. Xiao, Optimal decay rates for semi-linear non-autonomous evolut ion equa- tions with vanishing damping , Nonlinear Anal., 230(2023) 113247. [9] P. Martinez, A new method to obtain decay rate estimates for dissipative s ystems, ESAIM: Control Optim. Calc. Var., 4(1999), 419–444. [10] P. Martinez, Precise decay rate estimates for time-dependent dissipati ve systems , Israel J. Math.,119(2000), 291–324. [11] P. Pucci and J. Serrin, Asymptotic stability for non–autonomous dissipative wave systems, Comm. Pure Appl. Math., XLIX(1996), 177–216. [12] P. Pucci and J. Serrin, Asympptotic stablility for nonlinear parabolic systems //S.N. Antont- sev, J.I. Diaz, S.I. Shmarev. Energy Methods in Continuum Me chanics. Dordrecht: Kluwer Acad Publ, 1996: 66–74. [13] P. Pucci and J. Serrin, Local asymptotic stability for dissipative wave systems , Israel J. Math.,104(1998), 29–50.

2023-04-23

In this paper, a class of variable-coefficient wave equations equipped with time-dependent damping and the nonlinear source is considered. We show that the total energy of the system decays to zero with an explicit and precise decay rate estimate under different assumptions on the feedback with the help of the method of weighted energy integral.

Decay rates for a variable-coefficient wave equation with nonlinear time-dependent damping

2304.11522v1

Propagating spin waves in nanometer-thick yttrium iron garnet lms: Dependence on wave vector, magnetic eld strength and angle Huajun Qin,1,Sampo J. H am al ainen,1Kristian Arjas,1Jorn Witteveen,1and Sebastiaan van Dijken1,y 1NanoSpin, Department of Applied Physics, Aalto University School of Science, P.O. Box 15100, FI-00076 Aalto, Finland (Dated: October 12, 2018) We present a comprehensive investigation of propagating spin waves in nanometer-thick yttrium iron garnet (YIG) lms. We use broadband spin-wave spectroscopy with integrated coplanar waveg- uides (CPWs) and microstrip antennas on top of continuous and patterned YIG lms to characterize spin waves with wave vectors up to 10 rad/ m. All lms are grown by pulsed laser deposition. From spin-wave transmission spectra, parameters such as the Gilbert damping constant, spin-wave dis- persion relation, group velocity, relaxation time, and decay length are derived and their dependence on magnetic bias eld strength and angle is systematically gauged. For a 40-nm-thick YIG lm, we obtain a damping constant of 3 :5104and a maximum decay length of 1.2 mm. Our experiments reveal a strong variation of spin-wave parameters with magnetic bias eld and wave vector. Spin- wave properties change considerably up to a magnetic bias eld of about 30 mT and above a eld angle ofH= 20, whereH= 0corresponds to the Damon-Eshbach conguration. PACS numbers: I. INTRODUCTION Magnonics aims at the exploitation of spin waves for information transport, storage, and processing1{7. For practical devices, it is essential that spin waves propa- gate over long distances in thin lms. Because of its ul- tralow damping constant, ferrimagnetic YIG is a promis- ing material. Bulk crystals and m-thick YIG lms ex- hibit a Gilbert damping constant 3105at GHz frequencies. In recent years, nm-thick YIG lms with ultralow damping parameters have also been prepared successfully. High-quality YIG lms have been grown on Gd3Ga5O12(GGG) single-crystal substrates using liquid phase epitaxy8{11, magnetron sputtering12{15, and pulsed laser deposition (PLD)16{25. For thin YIG lms, damp- ing constants approaching the value of bulk crystals have been reported21,22. Meanwhile, YIG-based magnonic devices such as logic gates, transistors, and multiplex- ers have been demonstrated26{30. Spin-wave transmis- sion in nm-thick YIG lms24,31{36and the excitation of short-wavelength spin waves have been investigated as well37{40. To advance YIG-magnonics further, knowledge on the transport of spin waves in nm-thick YIG lms and its dependence on wave vector and external magnetic bias eld is essential. In this paper, we present a broadband spin-wave spec- troscopy study of PLD-grown YIG lms with a thickness of 35 nm and 40 nm. Spin-wave transmission spectra are recorded by patterning CPWs and microstrip antennas on top of continuous and patterned YIG lms. CPWs are used because they generate spin waves with well-dened wave vectors. This enables extraction of key parame- ters such as the Gilbert damping constant ( ), the spin- wave dispersion relation, group velocity ( g), relaxation time (), and decay length ( ld). For a 40 nm YIG lm, we nd3:5104and a maximum group velocity and decay length of 3.0 km/s and 1.2 mm, respectively. We show that spin-wave properties vary strongly withwave vector up to an in-plane external magnetic bias eld 0Hext= 30 mT and below a eld angle H= 20(H = 0 corresponds to the Damon-Eshbach geometry). Be- yond these eld parameters, the dependence of spin-wave properties on wave vector weakens. We demonstrate also that broadband spectroscopy with integrated CPWs and microstrip antennas provide similar spin-wave parame- ters. The paper is organized as follows. In Sec. II, we describe the PLD process, broadband spin-wave spectroscopy setup, and simulations of the CPW- and microstrip-antenna excitation spectra. In Sec. III, we present vector network analyzer ferromagnetic resonance (VNA-FMR) results and broadband spin-wave transmis- sion spectra for CPWs. In Sec. IV, we t the ex- perimental data and extract parameters of propagating spin waves. Spin-wave transmission measurements using CPWs and microstrip antennas are compared in Sec. V. Section VI summarizes the paper. II. EXPERIMENT A. PLD of YIG thin lms YIG lms with a thickness of 35 nm and 40 nm were grown on single-crystal GGG(111) substrates using PLD. Prior to loading into the PLD vacuum chamber, the substrates were ultrasonically cleaned in acetone, iso- propanol, and distilled water. The substrates were rst degassed at 550C for 15 minutes and then heated to 800C at a rate of 5C per minute in an O 2pressure of 0.13 mbar. YIG lms were deposited under these condi- tions by ablation from a stoichiometric target using an excimer laser with a pulse repetition rate of 2 Hz and a uence of 1.8 J/cm2. After deposition, the YIG lms were rst annealed at 730C for 10 minutes in 13 mbar O2before cooling down to room temperature at a rate ofarXiv:1810.04973v1 [cond-mat.mes-hall] 11 Oct 20182 -1.0 -0.5 0.0 0.5 1.0-100-50050100 49 50 51 52 53300 400 500 6000.00.51.0Ms (kA/m) Magnetic field (mT)GGG (444)Intensity (a.u.) 2oYIG (444) M/Ms Temperature (K)(a) (b) FIG. 1: (a) XRD 2scan of the (444) re ections from a PLD-grown YIG lm on a GGG(111) substrate. The period of Laue oscillations surrounding the (444) peaks corresponds to a lm thickness of 40 nm. (b) Room temperature VSM hysteresis loop of the same lm. The inset shows how the YIG saturation magnetization varies with temperature. 3C per minute. B. Structural and magnetic characterization The crystal structure of our YIG lms was inspected by high-resolution X-ray diraction (XRD) on a Rigaku SmartLab system. Figure 1(a) shows a XRD 2scan of a 40-nm-thick YIG lm on GGG(111). Clear (444) lm and substrate peaks are surrounded by Laue oscillations, signifying epitaxial and smooth lm growth. We used a vibrating sample magnetometer (VSM) in a PPMS Dy- nacool system from Quantum Design to characterize the magnetic properties. Figure 1(b) depicts a VSM hystere- sis loop of a 40-nm-thick YIG lm. At room temperature, the coercive eld of the YIG lm is only 0.1 mT and the saturation magnetization ( Ms) is 115 kA/m. The evo- lution ofMswith temperature is shown in the inset of Fig. 1(b). From these data, we derive a Curie temper- ature (TC) of around 500 K. The values of MsandTC are similar to those obtained in previous studies on nm- thick YIG lms14,22,23and about 10% smaller compared to values of YIG bulk crystals ( Ms= 139 kA/m, TC= 559 K). Minor o-stoichiometries in the YIG lm might be the reason for the small discrepancy41. C. Broadband spin-wave spectroscopy VNA-FMR and spin-wave transmission measurements were performed using a two-port VNA and a microwave probing station with a quadrupole electromagnet. In VNA-FMR experiments, the YIG lm was placed face- down onto a prepatterned CPW on a GaAs substrate. The signal line and ground lines of this CPW had a width of 50 m and 800 m, respectively, and were sep- arated by 30 m. Broadband spin-wave spectroscopyin transmission geometry was conducted by contacting two integrated CPWs or microstrip antennas on top of a continuous YIG lm or YIG waveguide. Most of the experiments were performed with CPWs consisting of 2 m-wide signal and ground lines with a separation of 1.6 m. For comparison measurements, we used CPWs and microstrip antennas with 4- m-wide signal lines. All an- tenna structures were fabricated by electron-beam lithog- raphy and were composed of 3-nm Ta and 120-nm Au. A microwave current provided by the VNA was used to generate a rf magnetic eld around one of the CPWs or microstrip antennas. We used CST microwave studio software to simulate the excitation spectra of the antenna structures (see next section). Spin waves that are excited by a rf magnetic eld pro- duce an inductive voltage across a nearby antenna. At the exciting CPW or microstrip antenna, this voltage is given by42: Vind/Z (!;k)j(k)j2dk; (1) where(!;k) is the magnetic susceptibility and j(k)j2 is the spin-wave excitation spectrum. Propagating spin waves arriving at the receiving CPW or microstrip an- tenna produce an inductive voltage: Vind/Z (!;k)j(k)j2exp(i(ks+  0))dk; (2) wheresis the propagation distance and  0is the initial phase of the spin waves. In the experiments, we used the rst and second port of the VNA to measure these induc- tive voltages by recording the S12scattering parameter. D. Simulations of CPW and microstrip antenna excitation spectra We used CST microwave studio software to simulate the spin-wave excitation spectra of the dierent antenna structures43. This commercial solver of Maxwell's equa- tions uses a nite integration method to calculate the rf magnetic eld 0hrfand its in-plane ( 0hrf x,0hrf y) and out-of-plane ( 0hrf z) components. Since the excitation eld along the CPW or antenna ( 0hrf x) is nearly uni- form and0hrf zis much smaller than 0hrf y, we Fourier- transformed only the latter component. Figure 2 depicts several CPW and antenna congurations used in the ex- periments together with their simulated spin-wave exci- tation spectra. The large prepatterned CPW on a GaAs substrate (Fig. 2(a)), which is used for VNA-FMR mea- surements, mainly excites spin waves with k0 rad/m (Fig. 2(d)). The excitation spectrum of the smaller in- tegrated CPW with a 2- m-wide signal line (Fig. 2(b)) includes one main spin-wave mode with wave vector k1 = 0.76 rad/ m and several high-order modes k2k7 (Fig. 2(e)). The 4- m-wide microstrip antenna (Fig. 2(c)) mainly excites spin waves with k1ranging from 03 (a) GGS 02468 1 0 1 20.00.51.0 -20 -10 0 10 20Amplitude (Normalized) Wave vector (rad/ m)0Hy (a. u.) y (m)S 02468 1 0 1 20.00.51.0-100 -50 0 50 100Amplitude (Normalized) Wave vector (rad/ m)GG0Hy (a. u.) y (m)S 02468 1 0 1 20.00.51.0 -10 -5 0 5 10Amplitude (Normalized) Wave vector (rad/ m)G G0Hy (a. u.) y (m)S (b) (c) (d) (e) (f) k1 k2k3k4k5k6k7k1 k2k3xyz xyz x yz FIG. 2: (a-c) Schematic illustrations of several measurement congurations used in this study. (a) VNA-FMR measurements are performed by placing the YIG/GGG sample face-down onto a CPW. The CPW consists of a 50 m-wide signal line and two 800m-wide ground lines. The gap between the signal and ground lines is 30 m. (b-c) Spin-wave transmission through the YIG lm is characterized by patterning two CPWs (b) or two microstrip antennas (c) on top of a YIG lm. The signal and ground lines of the CPWs in (b) are 2 m wide and separated by 1.6 m gaps. The microstrip antennas, which are marked by red arrows in (c), are 4 m wide. (d-f) Simulated spin-wave excitation spectra of the dierent antenna structures. The in-plane rf magnetic elds ( 0hrf y) that are produced by passing a microwave current through the CPWs in (a) and (b) or the microstrip antenna in (c) are shown in the insets. to 1.5 rad/m and some higher order modes at k22:0 rad/m andk33:8 rad/m (Fig. 2(f)). The insets of Figs. 2(d-f) show the simulated rf magnetic elds 0hrf y along they-axis for each antenna structure. III. RESULTS A. VNA-FMR We recorded FMR spectra for various in-plane exter- nal magnetic bias elds by measuring the S12scatter- ing parameter on a 40-nm-thick YIG lm. As an ex- ample, the imaginary part of S12recorded with a mag- netic bias eld 0Hext= 80 mT is shown in Fig. 3(a). The spectrum was subtracted a reference measured at a bias eld of 200 mT for enhancing signal-to-noise ra- tio. The resonance at f= 4:432 GHz is tted by a Lorentzian function. From similar data taken at other bias elds, we extracted the eld-dependence of FMR frequency and the evolution of resonance linewidth (
f) with frequency. Figures 3(b) and 3(c) summarize our results. Fitting the data of Fig. 3(b) to the Kittel for- mulafres= 0 2p Hext(Hext+Meff) using =2= 28 GHz/T, we nd Meff= 184 kA/m. The measured value ofMeffis comparable to those of other PLD-grown YIG thin lms23,24, but it is large compared to Ms(115 kA/m). Since Meff=Ms-Hani, this means that the anisotropy eld Hani=69 kA/m in our lm. The negative anisotropy eld is caused by a lattice mismatch between the YIG lm and GGG substrate23. Fitting the 4.42 4.44 3456681012 05 0 1 0 0 1 5 00246Im S12 Frequency (GHz) Frequency (GHz) Frequency (GHz) 0Hext (mT)(a) (b) = 3.5 × 10-4f(c)FIG. 3: (a) Imaginary part of the S12scattering parameter showing FMR for an in-plane external magnetic bias eld of 80 mT along the CPW. The orange line is a Lorentzian func- tion t. (b) FMR frequency as a function of external magnetic bias eld. The orange line represents a t to the experimen- tal data using the Kittel formula. (c) Dependence of FMR linewidth (
f) on resonance frequency. From a linear t to the data, we derive = 3:5104. data of Fig. 3(c) using
f= 2f+g
kgives a Gilbert damping constant = 3:5104, which is comparable to other experiments on PLD-grown lms17,18,20. In the t- ting formula, gand
kare the spin-wave group velocity and excitation-spectrum width, respectively44.4 1.8 2.1 2.4 2.7 3.0 10 20 30 40 501234 0Hext (mT)Frequency (GHz)Im S12k7k5k4 k6k3k2 Frequency (GHz)k1 -60 -30 0 30 601.82.12.42.73.03.3 H (O)Frequency (GHz)(a) (b) (c) k1k7 k1k7 CPW 1 CPW 2 0Hext45 m FIG. 4: (a) Spin-wave transmission spectrum (imaginary part of S12) recorded on a 40-nm-thick YIG waveguide with an external magnetic bias eld 0Hext= 15:5 mT along the CPWs. The inset shows a top-view schematic of the measurement geometry. (b) 2D map of spin-wave transmission spectra measured as a function of magnetic bias eld strength. (c) Angular dependence of spin-wave transmission spectra for a constant bias eld of 15.5 mT. The eld angle H= 0corresponds to the Damon-Eshbach conguration. B. Propagating spin waves We measured spin-wave transmission spectra on a 40- nm-thick YIG lm. The measurement geometry con- sisted of two CPWs on top of YIG waveguides with 45 edges (see the inset of Fig. 4(a)). The CPW parame- ters were identical to those in Fig. 2(b) and their sig- nal lines were separated by 45 m. During broadband spin-wave spectroscopy, spin waves with characteristic wave vectors ki(i= 1, 2...) were excited by passing a rf current through one of the CPWs. After propaga- tion through the YIG lm, the other CPW inductively detected the spin waves. Figure 4(a) shows the imagi- nary part of the S12scattering parameter for an external magnetic bias eld 0Hext= 15:5 mT parallel to the CPWs (Damon-Eshbach conguration). The graph con- tains seven envelope-type peaks ( k1k7) with clear pe- riodic oscillations. The peak intensities decrease with frequency because of reductions in the excitation e- ciency and spin-wave decay length. The oscillations sig- nify spin-wave propagation between the CPWs44. Fig- ure 4(b) shows a 2D representation of spin-wave trans- mission spectra recorded at dierent bias elds. As the eld strengthens, the frequency gaps between spin-wave modes become smaller. Figure 4(c) depicts the angu- lar dependence of S12spectra at a constant magnetic bias eld of 15.5 mT. In this measurement, the in-plane magnetic bias eld was rotated from -72to 72, where H= 0corresponds to the Damon-Eshbach congura- tion. As the magnetization rotates towards the wave vec- tor of propagating spin waves, the frequency and inten- sity of thek1k7modes drop. The frequency evolutions of the spin-wave modes in Figs. 4(b) and 4(c) are ex- plained by a attening of the dispersion relation with increasing magnetic bias eld strength and angle. 1 . 71 . 81 . 92 . 02 . 12 . 22 . 3-101Im S12 (Normalized) Fit Exp. Frequency (GHz)k1 k2FIG. 5: A t to the spectrum for 0Hext= 15:5 mT and H= 0(blue squares) using Eq. 3 (orange line). IV. DISCUSSION A. Fitting of spin-wave transmission spectra We used Eq. 2 to t spin-wave transmission spectra. In this equation, (!;k) is described by a Lorentzian function, while the excitation spectrum j(k)j2is ap- proximated by a Gaussian function (see Fig. 2(e)). For Damon-Eshbach spin waves with kd1, the wave vec- tor is given by k=2 d(2f)2(2fres)2 ( 0Ms)2, wheredis the lm thickness. Based on these approximations, we rewrite Eq. 2 as: ImS 12/
f (ffres)2+ (
f)2e4ln2(kk0)2=
k2 sin(ks+ );(3)5 02468 1 0 1 201234 0369 1 2 1 51.52.02.53.040 mT 15.5 mTFrequency (GHz) Wave vector (rad/ m)1 mT 9070604836H = 0Frequency (GHz) Wave vector (rad/ m)18(a) (b) FIG. 6: Spin-wave dispersion relations for dierent external magnetic bias elds (a) and eld angles (b). In (a) H= 0 and in (b)0Hext= 15.5 mT. The colored lines represent ts to the disperion relations using Eq. 4. where
fis theS12envelope width,
kis the width of the spin-wave excitation spectrum,  is the initial phase, andsis the propagation distance. Figure 5 shows a t- ting result for a spin-wave transmission spectrum with 0Hext= 15.5 mT and H= 0. As input parameters, we usedfres= 1.75 GHz, d= 40 nm,s= 45m, and Meff= 184 kA/m, which are either determined by ge- ometry or extracted from measurements.
f,
k,k0 are tting parameters. For the k1peak, we obtained the best t for
f= 0.25 GHz,
k= 0.6 rad/m, and k1= 0:72 rad/m. Thek2peak was tted with k2= 1:87 rad/m. The values of
k,k1, andk2are in good agree- ment with the simulated excitation spectrum of the CPW (Fig. 2(e)) and
fmatches the width of the envelope peak in the experimental S12spectrum. B. Spin-wave dispersion relations We extracted spin-wave dispersion relations for dier- ent magnetic bias elds and eld angles by tting the S12 transmission spectra shown in Figs. 4(b) and 4(c). The symbols in Fig. 6 summarize the results. We also cal- culated the dispersion relations using the Kalinikos and Slavin formula45: f= 0 2 Hext Hext+Meff 1Fsin2H +Meff HextF(1F) cos2H
1=2 ;(4) withF= 11exp(kd) kd. The calculated dispersion re- lations for =2= 28 GHz/T, Meff= 184 kA/m, and d = 40 nm are shown as lines in Fig. 6. The dispersion curves atten with increasing magnetic bias eld. For instance, at 0Hext= 1 mT, the frequency of propagating spin waves changes from 0.5 GHz to 2.4 GHz for wave vectors ranging from 0 to 10 rad/ m. At 0Hext= 40 mT, the frequency evolution with wave vec- 0 1 02 03 04 05 00.51.01.52.02.53.0 02 0 4 0 6 00.30.60.91.21.5g (k1) g (k2) g (k3)Group velocity g (km/s) ext (mT) Group velocity g (km/s) H (o)g (k1) g (k2) g (k3)(a) (b)FIG. 7: Spin-wave group velocity gofk1k3modes as a function of external magnetic bias eld (a) and eld angle (b). In (a)H= 0and in (b)0Hext= 15.5 mT. tor is reduced to 3 3:7 GHz. This magnetic-eld de- pendence of the dispersion relation narrows the spin-wave transmission bands in Fig. 4(b) at large 0Hext. The angular dependence of the spin-wave dispersion curves in Fig. 6(b) is explained by in-plane magneti- zation rotation from M?k(H= 0) towardsMkk (H= 90). AtH= 0, dispersive Damon-Eshbach spin waves with positive group velocity propagate between the CPWs. The character of excited spin waves changes gradually with increasing Huntil it has fully trans- formed into a backward-volume magnetostatic mode at H= 90. This mode is only weakly dispersive and ex- hibits a negative group velocity. C. Group velocity The phase relation between signals from the two CPWs is given by  = ks31,44. Since the phase shift between two neighboring maxima ( f) in broadband spin-wave transmission spectra corresponds to 2 , the group veloc- ity can be written as: g=@! @k=2f 2=s=fs; (5) wheres= 45m in our experiments. Using this equa- tion, we extracted the spin-wave group velocity for wave vectorsk1k3from the transmission spectra shown in Figs. 4(b) and 4(c). Figure 7 summarizes the variation ofgwith external magnetic bias eld and eld angle. For weak bias elds ( 0Hext<30 mT), the group ve- locity decreases swiftly, especially if kis small. For in- stance,g(k1) reduces from 3.0 km/s to 1.0 km/s in the 030 mT eld range, while g(k3) only changes from 1.2 km/s to 0.8 km/s. At larger external magnetic bias elds,gdecreases more slowly for all wave vectors. Fig- ure 7(b) shows how gvaries as a function of eld angle at0Hext= 15.5 mT. For all wave vectors, the group velocity is largest in the Damon-Eshbach conguration (H= 0). At larger eld angles, gdecreases and its6 0 1 02 03 04 05 0200400600 02 0 4 0 6 0200250300Relaxation time (ns) ext (mT) (k1) (k2) k3) Relaxation time (ns) H (o) (k1) (k2) (k3)(a) (b) FIG. 8: Spin-wave relaxation time ofk1k3modes as a function of external magnetic bias eld (a) and eld angle (b). In (a)H= 0and in (b)0Hext= 15.5 mT. 0 1 02 03 04 05 0030060090012001500 0 1 02 03 04 05 06 00100200300400Decay length ld (m) 0Hext (mT) ld (k1) ld (k2) ld (k3) Decay length ld (m) o ld (k1) ld (k2) ld (k3)(a) (b) FIG. 9: Spin-wave decay length ldofk1k3modes as a function of external magnetic bias eld (a) and eld angle (b). In (a) H= 0and in (b)0Hext= 15.5 mT. dependence on wave vector diminishes. Variations of the spin-wave group velocity with wave vector and magnetic- eld strength or angle are explained by a attening of the dispersion relations, as illustrated by the data in Fig. 6. D. Spin-wave relaxation time and decay length We now discuss the relaxation time ( ) and decay length (ld) of spin waves in our YIG lms. Following Ref. 46, the relaxation time is estimated by = 1=2f. Using= 3:5104and spin-wave transmission data from Fig. 4, we determined for wave vectors k1k3. The dependence of on external magnetic bias eld and eld angle is shown in Fig. 8. The maximum spin-wave relaxation time in our 40-nm-thick YIG lms is approx- imately 500 ns. Resembling the spin-wave group veloc- ity,is largest for small wave vectors and it decreases with increasing bias eld (Fig. 8(a)). In contrast to g, the spin-wave relaxation time is smallest in the Damon- Eshbach conguration ( H= 0) and it evolves more strongly with increasing Hifkis large (Fig. 8(b)). This result is explained by /1=fand a lowering of thespin-wave frequency if the in-plane bias eld rotates the magnetization towards k(see Fig. 4(c)). The spin-wave decay length is calculated using ld= gand data from Figs. 7 and 8. Figure 9(a) shows the dependence of ldon0Hextfor wave vectors k1k3. The largest spin-wave decay length in our 40-nm-thick YIG lms is 1.2 mm, which we measured for k1= 0:72 rad/m and0Hext= 2 mT. The decay length decreases with magnetic bias eld to about 100 m at0Hext= 50 mT. Figure 9(b) depicts the dependence of ldon the direction of a 15.5 mT bias eld. The spin-wave decay length decreases substantially with Hfor smallk, but its angular dependence weakens for larger wave vectors. The decay of propagating spin waves between the ex- citing and detecting CPW in the broadband spectroscopy measurement geometry is given by exp( s=ld)46. Based on the results of Fig. 9, one would thus expect the in- tensity of spin waves to drop with increasing wave vector and in-plane bias eld strength or angle. The spin-wave transmission spectra of Fig. 4 conrm this behavior. E. CPWs versus microstrip antennas Finally, we compare broadband spin-wave spec- troscopy measurements on YIG thin lms using CPWs and microstrip antennas. In these experiments, the CPWs and antenna structures have 4- m-wide signal lines and they were patterned onto the same 35-nm-thick YIG lm. For comparison, we also recorded transmission spectra on 50- m wide YIG waveguides. The separation distance (s) between the CPWs or microstrip antennas was set to 110 m or 220m. Schematics of the dierent measurement geometries are depicted on the sides of Fig. 10. Transmission spectra that were obtained for Damon- Eshbach spin waves in each conguration are also shown. In all measurements, we used an in-plane external mag- netic bias eld of 10 mT. The plots focus on phase os- cillations in the rst-order excitation at k1(higher-order excitations were measured also, but are not shown). The dierently shaped outline of the S12peak for two CPWs (left) or two microstrip antennas (right) mimics the pro- le of their excitation spectra (Fig. 2). As expected from f=g=s, the period of frequency oscillations ( f) be- comes smaller if the separation between antennas ( s) is enhanced (Figs. 10(c) and 10(f)). We tted the spin-wave transmission spectra obtained with CPWs (Figs. 10(a)-(c)) using the same procedure as described earlier. Good agreements between experimen- tal data (blue squares) and calculations (orange lines) were obtained by inserting Meff= 1904 kA/m,
f= 0.18 GHz,k= 0.34 rad/ m, and
k= 0.33 rad/ m into Eq. 3. To t S12spectra measured by microstrip anten- nas, we approximated the wave vector of the excitation ask=2 d(2f)2(2fres)2 ( 0Meff)2H(ffres), whereHis a Heav- iside step function47. The best results were achieved for Meff= 1782 kA/m,
f= 0.25 GHz, k= 0 rad/m and
k= 0.65 rad/ m. From the data comparison in7 -101 -101 1.3 1.4 1.5-101-101 -101 1.3 1.4 1.5-101 Frequency (GHz) Frequency (GHz)110 m 220 mCPW1 CPW2 0Hext 110 m110 mantenna2 0Hextantenna1 220 m(a) (d) (b) (e) (c) (f)Im S12 (Normalized) Im S12(Normalized)110 m 220 m FIG. 10: (a)-(c) Spin-wave transmission spectra measured using CPWs on a continuous YIG lm (a) and 50- m-wide YIG waveguides ((b) and (c)). The YIG lm and waveguides are 35 nm thick and the CPWs are separated by 110 m ((a) and (b)) and 220 m (c). (d)-(f) Spin-wave transmission spectra measured using microstrip antennas on the same YIG lm and waveguides. The signal lines of the CPWs and microstrip antennas are 4 m wide. The orange lines represent ts to the experimental data using Eq. 3. The measurement geometry for each spectrum is illustrated next to the graphs. In the schematics, the green areas depict a continuous YIG lm or waveguide. Fig. 10, we conclude that broadband spin-wave spec- troscopy measurements with CPWs and microstrip an- tennas yield similar results for Meff. We also note that theS12peak width (
f) obtained from measurements on continuous YIG lms and YIG waveguides are nearly identical (
f= 0:18 GHz for CPWs,
f= 0:22 GHz for antennas). Thus, patterning of the YIG lm into waveg- uides does not deteriorate the Gilbert damping constant. From the oscillation periods ( f) in the transmission spectra of Fig. 10, we extracted the properties of prop- agating spin waves. By averaging fover the same fre- quency range in spectra measured by CPWs and mi- crostrip antennas, we obtained g= 1:67 km/s and g= 1:53 km/s, respectively. The spin-wave relax- ation time was determined as = 225 ns (CPW) and = 237 ns (antenna) and the decay length was extracted asld= 375m (CPW) and ld= 363m (antenna). These results clearly demonstrate that broadband spin- wave spectroscopy measurements on YIG thin lms us- ing CPWs or microstrip antennas provide comparable results. V. SUMMARY In conclusion, we prepared 35 40 nm thick epitaxial YIG lms with a Gilbert damping constant = 3:5104on GGG(111) substrates using PLD. The dependence of spin-wave transmission on the strength and angle of an in-plane magnetic bias eld was systematically gauged. We used the measurements to demonstrate strong tun- ing of the spin-wave group velocity ( g), relaxation time (), and decay length ( ld) up to a eld strength of about 30 mT and above a eld angle of 20. Maximum val- ues ofg= 3:0 km/s,= 500 ns, and ld= 1:2 mm were extracted for Damon-Eshbach spin waves with k1 = 0.72 rad/ m. Moreover, we demonstrated that broad- band spin-wave spectroscopy performed with integrated CPWs and microstrip antennas yield similar results. VI. ACKNOWLEDGEMENTS This work was supported by the European Re- search Council (Grant Nos. ERC-2012-StG 307502-E- CONTROL and ERC-PoC-2018 812841-POWERSPIN). S.J.H. acknowledges nancial support from the V ais al a Foundation. Lithography was performed at the Mi- cronova Nanofabrication Centre, supported by Aalto University. 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2018-10-11

We present a comprehensive investigation of propagating spin waves in nanometer-thick yttrium iron garnet (YIG) films. We use broadband spin-wave spectroscopy with integrated coplanar waveguides (CPWs) and microstrip antennas on top of continuous and patterned YIG films to characterize spin waves with wave vectors up to 10 rad/$\mu$m. All films are grown by pulsed laser deposition. From spin-wave transmission spectra, parameters such as the Gilbert damping constant, spin-wave dispersion relation, group velocity, relaxation time, and decay length are derived and their dependence on magnetic bias field strength and angle is systematically gauged. For a 40-nm-thick YIG film, we obtain a damping constant of $3.5 \times 10^{-4}$ and a maximum decay length of 1.2 mm. Our experiments reveal a strong variation of spin-wave parameters with magnetic bias field and wave vector. Spin-wave properties change considerably up to a magnetic bias field of about 30 mT and above a field angle of $\theta_{H} = 20^{\circ}$, where $\theta_{H} = 0^{\circ}$ corresponds to the Damon-Eshbach configuration.

Propagating spin waves in nanometer-thick yttrium iron garnet films: Dependence on wave vector, magnetic field strength and angle

1810.04973v1

arXiv:1209.3669v2 [cond-mat.mes-hall] 12 Dec 2012Nonlinear emission of spin-wave caustics from an edge mode o f a micro-structured Co 2Mn0.6Fe0.4Si waveguide T.Sebastian∗and T.Br¨ acher Fachbereich Physik and Forschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaiserslautern, Germany and Graduate School Materials Science in Mainz, Gottlieb-Daimler-Straße 47, 67663 Kaiserslautern, Germa ny P.Pirro, A.A.Serga, and B.Hillebrands Fachbereich Physik and Forschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany T.Kubota WPI Advanced Institute for Materials Research, Tohoku Univ ersity, Katahira 2-1-1, Aoba-ku, Sendai 980-8577, Japan H.Naganuma, M.Oogane, and Y.Ando Department of Applied Physics, Graduate School of Engineer ing, Tohoku University, Aoba-yama 6-6-05, Sendai 980-8579, Jap an (Dated: August 30, 2018) Abstract Magnetic Heusler materials with very low Gilbert damping ar e expected to show novel magnonic transport phenomena. We report nonlinear generation of hig her harmonics leading to the emission ofcausticspin-wave beamsinalow-damping, micro-structu redCo 2Mn0.6Fe0.4SiHeuslerwaveguide. The source for the higher harmonic generation is a localized edge mode formed by the strongly inhomogeneous field distribution at the edges of the spin-wa ve waveguide. The radiation character- istics of the propagating caustic waves observed at twice an d three times the excitation frequency are described by an analytical calculation based on the anis otropic dispersion of spin waves in a magnetic thin film. 1In the last years nonlinear spin dynamics in magnetic microstructure s made of metallic ferromagnetic thin films or layer stacks have gained large interest.[1 –5] The intrinsically nonlinear Landau-Lifshitz and Gilbert equation (LLG), that govern s the spin dynamics, gives rise to a variety of nonlinear effects.[6, 7] Among the metallic ferromagnets, the class of Cobalt-based Heusle r materials is very promising for future magnon spintronics devices and the observation of new phenomena of magnonic transport. The reasons for the interest in these mater ials are the small magnetic Gilbert damping, the high spin-polarization, and the high Curie temper ature.[8, 9] As shown recently, the full Heusler compound Co 2Mn0.6Fe0.4Si (CMFS) is a very suitable material to be used as a micro-structured spin-wave waveguide du e to the increased decay length which was observed for wave propagation in the linear regime.[ 10] The reason for this observation is the low Gilbert damping of α= 3×10−3of CMFS compared to Ni 81Fe19 withα= 8×10−3, which is the material commonly used in related studies.[9] The decre ased magnetic losses not only lead to an increase of the decay length but a lso to large precession angles of the magnetic moments and, thus, to the occurrence of n onlinear effects. Regarding future applications, the investigation and the thorough understa nding of phenomena related to the spin-wave propagation in the nonlinear regime in Heusler compo unds is crucial. In this Letter, we report nonlinear higher harmonic generation fro m a localized edge mode [11, 12] causing the emission of caustic spin-wave beams [13, 14 ] in a micro-structured Heusler waveguide. Spin-wave caustics are characterized by the s mall transversal aperture of a beam, which practically does not increase during propagation, a nd the well-defined direction of propagation. The investigated sample is a 5 µm wide spin-wave waveguide structured from a 30nm thick film of the Heusler compound CMFS. Details about the fabricatio n and the material properties can be found in Refs.15 and 16. The microfabrication of the waveguide was per- formedusing electron-beam lithography andion-milling. Forthe excit ation ofspin dynamics in the waveguide the shortened end of a coplanar waveguide made of copper was placed on top of it. The Oersted field created by a microwave current in this an tenna structure can be used to excite spin dynamics in the Gigahertz range. The antenna ha s a thickness of 400nm and a width of ∆ x= 1µm. All observations have been carried out using Brillouin light scattering microscopy (µBLS).[17] µBLS is a powerful tool to investigate spin dynamics in microstructur es with a 2FIG. 1. (Color online) Sketch of the sample design. The short ened end of a coplanar waveguide is used as an antenna structure to excite spin dynamics in a 5 µm wide CMFS waveguide with a thickness of 30nm. The waveguide is positioned in the x-y-pl ane with its long axis pointing along the x-direction. The external magnetic field is applied tran sversely to the waveguide in y-direction. The figure includes a µBLS spectrum taken at a distance of 4.5 µm from the antenna in the center of the waveguide (see laser beam in the sketch) for an excitat ion frequency of fe= 3.5GHz, a microwave power of 20mW, and an external field µ0Hext= 48mT. spatial resolution of about 250nm and a frequency resolution of up to 50MHz. In the following description, the waveguide is positioned in the x-y-pla ne with the long axis pointing in x-direction. The origin of the coordinate system is give n by the position of the antenna between x=−1µm andx= 0µm. An external magnetic field of µ0Hext= 48mT was applied transversely to the waveguide in y-direction result ing in Damon-Eshbach geometry [18] forspin waves propagatingalong thewaveguide. Ask etch of thesample layout is shown in Fig.1. In addition, Fig.1 includes a spectrum taken by µBLS for an excitation frequency of fe= 3.5GHz and a microwave power of 20mW at a distance of 4.5 µm from the antenna in the center of the waveguide. The spectrum shows not only a peak a tfe= 3.5GHz but also at 2fe= 7.0GHzand 3 fe= 10.5GHz. Furthermore, the intensity of the directly excited spin waveat3.5GHzislowerthanforthehigherharmonicsatthepointofo bservation. Aswewill see, the higher harmonics are excited resonantly by nonlinear magn on-magnon interactions and the intensity distribution is a consequence of the different prop agation characteristics of the observed spin-wave modes. A two-dimensional intensity distribution for the detection frequen cyfd=fe= 3.5GHz as well as the calculated dispersion relation [19] for the center of th e CMFS waveguide are 3012345 y position ( µm)123456789x position ( µm)fd=fe 0123456 wavevector ( µm−1)3456789101112frequency (GHz) fe2fe3fe (a) (b) FIG. 2. (Color online) (a) µBLS intensity distribution for the detection frequency fdbeing equal to the excitation frequency fe= 3.5GHz. The observed intensity of the edge mode is maximal near the edges of the CMFS waveguide. Please note that the excitin g antenna is positioned between x=−1µm andx= 0µm. The blue dots in the graph indicates the position of the mea surement presented in Fig.1. (b) Calculated dispersion relation for the center of the CMFS waveguide according to Ref.19 as well as excitation frequency and high er harmonics (dashed lines). shown in Fig.2. Figure2(a) reveals a strong localization of the intens ity at the edges of the waveguide. The non-vanishing intensity close to the antenna and be tween the edges of the waveguide ( y= 1−4µm) can be attributed to nonresonant, forced excitation by the Oe rsted field created by the microwave current. Figure2(b) shows the spin -wave dispersion for the CMFS waveguide calculated according to Ref.19 assuming a homogen eous magnetization oriented in y-direction by the external field. The material paramet ers used in all our calcula- tions were determined experimentally on the unstructured film via fe rromagnetic resonance (MS= 1003kA/m and Hani= 1kA/m) and via µBLS in the micro-structured waveguide (Aex= 13pJ/m) following a method described in [9]. The effective field of µ0Heff= 46mT used in the calculations was obtained by micromagnetic simulations. As can be seen, the lower cut-off frequency f= 6.9GHz is well above the excitation frequency of fe= 3.5GHz (see dashed line in Fig.2(b)). In the center of the waveguide, the a ssumption of a homoge- neous magnetizationis a very goodapproximation. Close to theedge s, demagnetizing effects are responsible for a strongly decreased effective field and an inhom ogeneous magnetization configuration. The inhomogeneity of the magnetization does not allo w for a quantitative 45 10 15 20 microwave power (mW)BLS intensity (arb. u.)fd=fe fd=2fe fd=3fe FIG. 3. (Color online) Power dependence of the BLS intensity for the detection frequencies fd= 3,5GHz, 7GHz, and 10.5GHz for the fixed excitation frequency fe= 3.5GHz. Please note the log-log presentation of the data. The lines in the graph corr espond to fits according to Eq.1. A least square fit of the data yields s1f= 0.9±0.1,s2f= 2.1±0.1, ands3f= 2.8±0.3. modeling of spin dynamics close to edges.[20] However, from previous work it is known that this field and magnetization configuration allow for the existence of lo calized spin waves - commonly referred to as edge modes - energetically far below the sp in-wave dispersion for propagating modes in the center.[11, 12] Therefore, we conclude t hat the spin-wave mode atfd=fe= 3.5GHz is excited resonantly by the microwave field. As can be seen from the spectrum in Fig.1, the applied microwave pow er of 20mW is sufficiently high to observe the nonlinear generation of higher harmo nics of the excitation frequency fein theµBLS spectra. These resonant frequency multiplications to f= 2fe= 7.0GHz and f= 3fe= 10.5GHz result in the excitation of propagating spin-wave modes in the waveguide energetically above the cut-off frequency of the dis persion shown in Fig.2(b). Figure3 shows the dependence of the directly excited mode and the higher harmonics on the applied microwave power. This data has been acquired close to the position of the edge mode and near the antenna at x= 0.7µm andy= 0.8µm. The data is presented on a log-log scale with fits according to In(p) =Anpsn+b, (1) whereInis theµBLS intensity, Ana coupling parameter, pthe applied microwave power, andbthe noise-level in our measurement. As expected, these process es do not show a 5threshold power level, but reliable detection on the background of t he noise is not possible for powers below 5mW for fd= 3fe= 10.5GHz. The different slopes of the curves for the different spin-wave modes nare caused by the different power-laws specified by the exponent sn. A least square fit of the data yields s1f= 0.9±0.1,s2f= 2.1±0.1, ands3f= 2.8±0.3. These experimental findings close to the integer values 1, 2, and 3 a re in accordance with both reported experimental data and theoretical predictions fo r the nonlinear generation of higher harmonics.[3, 7] The observation of the second harmonic can be understood qualita tively by considering the strong demagnetizing fields caused by the out-of-plane compo nentmz(t) during the magnetization precession. Due to the demagnetizing fields, the mag netization precession M(t) around the y-direction (defined by the effective field) follows an ellip tical trajectory rather than a circular one. In contrast to the case of a circular pr ecession, the resulting projection of Mon the y-axis is time dependent and oscillating with the frequency 2 fe. The resulting dynamic dipolar field |hy(t)| ∝m2 x−m2 zcan be regarded as the source for the frequency doubling. Similar considerations lead to the observation o f higher harmonics. A full quantitative derivation of higher harmonic generation and othe r nonlinear effects based on the expansion of the LLG in terms of the dynamic magnetization ca n be found in Ref.7. Figures4(a)and(b)showintensity mapsforthedetectionfreque nciesfd= 2fe= 7.0GHz andfd= 3fe= 10.5GHz, respectively. In both cases the intensity radiated from the position of the edge mode is strongly directed, has a small transver sal aperture, and shows nondiffractive behavior. These radiation characteristics recorde d for the three spin-wave modes at f= 3.5, 7.0 and 10.5GHz presented in Figs.2(a) and 4 are responsible for t he intensity distribution shown in the spectrum in Fig.1. The position of t his measurement is indicated in the corresponding intensity maps with a circle. Since the spin wave at fe= 3.5GHz is localized at the edges of the waveguide, its intensity is compar ably weak in the center. In contrast, the higher harmonics have frequencies above the cut-off frequency of the spin-wave dispersion and can propagate in the center of the wa veguide. The propagation anglesofthese modessupport theintensity distributionrecorded inourmeasurement. While an increased intensity can be found already for fd= 2fe, forfd= 3fethe two beams starting from both edges of the spin-wave waveguide even intersect at the measurement position resultinginthehighestintensityatthispoint. Becauseofthewell-de finedpropagationangles of the higher harmonics and the localization of the edge mode, the int ensity distribution is 6012345 y position ( µm)123456789x position ( µm)θ=78.0◦ HextvG θfd=2fe 012345 y position ( µm)123456789x position ( µm)θ=67.0◦ HextvG θfd=3fe (a) (b) FIG. 4. (Color online) µBLS intensity distribution for (a) fd= 2fe= 7.0GHz and (b) fd= 3fe= 10.5GHz. Both intensity maps show strongly directed spin-wave beams along the angle θ=∠(Hext,vG). The lines in the maps are guides to the eye to identify the pr opagation angle θ. The blue dots in the graphs indicate the position of the measu rement presented in Fig.1. strongly depending on the measurement position. This observation of spin-wave beams with small transversal apert ure is reminiscent of the results in Refs.21 and 22, where nonlinear three-magnon scatter ing in yttrium iron garnet is reported. However, in that case, the propagation direction of t he nonlinearly generated spin wave is given by momentum conservation in the scattering proce ss. In contrary, in our case, due to the strong localization of the edge mode, the assu mption of a well-defined initial wavevector and, thus, a strict momentum conservation is no t justified. In particular, it is not possible to find an initial wavevector that allows for the nonline ar generation of the second and third harmonic at the same time still respecting momentu m conservation. In the following, we will describe the observed propagationcharact eristics using the prop- erties of the anisotropic spin-wave dispersion in a magnetic thin film.[13 , 14, 19] Because of this anisotropy, the direction of the flow of energy, which is given by the direction of the group velocity vG= 2π∂f(k)/∂kof the investigated spin waves, can differ significantly from the direction of its wavevector k. To estimate the relevant range in k-space in our experiment, we have to consider the lateral dimensions of the sour ce for the nonlinear pro- cesses. Since the excitation by the oscillating Oersted field is most effi cient directly below the antenna, the edge mode has the highest intensity in this region g iven by the width of the antenna of ∆ x= 1µm. The spread of the edge mode in y-direction can be estimated 7from the intensity map in Fig.2(a) to be smaller than 1 µm. Because of this localization, the edge mode in our measurement at f= 3.5GHz can be regarded as a source for the nonlinear emission of the higher harmonics with lateral dimensions of approxima tely 1×1µm2. As a first approximation, the Fourier transformation of this geometry lets us estimate the maxi- mum wavevector that can be excited by the edge mode to be kmax≈6.3µm−1. As we will see, the direction of the group velocity can be assumed to be const ant for most wavevectors that can be excited. This finally leads to the formation of the caustic s in our experiment. For given frequency and external field, the iso-frequency curve f(kx,ky) =constcan be calculated analytically from the dispersion relation. Calculations for f= 2fe= 7.0GHz and f= 3fe= 10.5GHz are illustrated in Fig.5(a), where kyis shown as a function of kx. Using this data, we calculate the direction θof the flow of energy of the spin waves relative to the externally applied field by: θ=∠(Hext,vG) = arctan( vx/vy) = arctan( dky/dkx). (2) Figure5(b)showsthecalculatedpropagationangle θintheCMFSwaveguideasafunction ofky. The most important feature in the trend of θis the small variation of ∆ θ≤2◦in the range of ky= 2−7µm−1for both frequencies f= 2feandf= 3fe. While the wavevector kchanges, the direction of vG- and, thus, the flow of energy - keeps almost constant as a function of k. In this range, which includes the maximum wavevector kmax≈6.3µm−1 that can be emitted from the edge mode (see considerations above ), the calculations yield θcalc(2f) = 79◦andθcalc(3f) = 66◦as mean values, respectively. The dash-dotted lines in Fig.5(b) represent the propagation angles θof the spin-wave beams observed experimentally as shown in Fig.4 ( θexp(2f) = 78◦andθexp(3f) = 67◦). The comparison of experimental findings and analytical calculations shows an agreement within the ex pected accuracy of our measurement setup and is, therefore, supporting our conclu sion. Higher harmonics with ky≤2µm−1are emitted with strongly varying directions from the edge mode and can be regarded as a negligible background in our measurement. In summary, we reported nonlinear higher harmonic generation fro m a localized source in a micro-structured CMFS waveguide leading to the emission of stro ngly directed spin- wave beams or caustics. This observation results from the complex interplay of different phenomena in magnonic transport in magnetic microstructures. Th e localization of an edge mode due to demagnetizing fields in the waveguide leads to the format ion of a source for 80 2 4 6 8 kx (µm−1)02468ky (µm−1) k vG Hextθf=2fe f=3fe 0 2 4 6 8 ky (µm−1)657075808590θ (degree)f=2fe f=3fe (a) (b) FIG. 5. (Color online) Analytical calculations according t o Ref.19. (a) Iso-frequency curves fconst=f(kx,ky) forf= 2fe= 7.0GHz and f= 3fe= 10.5GHz. Based on these calculations ex- emplary directions for k,vG,Hextand the propagation angle θ=∠(Hext,vG) = arctan( dky/dkx) are shown in the graph. (b) Radiation direction θcalculated from the iso-frequency curves shown in (a). Dash-dotted lines correspond to the angles θexpobserved in the experiment. the following nonlinear processes. The nonlinear higher harmonic gen eration results in the resonant excitation and emission of propagating spin waves at 2 feand 3fein a wavevector range corresponding to the localization of the edge mode. The expe rimentally observed power dependencies of the different spin-wave modes show the exp ected behavior for direct resonant excitation and nonlinear higher harmonic generation. As s hown by our calculation, the anisotropic spin-wave dispersion yields a well-defined direction of the flow of energy of the emitted spin waves in the relevant range in k-space. The calcu lation is not only qualitatively in accordance with our experimental findings but does a lso show quantitative agreement. We gratefully acknowledge financial support by the DFG Research U nit 1464 and the Strategic Japanese-German Joint Research from JST: ASPIMATT . Thomas Br¨ acher is sup- ported by a fellowship of the Graduate School Materials Science in Ma inz (MAINZ) through DFG-fundingoftheExcellence Initiative(GSC266). Wethankourco lleaguesfromthe Nano Structuring Center of the TU Kaiserslautern for their assistance in sample preparation . ∗tomseb@physik.uni-kl.de [1] H.Schultheiss, X.Janssens, M. van Kampen, F.Ciubotaru , S.J.Hermsdoerfer, B.Obry, 9A.Laraoui, A.A.Serga, L.Lagae, A.N.Slavin, B.Leven, B.Hi llebrands, Phys. Rev. Lett. 103, 157202 (2009). [2] V.E.Demidov, M.P.Kostylev, K.Rott, P.Krzysteczko, G. Reiss, S.O.Demokritov, Phys. Rev. B83, 054408 (2011). [3] V.E.Demidov, H.Ulrichs, S.Urazhdin, S.O.Demokritov, V.Bessonov, R.Gieniusz, A.Maziewski, Appl. Phys. Lett. 99, 012505 (2011). [4] H.T.Nembach, K.L.Livesey, M.P.Kostylev, P.Martin-Pi mentel, S.J.Hermsdoerfer, B.Leven, J.Fassbender, B.Hillebrands, Phys. Rev. B 84, 184413 (2011). [5] H.Schultheiss, K.Vogt, B.Hillebrands, Phys. Rev. B 86, 054414 (2012). [6] T.L.Gilbert, IEEE T. Magn. 40, 3443 (2004). [7] A.G.Gurevich, G.A.Melkov, Magnetization Oscillations and Waves (CRC, Boca Raton, 1996). [8] K. Kobayashi, N. Inaba, N. Fujita, Y. Sudo, T. Tanaka, M. O htake, M. Futamoto, F. Kirino, IEEE Trans. Magn. 45, 2541 (2009). [9] S.Trudel, O.Gaier, J.Hamrle, B.Hillebrands, J. Phys. D : Appl. Phys. 43, 193001 (2010). [10] T.Sebastian, Y.Ohdaira, T.Kubota, P.Pirro, T.Br¨ ach er, K.Vogt, A.A.Serga, H.Naganuma, M.Oogane, Y.Ando, B.Hillebrands, Appl. Phys. Lett. 100, 112402 (2012). [11] G.Gubbiotti, M.Conti, G.Carlotti, P.Candeloro, E.D. Fabrizio, K.Y.Guslienko, A.Andr´ e, C.Bayer, N.Slavin, J. Phys. Cond. Matt. 16, 7709 (2004). [12] C.Bayer, J.P.Park, H.Wang, M.Yan, C.E.Campbell, P.A. Crowell, Phys. Rev. B 69, 134401 (2004). [13] V.E.Demidov, S.O.Demokritov, D.Birt, B.O’Gorman, M. Tsoi, X.Li, Phys. Rev. B 80, 014429 (2009). [14] T.Schneider, A.A.Serga, A.V.Chumak, C.W.Sandweg, S. Trudel, S.Wolff, M.P.Kostylev, V.S.Tiberkevich, A.N.Slavin, B.Hillebrands, Phys. Rev. L ett.104, 197203 (2010). [15] T.Kubota, S.Tsunegi, M.Oogane, S.Mizukami, T.Miyaza ki, H.Naganuma, Y.Ando, Appl. Phys. Lett. 94, 122504 (2009). [16] M.Oogane, T.Kubota, Y.Kota, S.Mizukami, H.Naganuma, A.Sakuma, Y.Ando, Appl.Phys. Lett.96, 252501 (2010). [17] V.E.Demidov, S.O.Demokritov, B.Hillebrands, M.Lauf enberg, P.P.Freitas, Appl. Phys.Lett. 85, 2866 (2004). 10[18] R.W.Damon, J.R.Eshbach, J. Phys. Chem. Solids 19, 308 (1961). [19] B.Kalinikos, A.Slavin, J. Phys. C: Solid State 19, 7013 (1986). [20] M.P. Kostylev, G. Gubbiotti, J.-G. Hu, G. Carlotti, T. O no, R. Stamps, Phys. Rev. B 76, 054422 (2007). [21] C.Mathieu, V.T.Synogatch, C.E.Patton, Phys. Rev. B 67, 104402 (2003). [22] C.L.Ord´ o˜ nez-Romero, B.A.Kalinikos, P.Krivosik, W .Tong, P.Kabos, C.E.Patton, Phys. Rev. B79, 144428 (2009). 11

2012-09-17

Magnetic Heusler materials with very low Gilbert damping are expected to show novel magnonic transport phenomena. We report nonlinear generation of higher harmonics leading to the emission of caustic spin-wave beams in a low-damping, micro-structured Co2Mn0.6Fe0.4Si Heusler waveguide. The source for the higher harmonic generation is a localized edge mode formed by the strongly inhomogeneous field distribution at the edges of the spin-wave waveguide. The radiation characteristics of the propagating caustic waves observed at twice and three times the excitation frequency are described by an analytical calculation based on the anisotropic dispersion of spin waves in a magnetic thin film.

Nonlinear emission of spin-wave caustics from an edge mode of a micro-structured Co2Mn0.6Fe0.4Si waveguide

1209.3669v2

epl draft Transport properties of L evy walks: an analysis in terms of mul- tistate processes Giampaolo Cristadoro1, Thomas Gilbert(a)2, Marco Lenci1;3andDavid P. Sanders4 1Dipartimento di Matematica, Universit a di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy. 2Center for Nonlinear Phenomena and Complex Systems, Universit e Libre de Bruxelles, C. P. 231, Campus Plaine, B-1050 Brussels, Belgium. 3Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126 Bologna, Italy. 4Departamento de F sica, Facultad de Ciencias, Universidad Nacional Aut onoma de M exico, Ciudad Universitaria, 04510 M exico D.F., Mexico. PACS 05.40.Fb { Random walks and L evy ights PACS 05.60.-k { Transport processes PACS 02.50.-r { Probability theory, stochastic processes, and statistics PACS 02.30.Ks { Delay and functional equations Abstract { Continuous time random walks combining diusive and ballistic regimes are intro- duced to describe a class of L evy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker, we are led to a description of such L evy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay dierential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coecients in terms of the parameters of the models. Random walks described by L evy ights give rise to complex diusive processes [1{4] and have found many applications in physics and beyond [5{8]. Whereas the random walks associated with Brownian motion are char- acterized by Gaussian propagators whose variance grows linearly in time, the propagators of L evy ights have in- nite variance [9{11]; they occur in models of random walks such that the probability of a long jump decays slowly with its length [12]. By considering the propagation time between the two ends of a jump, one obtains a class of models known as L evy walks [13{19]. A L evy walker thus follows a contin- uous path between the two end points of every jump, per- forming each in a nite time; instead of having an innite mean squared displacement, as happens in a L evy ight whose jumps take place instantaneously, a L evy walker moves with nite velocity and, ipso facto, has a nite mean squared displacement, although it may increase faster than linearly in time. A L evy ight is characterized by its probability density (a)E-mail: thomas.gilbert@ulb.ac.beof jump lengths x, orfree paths , which we denote (x). It is assumed to have the asymptotic scaling, (x)x1, whose exponent, >0, determines whether the moments of the displacement are nite. In particular, for 2, the variance diverges. In the framework of continuous time random walks [5, Chs. 10 & 13], a probability distribution ( r;t) of mak- ing a displacement rin a timetis introduced, such that, for instance, in the so-called velocity picture, ( r;t) = (jrj)D(tjrj=v), wherevdenotes the constant speed of the particle and D(:) is the Dirac delta function. Consid- ering the Fourier-Laplace transform of the propagator of this process, one obtains, in terms of the parameter , the following scaling laws for the mean squared displacement after timet[13,20], hr2it8 >>>>>>< >>>>>>:t2; 0<< 1; t2=logt;  = 1; t3; 1<< 2; tlogt;  = 2; t; > 2:(1) p-1arXiv:1407.0227v2 [cond-mat.stat-mech] 10 Nov 2014G. Cristadoro et al. In this Letter, we consider L evy walks on lattices and generalize the above description, according to which a new jump event takes place as soon as the previous one is completed, to include an exponentially-distributed wait- ing time which separates successive jumps. This induces a distinction between the states of particles which are in the process of completing a jump and those that are waiting to start a new one. As shown below, such considerations lead to a theoretical formulation of the model as a multistate generalized master equation [21{23], which translates into a set of coupled delay dierential equations for the corre- sponding distributions. The physical motivation for the inclusion of an exponentially-distributed waiting time between successive jump events stems, for instance, in the framework of chaotic scattering, from the time required to escape a frac- tal repeller [24, 25], or, more generally, the time spent in a chaotic transient [26]. In the framework of active trans- port, such as dealing with the motion of particles embed- ded within living cells [27], such waiting times may help model the complex process related to changes in the di- rection of propagation of such particles. This is also rele- vant to laser cooling experiments [28], where a competition in the damping and increase of atomic momenta induces a form of random walk in momentum space. The times spent by atoms in small momenta states typically follow exponential distributions. The stop-and-go patterns of random walkers thus gen- erated have been studied in the context of animal foraging [29]. Such search strategies have been termed saltatory. In contrast to classical strategies, according to which animals either move while foraging or stop to ambush their prey, a saltatory searcher alternates between scanning phases, which are performed diusively on a local scale, and re- location phases, during which motion takes place with- out search. Examples of such intermittent behaviour have been identied in a variety of animal species [30, 31], as well as in intracellular processes such as proteins binding to DNA strands [32]. Visual searching patterns whereby information is extracted through a cycle of brief xations interspersed with gaze shifts [33] provide another illustra- tion in the context of neuroscience. One can also think of applications to sociological processes, for instance when interactions between individuals is sampled at random times, independent of the underlying process [34]. From a mathematical perspective, an important ques- tion that arises in the framework of foraging is that of optimal strategies [35]. As reported in [36], L evy ight motion can, under some conditions on the nature and dis- tribution of targets, emerge as an optimal strategy for non-destructive search, i.e., when targets can be visited innitely often. For destructive searches on the other hand, intermittent search strategies with exponentially- distributed waiting times provide an alternative to L evy search strategies, which turns out to minimize the search time [37]. The processes we analyze in this Letter, al- though they are restricted to motion on lattices, can bethought of as extensions of intermittent search processes to power-law distributed relocation phases which are typical of L evy search strategies, thus opening a new perspective. We show below that, inasmuch as the dispersive proper- ties are concerned, a complete characterisation of the pro- cess can be obtained, which reproduces the scaling laws (1), as well as yields the corresponding transport coe- cients, whether normal or anomalous. These results also elucidate the incidence of exponential waiting times on these coecients. L evy walks as multistate processes. { We call propagating the state of a particle which is in the process of completing a jump. In contrast, the state of a par- ticle waiting to start a new jump is called scattering1. Whereas particles switch from propagating to scattering states as they complete a jump, particles in a scattering state can make transitions to both scattering and propa- gating states; as soon as their waiting time has elapsed, they move on to a neighbouring site and, doing so, may switch to a propagating state and carry their motion on to the next site, or start anew in a scattering state. We consider a d-dimensional cubic lattice of individual cells n2Zd. The state of a walker at position nand timetcan take on a countable number of dierent values, specied by two integers, k0 andj2f1;:::;zg, where z2dis the coordination number of the lattice. Scatter- ing states are labeled by the state k= 0 and propagating states by the pair ( k;j), such that k1 counts the re- maining number of lattice sites the particle has to travel in direction jto complete its jump. Time-evolution proceeds as follows. After a random waiting time t, exponentially-distributed with mean R, a particle in the scattering state k= 0 changes its state to (k;j) with probability k=z, moving its location from site nto site n+ej. Conversely, particles which are at site n in a propagating state ( k;j),k1, jump to site n+ej, in timeB, changing their state to ( k1;j). The waiting time density of the process is the function k(t) =( 1 Ret=R; k = 0; D(tB); k6= 0:(2) When a step takes place, the transition probability to go from state ( k;j) to state (k0;j0) is p(k;j);(k0;j0)=( k0=z; k = 0; k1;k0j;j0; k6= 0;(3) where:;:is the Kronecker symbol. For deniteness, we consider below the following simple parameterisation of the transition probabilities, k=( 1; k = 0; [k(k+ 1)]; k1;(4) 1B enichou et al.[37] refer to these two states as respectively bal- listic and diusive. p-2Transport properties of L evy walks in terms of multistate processes in terms of the parameters 0 << 1, which weights scat- tering states relative to propagating ones, and >0, the asymptotic scaling parameter of free path lengths. Master equation. { The probability distribution of particles at site nand timet,P(n;t), is a sum of the dis- tributions over the scattering states, P0(n;t), and propa- gating states, Pk;j(n;t),k1 and 1jz. According to eqs. (2) and (3), changes in the distribution of ( k;j)- states,k1, in cell narise from particles located at cell nejwhich make a transition from either state 0 or state (k+ 1;j). Since the latter transitions can be traced back to changes in the distribution of ( k+ 1;j)-states in cell nejat timeBearlier, we can write2 @tPk;j(n;t)@tPk+1;j(nej;tB) =k zR[P0(nej;t)P0(nej;tB)];(5) which accounts for the fact that a positive 0-state contri- bution at time tbecomes a negative one at time t+B. Applying this relation recursively, we have See eq. (6)next page. Terms lost by (1 ;j)-states in cells nej,j= 1;:::;z , are gained by the 0-state in cell n, which also gains contribu- tions from 0-state transitions. Since the scattering state also loses particles at exponential rate 1 =R, we have @tP0(n;t) =1 zRzX j=11X k=0kP0(n(k+ 1)ej;tkB) 1 RP0(n;t): (7) It is straightforward to check that eqs. (6) and (7) are con- sistent with conservation of probability3,P nP(n;t) = 1. Fraction of scattering particles. { As discussed below, an important role is played by the overall fraction of particles in the scattering state, S0(t)P n2ZdP0(n;t). From eq. (7), this quantity is found to obey the following linear delay dierential equation, R_S0(t) =1X k=1kS0(tkB)S0(t): (8) Given initial conditions, e.g. S0(t) = 0,t<0, andS0(0) = 1 (all particles start in a scattering state), this equation can be solved by the method of steps [38]. Because the sum of the coecients on the right-hand side of eq. (8) is zero, the solutions are asymptotically constant and can be classied in terms of the parameter . 2The possible addition of source terms into this expression will not be considered here. 3A simplication occurs if one considers the distribution of prop- agating states in direction j,Pj(n;t) =P1 k=1Pk;j(n;t). Us- ing eq. (4), the time-evolution of this quantity is @tPj(n;t) = =(zR)P1 k=1k[P0(nkej;t(k1)B)P0(nkej;tkB)].For > 1, the average return time to the 0-state,P1 k=0k(R+kB), is nite and given in terms of the Rie- mann zeta function, sinceP1 k=0kk=(). The process is thus positive-recurrent and we have lim t!1S0(t) =R R+B()(>1): (9a) In the remaining range of parameter values, 0 < 1, the process is null-recurrent: the average return time to the 0-state diverges and lim t!1S0(t) = 0. If6= 1, the decay is algebraic, lim t!1(t=B)1S0(t) =sin() R B(0<< 1);(9b) which can be obtained from a result due to Dynkin [39]; see also Refs. [40, Vol. 2, xXIV.3] and [28,x4.4]. The case= 1 is a singular limit with logarithmic decay, lim t!1log(t=B)S0(t) =1 R B(= 1): (9c) Mean squared displacement. { Assuming an ini- tial position at the origin, the second moment of the dis- placement ishn2it=P n2Zdn2P(n;t). Its time-evolution is obtained by dierentiating this expression with respect to time and substituting eqs. (6) and (7), Rd dthn2it=S0(t) +1X k=12k+ 1 kS0(tkB);(10) where, using eq. (4), we made use of the identityP1 j=kj= 1 fork= 0 andkotherwise. The time- evolution of the second moment is thus obtained by inte- grating the fraction of 0-state particles, Rhn2it=Zt 0dsS0(s) +bt=BcX k=12k+ 1 kZtkB 0dsS0(s); (11) where, assuming the process starts at t= 0, we set S0(t) = 0 fort<0. As emphasized earlier, equation (8) can be solved an- alytically given initial conditions on the state of walkers. By extension, so can equation (11), thus providing an ex- act time-dependent expression of the mean squared dis- placement. This is particularly useful when one wishes to study transient regimes and the possibility of a crossover between dierent scaling behaviours, or indeed when the asymptotic regime remains experimentally or numerically unaccessible. The analytic expression of the mean squared displacement and the issue of the transients will be stud- ied elsewhere. Here, we focus on the asymptotic regime, i.e.,tB. Substituting the asymptotic expressions (9), into eq. (11), we retrieve the regimes described by eq. (1) and obtain the corresponding coecients. p-3G. Cristadoro et al. @tPk;j(n;t) =1 zR1X k0=1k+k01h P0(nk0ej;t(k01)B)P0(nk0ej;tk0B)i : (6) a=2.5 e=0.5 1011021031041.92.02.12.22.32.42.52.6 tXn2\tt 1011021031040.59850.59900.59950.60000.60050.60100.6015S0HtL (a) Normal diusion, = 5=2 a=2. e=0.5 1011021031040.60.70.80.91.0 tXn2\t@tlogHtLD 1011021031040.5500.5520.5540.5560.5580.560S0HtL (b) Weak super-diusion, = 2 a=1.5 e=0.5 1011021031040.600.650.700.750.80 tXn2\tt1.5 1011021031040.440.450.460.470.480.49S0HtL (c) Super diusion, = 3=2 Fig. 1: Examples of numerical computations of hn2itfor pa- rameters values >1, rescaled by their respective asymptotic scalings with respect to time ( = 1=2 in all cases). The dotted lines correspond to eq. (12a). The insets show the evolution of the fraction of scattering states towards their asymptotic values, given by eq. (9a). Starting with the positive-recurrent regime,  > 1, eq. (9a), we have the three asymptotic regimes, tB, hn2it't R+B() a=0.5 e=0.5 1011021031040.460.470.480.490.50 tXn2\tt2 1011021031040.6360.6380.6400.6420.6440.6460.648t0.5S0HtL(a) Ballistic diusion, = 1=2 a=1. e=0.5 1011021031040.800.850.900.951.00 tXn2\tlogHtLt2 1011021031041.01.21.41.61.82.0logHtLS0HtL (b) Sub-ballistic diusion, = 1 Fig. 2: Same as g. 1 for the range of parameters 0 < 1, with comparisons to eq. (12b) and, in the insets, eqs. (9b) and (9c). 8 >< >:1 +[() + 2(1)]; > 2; 2log(t=B);  = 2; 2 (2)(3)(t=B)2; 1<< 2:(12a) Whereas the rst regime,  > 2, yields normal diusion, the other two correspond, for = 2, to a weak form of super-diusion, and, for 1 <  < 2, to super-diusion, such that the mean squared displacement grows with a power of time 3>1, faster than linear4. Ballistic diusion occurs in the null-recurrent regime of the parameter, 0 < 1. Using eqs. (9b) and (9c), we nd hn2it't2 2 B( 1=log(t=B);  = 1; 1; 0<< 1:(12b) The asymptotic regimes described by eqs. (12) gener- alize to continuous-time processes similar results found in 4Equation (12a) assumes  >0. If one takes the limit !0, sub-leading terms may become relevant. In particular, when = 0, normal diusion is recovered and the right-hand side of (12a) is t=R for all. p-4Transport properties of L evy walks in terms of multistate processes the context of countable Markov chains applied to discrete time processes [41]. They can also be compared to results obtained in Ref. [42]. Although the L evy walks considered by these authors do not include exponentially-distributed waiting times separating successive propagating phases, our results are rather similar to theirs; the only actual dierences arise in the regime of normal diusion, >2. In gs. 1 and 2, the asymptotic results (12) are com- pared to numerical measurements of the mean squared displacement of the process dened by eqs. (2), (3) and the transition probabilities (4). Timescales were set to RB1 and the lattice dimension to d= 1. The algorithm is based on a classic kinetic Monte Carlo algo- rithm [43], which incorporates the possibility of a ballistic propagation of particles after they undergo a transition from a scattering to a propagating state. For each realisa- tion, the initial state is taken to be scattering. Positions are measured at regular intervals on a logarithmic time scale for times up to t= 104R. Averages are performed over sets of 108trajectories. Concluding remarks. { The specicity of our ap- proach to L evy walks lies in the inclusion of exponentially- distributed waiting times that separate successive jumps. This additional feature induces a natural description of the process in terms of multiple propagating and scatter- ing states whose distributions evolve according to a set of coupled delay dierential equations. The mean squared displacement of the process depends on the distribution of free paths and boils down to a simple expression involving time-integrals of the fraction of scat- tering states. Using straightforward arguments, precise asymptotic expressions were obtained for this quantity, which reproduce the expected scaling regimes [13,20], and provide values of the diusion coecients, whether normal or anomalous. Our results conrm that, in the null-recurrent regime of ballistic transport, scattering events, are unimportant. Furthermore, these events do not modify the exponent of the mean squared displacement in the positive recur- rent regime; in other words, the addition of a scattering phase has no incidence on the scaling exponents. In this regime, however, the transport coecients, whether nor- mal or anomalous, depend on the details of the model, underlying the relevance of pausing times that separate long ight events, for example, in the context of animal foraging [36]. Although the results we reported are limited to walks with exponentially distributed waiting times, our formal- ism can be easily extended to include the possibility of waiting times with power law distributions such as ob- served in Ref. [44]. Such processes are known to allow for sub-diusive transport regimes [11]. The combination of two power law scaling parameters, one for the waiting time and the other for the duration of ights, indeed yields a richer set of scaling regimes [45], which can be studied within our framework.Our results can on the other hand be readily applied to the regime R=B1, i.e., such that the waiting times in the scattering state are typically negligible compared to the ballistic timescale. This is the regime commonly studied in reference to L evy walks. Our investigation simultaneously opens up new avenues for future work. Among results to be discussed elsewhere, our formalism can be used to obtain exact solutions of the mean squared displacement as a function of time. As dis- cussed already, this is particularly useful to study transient regimes, such as can be observed when the distribution of free paths has a cut-o or, more generally, when it crosses over from one regime to another, e.g. from a power law for small lengths to exponential decay for large ones, or when the anomalous regime is masked by normal sub-leading contributions which may nonetheless dominate over time scales accessible to numerical computations [46]. One can also apply these ideas to the anomalous photon statistics of blinking quantum dots [47,48]. The on/o switchings of a quantum dot typically exhibit power law distributions. In the limit of strong elds, however, the on-times display exponential cutos. Another interesting regime occurs when, in the positive- recurrent range of the scaling parameter, >1, the like- lihood of a transition from a scattering to a propagating state is small, 1. A similar perturbative regime arises in the innite horizon Lorentz gas in the limit of narrow corridors [49]. As is well-known [50], the scaling parame- ter of the distribution of free paths has the marginal value = 2, such that the mean squared displacement asymp- totically grows with tlogt. Although it has long been ac- knowledged that the innite horizon Lorentz gas exhibits features similar to a L evy walk [51, 52], we argue that a consistent treatment of this model in such terms is not possible unless exponentially-distributed waiting times are taken into account that separate successive jumps. In- deed, the parameter , which weights the likelihood of a transition from scattering to propagating states, is the same parameter that separates the average relaxation time of the scattering state from the ballistic timescale, i.e., B=R/1. This is the subject of a separate publica- tion [53].  We wish to thank Eli Barkai for useful comments and suggestions. This work was partially supported by FIRB- Project No. RBFR08UH60 (MIUR, Italy), by SEP- CONACYT Grant No. CB-101246 and DGAPA-UNAM PAPIIT Grant No. IN117214 (Mexico), and by FRFC convention 2,4592.11 (Belgium). T.G. is nancially sup- ported by the (Belgian) FRS-FNRS. REFERENCES [1]Haus J. W. andKehr K. W. ,Physics Reports ,150 (1987) 263. p-5G. Cristadoro et al. [2]Weiss G. H. ,Aspects and Applications of the Random Walk (North-Holland, Amsterdam) 1994. [3]Krapivsky P. L., Redner S. andBen-Naim E. ,A Ki- netic View of Statistical Physics (Cambridge University Press, Cambridge, UK) 2010. [4]Klafter J. andSokolov I. M. ,First steps in random walks: from tools to applications (Oxford University Press, Oxford, UK) 2011. [5]Shlesinger M. F., Zaslavsky G. M. andFrisch U. , (Editors) L evy Flights and Related Topics in Physics Vol. 450 of Lecture Notes in Physics (Springer, Berlin, Heidel- berg) 1995. [6]Klages R., Radons G. andSokolov I. M. ,Anoma- lous transport: Foundations and applications (Wiley-VCH Verlag, Weinheim) 2008. [7]Denisov S., Zaburdaev V. Y. andHanggi P. ,Physical Review E ,85(2012) 31148. [8]Mendez V., Campos D. andBartumeus F. ,Stochas- tic Foundations in Movement Ecology (Springer, Berlin, Heidelberg) 2013. [9]Shlesinger M. F., Zaslavsky G. M. andKlafter J. , Nature ,363(1993) 31. [10]Klafter J., Shlesinger M. F. andZumofen G. , Physics Today ,49(1996) 33. [11]Metzler R. andKlafter J. ,Physics Reports ,339 (2000) 1. [12]Weiss G. H. andRubin R. J. ,Advances in Chemical Physics ,52(1983) 363. [13]Geisel T., Nierwetberg J. andZacherl A. ,Physical Review Letters ,54(1985) 616. [14]Shlesinger M. F. andKlafter J. ,Physical Review Let- ters,54(1985) 2551. [15]Shlesinger M. F., West B. J. andKlafter J. ,Phys- ical Review Letters (ISSN 0031-9007) ,58(1987) 1100. [16]Klafter J., Blumen A. andShlesinger M. F. ,Phys- ical Review A ,35(1987) 3081. [17]Blumen A., Zumofen G. andKlafter J. ,Physical Re- view A ,40(1989) 3964. [18]Zumofen G. andKlafter J. ,Physical Review E ,47 (1993) 851. [19]Shlesinger M. F., Klafter J. andZumofen G. , American Journal of Physics ,67(1999) 1253. [20]Wang X.-J. ,Physical Review A ,45(1992) 8407. [21]Montroll E. W. andWeiss G. H. ,Journal of Mathe- matical Physics ,6(1965) 167. [22]Kenkre V. M., Montroll E. W. andShlesinger M. F. ,Journal of Statistical Physics ,9(1973) 45. [23]Landman U., Montroll E. W. andShlesinger M. F. , Proceedings of the National Academy of Sciences of the United States of America ,74(1977) 430. [24]Ott E. andTel T. ,Chaos ,3(1993) 417. [25]Gaspard P. ,Chaos, Scattering and Statistical Mechanics (Cambridge University Press, Cambridge, UK) 1998. [26]Kantz H. andGrassberger P. ,Physica D ,17(1985) 75. [27]Gal N. andWeihs D. ,Physical Review E ,81(2010) 020903. [28]Bardou F., Bouchaud J.-P., Aspect A. andCohen- Tannoudji C. ,L evy Statistics and Laser Cooling (Cam- bridge University Press, Cambridge, UK) 2001. [29]OBrien W. J., Browman H. I. andEvans B. I. ,Amer- ican Scientist ,78(1990) 152.[30]Kramer D. L. andMcLaughlin R. L. ,American Zool- ogist,41(2001) 137. [31]Mashanova A., Oliver T. H. andJansen V. A. ,Jour- nal of The Royal Society Interface ,7(2010) 199. [32]Halford S. E. andMarko J. F. ,Nucleic acids research , 32(2004) 3040. [33]Ludwig C. J., Davies J. R. andEckstein M. P. ,Pro- ceedings of the National Academy of Sciences ,111(2014) E291. [34]Brockmann D., Hufnagel L. andGeisel T. ,Nature , 439(2006) 462. [35]Viswanathan G. M., da Luz M. G. E., Raposo E. P. andStanley H. E. ,The Physics of Foraging: an Intro- duction to Random Searches and Biological Encounters (Cambridge University Press, Cambridge, UK) 2011. [36]Viswanathan G., Buldyrev S. V., Havlin S., Da Luz M., Raposo E. andStanley H. E. ,Nature ,401(1999) 911. [37]Benichou O., Loverdo C., Moreau M. andVoituriez R.,Reviews of Modern Physics ,83(2011) 81. [38]Driver R. D. ,Ordinary and delay dierential equations (Springer-Verlag, New York, NY) 1977. [39]Dynkin E. B. ,IMS-AMS Selected Translations in Math. Stat. and Prob. ,1(1961) 171. [40]Feller W. ,An introduction to probability theory and its applications 3rd Edition (Wiley) 1968. [41]Wang X.-J. andHu C.-K. ,Physical Review E ,48(1993) 728. [42]Zumofen G. andKlafter J. ,Physica D ,69(1993) 436. [43]Gillespie D. T. ,J. Comput. Phys ,22(1976) 403. [44]Solomon T. H., Weeks E. R. andSwinney H. L. , Physical Review Letters ,71(1993) 3975. [45]Portillo I. G., Campos D. andMendez V. ,Journal of Statistical Mechanics ,2011 (2011) P02033. 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2014-07-01

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of L\'evy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker, we are led to a description of such L\'evy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay differential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coefficients in terms of the parameters of the models.

Transport properties of Lévy walks: an analysis in terms of multistate processes

1407.0227v2

1 Ultra -low damping insulating magnetic thin films get perpendicular Lucile Soumah1, Nathan Beaulieu2, Lilia Qassym3, Cécile Carrétero1, Eric Jacquet1, Richard Lebourgeois3, Jamal Ben Youssef2, Paolo Bortolotti1, Vincent Cros1, Abdelmadjid Anane1* 1 Unité Mixte de Physique CNRS , Thales, Univ. Paris -Sud, Université Paris Saclay, 91767 Palaiseau, France 2LABSTICC, UMR 6285 CNRS, Université de Bretagne Occidentale, 29238 Brest, France 3 Thales Research and Technology, Thales 91767 Palaiseau , France * Email : madjid.anane@u -psud.fr A magnetic material combining low losses and large Perpendicular Magnetic Anisotropy (PMA) is still a missing brick in the magnonic and spintronic field s. We report here on the growth of ultrathin Bismuth doped Y 3Fe5O12 (BiYIG ) films on Gd 3Ga 5O12 (GGG) and substituted GGG (sGGG) (111) oriented substrates. A fine tuning of the PMA is obtained using both epitaxial strain and growth induced anisotrop ies. Both spontaneously in -plane and out -of-plane magnetized thin films can be elaborated . Ferromagnetic Resonance (FMR) measurement s demonstrate the high dynamic quality of these BiYIG ultrathin films , PMA films with Gilbert damping values as low as 3 10-4 and FMR linewidth of 0.3 mT at 8 GHz are achieved even for films that do not exceed 30 nm in thickness . Moreover, w e measure Inverse Spin Hall Effect (ISHE) on Pt/BiYIG stack s showing that the magnetic insulator ’s surface is transparent to spin current making it appealing for spintronic applications . 2 Introduction. Spintronic s exploit s the electron’s spin in ferromagnetic transition metal s for data storage and data processing. Interestingly, as spintronics codes information in the angular momentum degre es of freedom , charge transport and therefore the use of conducting materials is not a requirement, opening thus electronics to insulators . In magnetic insulators (MI), pure spin currents are described using excitation states of the ferromagnetic background named magnons (or spin waves). Excitation, propagation and detection of magnons are at the confluent of the emerging concepts of magnonics 1,2, caloritronics3 and spin -orbitronics4. Magnons, and their classical counterpart , the spin waves (SWs) can carry information over distances as large as millimeters in high quality thick YIG films, with frequencies extending from the GHz to the THz regime5–7. The main figure of merit for magnonic materials is the Gilbert damping 1,5,8 which has to be as small as possible. This makes the number of relevant materials for SW propagation quite limited and none of them has yet been found to possess a large enough perpendicular magnetic anisotropy (PMA ) to induce spontaneous out -of-plane magnetization . We report here on the Pulsed Laser Deposition (PLD) growth of ultra -low loss MI nanometers -thick films with large PMA : Bi substituted Yttrium Iron Garnet ( BixY3-xFe5O12 or BiYIG ) where tunability of the PMA is achieved through epitaxial strain and Bi doping level. The peak -to-peak FMR linewidth (that characterize the losses) can be as low as 𝜇0𝛥𝐻pp=0.3 mT at 8 GHz for 30 nm thick films. This material thus opens new perspectives for both spintronic s and magnonic s fields as the SW dispersion relation can now be easily tuned through magnetic anisotropy without the need of a large bias magnetic field. Moreover, energy efficient data storage devices based on magn etic textures existing in PMA materials like magnetic bubble s, chiral domain walls and magnetic skyrmions would benefit from such a low loss material for efficient operation9. The study of micron -thick YIG films grown by liquid phase epitaxy (LPE) was among the hottest topics in magnetism few decades ago . At this time, it has been already noticed that unlike rare earths (Thulium, Terbium, Dysprosium …) substitutions, Bi substitution does not overwhelmingly increase the magnetic losses10,11 even though it induces high uniaxial magnetic anisotropy12–14 . Very recently, ultra -thin MI films showing PMA have been the subject of an increasing interes t 15,16: Tm 3Fe5O12 or BaFe 12O19 (respectively a garnet and an hexaferrite) have been used to demonstrate spin -orbit -torque magnetization reversal using a Pt over -layer as a source of spin current 4,17,18. However, their large magnetic losses prohibit their use as a spin -wave medium (reported va lue o f 𝜇0𝛥𝐻pp of TIG is 16.7 mT at 9.5 GHz)19. Hence, whether it is possible to fabricate ultra -low loss thin films with a large PMA that can be used for both magnonics and spintronics applications remains to be demonstrated . Not only l ow 3 losses are important for long range spin wave propagation but they are also necessary for spin transfer torque oscillators (STNOs) as the threshold current scales with the Gilbert damping20. In the quest for the optimal material platform , we explore here the growth of Bi doped YIG ultra -thin films using PLD with different substitution; BixY3-xIG (x= 0.7, 1 and 1.5) and having a thickness ranging between 8 and 50 nm. We demonstrate fine tuning of the magnetic anisotropy using epitaxial strain and measure ultra low Gilbert damping values ( 𝛼=3∗10−4) on ultrathin films with PMA . Results Structural and magnetic characterization s The two substrates that are used are Gallium Gadolinium Garnet (GGG) which is best lattice matched to pristine YIG and substituted GGG (sGGG) which is traditionally used to accommodate substituted YIG films for photonics applications . The difference between Bi and Y ionic radii ( rBi = 113 pm and rY = 102 pm)21 leads to a linear increase of the BixY3-xIG bulk lattice parameter with Bi content (Fig. 1 -(a) and Fig. 1-(b)). In Fig. 1, we present the (2−) X-ray diffraction patterns (Fig.1 -(c) and 1 -(d)) and reciprocal space maps (RSM) (Fig.1 -(e) and 1 -(f)) of BiYIG on sGGG(111) and GGG(111) substrates respectively . The presence of ( 222) family peaks in the diffraction spectra shown in Fig. 1 -(b) and 1 -(c) is a signature of the films’ epitaxial quality and the presence of Laue fringes attest s the coherent crystal structure existing over the whole thickness. As expected, all films on GGG are under compressive strain, whereas films grown on sGGG exhibit a transition from a tensile (for x= 0.7 and 1) towards a compressive ( x= 1.5) strain . Reciprocal Space Mapping of these BiYIG samples shown in Fig.1 -(e) and 1 -(f) evidences the pseudomorphic nature of the growth for all films , which confirms the good epitaxy. The static magnetic properties of the films have been characterized using SQUID ma gnetometry, Faraday rotation measurements and Kerr microscopy. As the Bi doping has the effect of enhancing the magneto - optical response 22–24, we measure on average a large Faraday rotation coefficients reaching up to 𝜃F = −3 °.𝑚−1 @ 632 nm for x= 1 Bi doping level and 15 nm film thickness . Chern et al .25 performed PLD growth of BixY3-xIG on GGG and reported an increase of 𝜃F 𝑥= −1.9 °.𝜇𝑚−1 per Bi substitution x @ 632 nm. The Faraday rotation coefficients we find are slightly larger and m ay be due to the much lower thickness of our films as 𝜃F is also dependent on the film thickness26. The saturation magnetization ( Ms) remains constant for all Bi content (see Table 1) within the 10% experimental errors . We observe a clear correlation between the strain and the shape of the in -plane and out -of-plane hysteresis loop s reflecting changes in the magnetic anisotropy. Wh ile films under compressive strain exh ibit in -plane anisotropy, those under tensile strain show a large out -of-plane anisotrop y that can eventually lead to an out -of- plane easy axis for x= 0.7 and x= 1 grown on sGGG. The transition can be either induced by ch anging the 4 substrate (Fig.2 -(a)) or the Bi content ( Fig. 2-(b)) since both act on the misfit strain. We ascribe the anisotropy change in our films to a combination of magneto -elastic anisotropy and growth induced anisotropy, this later term being the domin ant one (see Supplementary Note 1). In Fig. 2 -(c), we show the magnetic domains structures at remanance observed using polar Kerr microscopy for Bi1Y2IG films after demagnetization : µm-wide maze -like magnetic domains demonstrate s unambiguously that the magnetic easy axis is perpendicular to the film surface . We observe a decrease of the domain width (Dwidth) when the film thickness ( tfilm) increases as expected from magnetostatic energy considerations. In fact, a s Dwidth is severa l order s of magnitude larger than tfilm, a domain wall energy of σDW 0.7 and 0.65 mJ.m-2 (for x= 0.7 and 1 Bi doping) can inferred using the Kaplan and Gerhing model27 (the fitting procedure is detailed in the Supplementary Note 2). Dynamical characterization and spin transparency . The most striking feature of these large PMA films is their extremely low magne tic losses that we characterize using Ferromagnetic Resonance (FMR) measurements. First of all, we quantify by in-plane FMR the anisotropy field HKU deduced from the effective magnetization ( Meff): HKU = M S – Meff (the procedure to derive Meff from in plan e FMR is presented in Supplementary Note 3 ). HKU values for BiYIG films with different doping levels grown on various substrates are summarized in Table 1. As expected from out-of-plane hysteresis curves , we observe different signs for HKU. For spontaneously out -of-plane magnetized samples , HKU is positive and large enough to fully compensate the demagnetizing field while it is negative for in -plane magnetized films. From these results , one can expect that fine tuning of the Bi content allow s fine tuning of the effective magnetization and consequently of the FMR resonan ce conditions. We measure magnetic losses on a 30nm thick Bi1Y2IG//sGGG film under tensile strain with PMA (Fig. 3 -(a)). We use the FMR absorption line shape by extracting the peak -to-peak linewidt h (𝛥𝐻pp) at different out-of-plane angle for a 30nm thick perpendicularly magnetized Bi 1Y2IG//sGGG film at 8 GHz (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 angle. We stress here that state of the art PLD grown YIG//GGG films exhibit similar values for 𝛥𝐻pp at such resonant conditions28. This angular dependence of 𝛥𝐻pp that shows pronounced variations at specific angle is characteristic of a two magnons scattering relaxation process with few inhomogenei ties29. The value of this angle is sample dependent as it is related to the distribution of the magnetic inhomogeneities . The dominance in our films of those two i ntrinsic relaxation processes (Gilbert damping and two magnons scattering) confirms the high films quality . We also derived the damping value of th is film (Fig. 3-(d)) by selecting the lowest linewidth (corresponding to a specific out of 5 plane angle) at each frequency, the spread of the out of plane angle is ±3.5 ° around 30.5 °. The obtained Gilbert damping value is α = 3.10-4 and the peak -to-peak extrinsic linew idth 𝜇0𝛥𝐻0 =0.23 mT a re comparable to the one obtained for the best PLD grown YIG//GGG nanometer thick films28 (α =2.10-4). For x= 0.7 Bi doping, the smallest observed FMR linewidth is 0.5 mT at 8 G Hz. The low magnetic losses of BiYIG films could open new perspectives for magnetization dynamics control using spin-orbit torques20,30,31. For such phenomenon interface transparency to spin curr ent is then the critical parameter which is defined using the effective spin -mixing conductance ( 𝐺↑↓). We use spin pumping experiments to estimate the increase of the Gilbert damping due to Pt deposition on Bi1Y2IG films. The spin mixing conductance can thereafter be calculated using 𝐺↑↓=4𝜋𝑀s𝑡film 𝑔eff𝜇B(𝛥𝛼) where 𝑀s and 𝑡film are the BiYIG magnetization saturation and thickness, 𝑔eff is the effective Landé factor ( 𝑔eff=2), 𝜇B is the Bohr magneton and 𝛥𝛼 is the increase in the Gilbert damping constant induced by the Pt top layer. We obtain 𝐺↑↓=3.9 1018m−2 which is comparable to what is obtained on PLD grown YIG//GGG systems 28,32,33. Consequently , the doping in Bi should not alter the spin orbit -torque efficiency and spin torque devices made out of BiYIG will be as energy efficient as their YIG counterpart . To further confirm that spin current cross es the Pt/BiYIG interface , we measure Inverse Spin Hall Effect (ISHE) in Pt for a Pt/ Bi1.5Y1.5IG(20nm)/ /sGGG in -plane magnetized film (to fulfill the ISHE geometry requirements the magnetization needs to be in -plane and perp endicular to the measured voltage ). We measure a characteristic voltage peak due to ISHE that reverses its sign when the static in-plane magnetic field is reversed (Fig. 4). We emphasize here that the amplitude of the s ignal is similar to that of Pt/ YIG//GGG in the same experimental conditions. Conclusion In summary, this new material platform will be highly beneficial for magnon -spintronics and related research fields like caloritronics. In many aspects , ultra -thin BiYIG films offer new leverages for fine tuning of the magnetic properties with no drawbacks compared to the reference materials of th ese fields: YIG. BiYIG with its higher Faraday rotation coefficient (almost two orders of magnitude more than that of YIG) will increase the sensitivity of light based detection technics that can be used (Brillouin light spectroscopy (BLS) or time resolved Kerr microscopy34). Innovative scheme s for on -chip magnon -light coupler could be now developed bridging the field of magnonics to th e one of photonics. From a practical point of view , the design of future active devices will be much more flexible as it is possible to easily engineer the spin waves dispersion relation through magnetic anis otropy tuning without the need of large bias magnetic fields. For instance, working in the forward volume waves configuration comes 6 now cost free, whereas in sta ndard in -plane magnetized media one has to overcome the demagnetizing field. As the development of PMA tunnel junctions was key in developing today scalable MRAM technology , likewise, we believe that P MA in nanometer -thick low loss insulator s paves the path to new approaches where the magnonic medium material could also be used to store information locally combining therefore the memory and computational functions, a most desirable feature for the brain - inspired neuromorphic paradigm . 7 Methods Pulsed Laser Deposition (PLD) growth The PLD growth of BiYIG films is realized using stoichiometric BiYIG target. The laser used is a frequency tripled Nd:YAG laser ( λ =355nm), of a 2.5Hz repetition rate and a fluency E varying from 0.95 to 1.43 J.cm-2 depending upon the Bi doping in the target. The distance between target and substrate is fixed at 44mm. Pri or to the deposition the substrate is annealed at 700°C under 0.4 mbar of O 2. For the growth, the pressure is set at 0.25 mbar O 2 pressure. The optimum growth temperature varies with the Bi content from 400 to 550°C. At the end of the growth, the sample is cooled down under 300 mbar of O 2. Structural characterization An Empyrean diffractometer with Kα 1 monochromator is used for measurement in Bragg -Brentano reflection mode to derive the (111) interatomic plan distance. Reciprocal Space Mapping is performed on the same diffractometer and we used the diffraction along the (642) plane direction which allow to gain information on the in-plane epitaxy relation along [20 -2] direc tion. Magnetic characterization A quantum design SQUID magnetometer was used to measure the films’ magnetic moment ( Ms) by performing hysteresis curves along the easy magnetic direction at room temperature. The linear contribution of the paramagnetic (sGG G or GGG) substrate is linearly subtracted. Kerr microscope (Evico Magnetics) is used in the polar mode to measure out-of-plane hysteresis curves at room temperature. The same microscope is also used to image the magnetic domains structure after a demagn etization procedure. The spatial resolution of the system is 300 nm. A broadband FMR setup with a motorized rotation stage was used. Frequencies from 1 to 20GHz have been explored. The FMR is measured as the derivative of microwave power absorption via a low frequency modulation of the DC magnetic field. Resonance spectra were recorded with the applied static magnetic field oriented in different geometries (in plane or tilted of an angle 𝜃 out of the strip line plane). For o ut of plane magnetized samples the Gilbert damping parameter has been obtained by studying the angular linewidth dependence. The procedure assumes that close to the minimum linewidth (Fig 3a) most of the linewidth angular dependence is dominated by the inhomogeneous broadening, thus opt imizing the angle for each frequency within few degrees allows to estimate better 8 the intrinsic contribution. To do so we varied the out of plane angle of the static field from 2 7° to 34 ° for each frequency and w e select the lowest value of 𝛥𝐻pp. For Inverse spin Hall effect measurements, the same FMR setup was used, however here the modulation is no longer applied to the magnetic field but to the RF power at a frequency of 5kHz. A Stanford Research SR860 lock -in was used a signal demodulator. Data availability : The data that support the findings of this study are available within the article or from the corresponding author upon reasonable request . Acknowledgements: We acknowledge J. Sampaio for preliminary Faraday rotation measurements and N. Rey ren and A. Barthélémy for fruitful discussions. This research was supported by the ANR Grant ISOLYIG (ref 15 -CE08 - 0030 -01). LS is partially supported by G.I.E III -V Lab. France. Author Contributions : LS performed the growth, all the measurements, the da ta analysis and wrote the manuscript with AA . NB and JBY conducted the quantitative Faraday Rotation measurements and participated in the FMR data analysis. LQ and fabricated the PLD targets . RL supervised the target fabrication and participated in the design of the study . EJ participated in the optimization of the film growth conditions. CC supervised the structural characterization experiments. AA conceived the study and w as in charge of overall direction . PB and VC contributed to the design and implem entation of the research . All authors discussed the results and commented on the manuscript. 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Electrical determination of spin mixing conductance at metal/ins ulator interface using inverse spin Hall effect. J. Appl. Phys. 111, 07C307 (2012). 33. Heinrich, B. et al. Spin Pumping at the Magnetic Insulator (YIG)/Normal Metal (Au) Interfaces. Phys. Rev. Lett. 107, 66604 (2011). 34. Stigloher, J. et al. Snell’s Law for Spin Waves. Phys. Rev. Lett. 117, 1–5 (2016). 11 Figures Captions : Figure 1 -Structural properties of ultra -thin BiYIG films . (a) and (b) : Evolution of the target cubic lattice parameter of BixY3-xIG, the dashed line represents the substrate (sGGG and GGG respectively) lattice parameter and allow s to infer the expected tensile or compressive strain arising for each substrate/target combination. (c) and ( d): 2𝜃−𝜔 X-Ray diffraction scan along the (111) out-of-plane direction for BixY3-xIG films gr own on sGGG (111) and GGG (111) respectively. From the film and substrate diffraction peak position , we can conclude about the nature of the strain. Compressive strain is observed for 1.5 doped films grown on sGGG substrate and for all films grown on GGG w hereas tensile strain occurs for films with x= 0.7 and x= 1 Bi content grown on sGGG. (e) and (f) : RSM along the evidence the (642) oblique plan showing pseudomorphic growth in films: both substrate and film the diffraction peak are aligned along the qx\\[20-2] direction. The relative position of the diffraction peak of the film (up or down) along qx is related to the out-of-plane misfit between the substrate and the film (tensile or compressive). Figure 2 -Static magnetic properties . (a) Out-of plane Kerr hysteresis loop performed in the polar mode for Bi0.7Y2.3IG films grown on the two substrates: GGG and sGGG (b) Same measurement for BixY3-xIG grown on sGGG with the three different Bi doping ( x= 0.7, 1 and 1.5) . Bi0.7Y2.3IG//GGG is in -plane magnetized whereas perpendicular magnetic anisotropy (PMA) occurs for x= 0.7 and x= 1 films grown on sGGG: square shaped loops with low saturation field ( µ0Hsat about 2.5 mT) are observed. Those two films are experiencing tensile strain . Whereas the inset shows that the Bi1.5Y1.5IG film saturates at a much higher field wi th a curve characteristic of in -plane easy magnetization direction. Note that for Bi1.5Y1.5IG//sGGG µ0Hsat ≈290mT >µ0Ms≈162mT which points toward a negative uniaxial anisotropy term ( µ0HKU) of 128mT which is coherent with the values obtained from in plane FMR measurement . (c) Magnetic domains structure imaged on Bi 1Y2IG//sGGG films of three different thicknesses at reman ant state after demagnetization . The scale bar, displayed in blue , equal s 20 µm . Periods of the magnetic domains structure ( Dwidth) are derived using 2D Fast Fourier Transform . We obtained Dwidth =3.1, 1.6 and 0.4 µm for tBi1Y2IG= 32, 47 and 52 nm respectively. We note a decrease of Dwidth with increasing tBi1Y2IG that is coherent with the Kaplan and Gehring model valide in the case Dwidth>>tBiYIG. 12 Figure 3-Dynamical properties of BiYIG films with PMA. (a) Sketch of the epitaxial configuration for Bi1Y2IG films , films are grown under tensi le strain giving rise to tetragonal distortion of the unit cell. (b) Out-of-plane angular depend ence of the peak -to-peak FMR linewidth ( 𝛥𝐻pp) at 8 GHz on a 30 nm thick Bi1Y2IG//sGGG with PMA (the continuous line is a guide for the eye) . The geometry of the measurement is shown in top right of the graph. The wide disparity of the value for the peak to peak linewidth 𝛥𝐻pp is attributed to the two magnons scattering process and inhomogeneties in the sample . (c) FMR absorption linewidth o f 0.3 mT for the same film at measured at 𝜃=27°. (d) Frequency dependence of the FMR linewidth . The calculated Gilbert damping parameter and the extrinsic linewidth are displayed on the graph . Figure 4- Inverse Spin Hall Effect of BiYIG films with in plane magnetic anisotropy. Inverse Spin Hall Effect (ISHE) voltage vs magnetic field measured on the Pt / Bi1.5Y1.5IG//sGGG sample in the FMR resonant condition at 6 GHz proving the interface transparency to spin current. The rf excitation field is about 10-3 mT which corresponds to a linear regime of excitation. Bi1.5Y1.5IG//sGGG present s an in- plane easy magnetization axis due to a growth under compressive strain. 13 Table 1 - Summary of the magnetic properties of BixY3-xIG films on GGG and sGGG substrates. The saturation magnetization is roughly unchanged. The effective magnetization Meff obtained through broad -Band FMR measurements allow to deduce the out -of-plane anisotropy fields HKU (HKU =Ms-Meff) confirming the dramatic changes of the out-of-plane magnetic anisotropy variations observed in the hysteresis curves . Bi doping Substrate µ0MS(mT) µ0Meff(mT) µ0HKU(mT) 0 GGG 157 200 -43 0.7 sGGG 180 -151 331 0.7 GGG 172 214 -42 1 sGGG 172 -29 201 1 GGG 160 189 -29 1.5 sGGG 162 278 -116 14 Figure 1 (f) 0.5 1.0 1.51.2351.2401.2451.2501.255 Cubic Lattice parameter in nmBi content sGGG 45 50 55103106109 Intensity (cps) 2 angle(°)0.711.5 0.711.5 1.70 1.724.254.304.35qz //[444] (rlu)*10 qx//[2-20] (rlu)*101E+01 1E+05 1.74 1.754.234.274.30qz //[444] (rlu)*10 qx//[2-20] (rlu)*101E+02 1E+05 1.73 1.764.214.254.29qz //[444] (rlu)*10 qx//[2-20] (rlu)*101E+02 1E+05 0.7 1 1.5 sGGGBi0.7Y2.3IG sGGGBi1Y2IG Bi1.5Y1.5IGsGGG 45 50 55102105108 Intensity (cps) 2 angle (°) 0.5 1.0 1.51.2351.2401.2451.2501.255 Cubic Lattice parameter in nmBi content GGG0.711.5 0.7511.5 1.75 1.774.204.304.40qz//[444] (rlu)*10 qx//[2-20] (rlu)*101E+01 1E+05 1.750 1.7754.284.324.35qz//[444] (rlu)*10 qx//[2-20] (rlu)*101E+02 1E+05 0.7 GGG Bi0.7Y2.3IGGGG Bi1Y2IG1sGGG GGG(a) (b)(c) (d)(e) (f)15 Figure 2 16 Figure 3 17 Figure 4 -130 -120 -110 110 120 130-4000400800V (nV) µ0H (mT)2 Supplementary Notes 1- Derivation of the magneto -elastic anisotropy The out -of-plan e anisotropy constant KU is ascribed to be a result of, at least , two contributions : a magneto -elastic anisotropy term induced by strain (𝐾MO) and a term that is due to preferential occupation of Bi atoms of non equivalent dodecahedral sites of the cubic u nit cell. This last term is known as the growth induced anisotropy term 𝐾GROWTH . From X -ray characterizations and f rom the known properties of the thick BiYIG LPE grown films, it is possible to calculate the expected values of 𝐾MO in each doping/substra te combination. We thereafter deduce 𝐾GROWTH from the relation 𝐾U=𝐾MO+ 𝐾GROWTH . KMO is directly proportional to the misfit between the film and the substrate: 𝐾MO=3 2∙𝐸 1−𝜇∙𝑎film −𝑎substrate 𝑎film∙𝜆111 Where E, µ and λ111 are respectively the Young modulus, the Poisson coefficient, and the magnetostrictive constant along the (111) direction. Those constant are well established for the bulk1: E= 2.055.1011 J.m-3, µ= 0.29 . The magnetostriction coefficient λ111 for the thin film case is slightly higher than that of the bulk and depends upon the Bi rate x: 𝜆111(𝑥)=−2.819 ∙10−6(1+0.75𝑥) 2. The two lattice par ameter entering in the equation: afilm and asubstrate correspond to the lattice parameter of the relaxed film structure and of the subs trate. Under an elastic deformation afilm can be derived with the Poisson coefficient: 𝑎film =𝑎substrate −[1−𝜇111 1+𝜇111]𝛥𝑎⊥ with ∆𝑎⊥=4√3𝑑444film−𝑎substrate All values for the different target/substrate combinations are displayed in the Table S-1. We note here that a negative (positive) misfit corresponding to a tensile (compressive) strain will favor an out-of-plane (in-plane ) magnetic anisotropy which is coherent with what is observed in our samples. To estimate the contribution to the magnetic energy of the magneto elastic anisotropy term we compare it to the demagnetizing field 𝜇0𝑀s that favors in-plane magnetic anisotropy in thin films. Interestingly the magneto elastic field ( 𝜇0𝐻MO) arising from 𝐾MO (𝜇0𝐻MO=2𝐾MO 𝑀s) never exceed 30% o f the demagnetizing fields and therefore cannot alon e be responsible of the observed PMA. Studies on µm-thick BiYIG films grown by LPE showed that PMA in BiYIG arises due to the growth induced anisotropy term 𝐾GROWTH , this term is positive 3 for the case of Bi substitution . We have inferred 𝜇0𝐻GROWTH values for all films using 𝐾U constants measured by FMR. The results are summarized in Supplementary Figure 1. One can clearly see that 𝐾GROWTH is strongly substrate dependent and therefore does not depend sol ely on the Bi content. We conclude that strain play s a role in Bi3+ ion ordering within the unit cell . 3 Supplementary Figure 1- Summary of the inferred values of the effective magnetic anisotropy out -of-plan fields. Horizontal dash lines represent the magnitude of the demagnetization field µ0Ms. When µ0HKU is larger than µ0Ms (dot line) films have a PMA, they are in -plane magnetized otherwise . 0200400µ0HKuvs Bi content HKu = Hstrain+ Hgrowth µ0Ms 1.50.7µ0HKu (mT) µ0 Hstrain µ0 Hgrowth 1 µ0Ms -1000100200 0.7 µ0HKu (mT) µ0Hstrain µ0Hgrowth 14 Supplementary Table 1 - Summary of the films’ calculated magneto -elastic anisotropy constant ( KMO) and the corresponding anisotropy field HMO Bi content substrate afilm(Å) Δ a┴/afilm KMO (J.m-3) µ0 HMO(mT) µ0 Hdemag (mT) 0.7 sGGG 12.45 0.6 5818 81 179 0.7 GGG 12.41 -0.4 -4 223 -61 172 1.0 sGGG 12.47 0.3 3 958 57 172 1.0 GGG 12.42 -0.6 -6 500 -102 160 1.5 sGGG 12.53 -0.6 -8 041 -124 157 5 Supplementary Notes 2- Derivation of the domain wall energy To derive the characteristic domain wall energy σDW for the maze shape like magnetic domains , we use the Kaplan et al. model4. This model applies in our case as the ratio of the film thickness ( tfilm) to the magnetic domain width ( Dwidth) is small (𝑡film 𝐷width~ 0.01). The domain wall width and the film thickness are then expected to be linked by: 𝐷width =𝑡filme−π1.33𝑒𝜋𝐷0 2𝑡film where 𝐷0=2𝜎w 𝜇0𝑀s2 is the dipolar length. Hence we expect a linear dependence of ln(𝐷width 𝑡film) vs 1 𝑡film : ln(𝐷width 𝑡film)=π𝐷0 2∙ 1 𝑡film+Cst. (1) The magnetic domain width of Bi xY3-xIG//sGGG ( x = 0.7 and 1) for several thicknesses are extracted from 2D Fourier Transform of the Kerr microscopy images at remanence. In Supplementary Fig ure 2, we plot ln(𝐷width 𝑡film) vs 1 𝑡film which follow s the expected linear dependence of Equation (1) . We infer from the zero intercept 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 and D0 x=1= 18.9µm. The corresponding domain wall energy are respectively 0.7 mJ .m-2 and 0.65 mJ .m-2. Even if the small difference in domain wall energy between the two Bi content may not be significant regarding the statistical fitting errors, it correlates to the decrease of the out-of-plane anisotropy ( KU) with increasing the Bi content. 6 Supplementary Figure 2- Evolution of the domain width vs film thickness ln(𝐷width 𝑡film) vs 1 𝑡film for Bi xY3-xIG//sGGG films doped at 0.7 (a) and 1 (b) in Bi. Dots correspond to the experimental values. The dashed line is the linear fit that allows to extract the D0 parameter. 25 50 752468 ln(Dwidth/tfilm) tfilm(µm-1)50 1002468 ln(Dwidth/tfilm) /tfilm(µm-1)Bi0.7Y2.3IG//sGGG D0=16.5µm Bi1Y2IG//sGGG D0=18.9µm (a) (b) 7 Supplementary Notes 3 - Damping and effective magnetic field derivation From In Plane frequency dependent of FMR we can derive the effective magnetization (𝑀eff) using the Kittel law: 𝑓res=𝜇0𝛾√𝐻res(𝐻res+𝑀-eff) Where γ is the gyromagnetic ratio of the BiYIG (assumed to be same as the one of the YIG) : γ=28 GHz.T-1. 𝐻res and 𝑓res are respectively the FMR resonant field and frequency . The uniaxial magnetic anisotropy can thereafter be derived using the saturation magnetizatio n form squid magnetometry using : Meff=Ms-HKU. The Gilbert damping ( α) and the inhomogeneous linewidth ( ΔH 0) which are the two paramaters defining the magnetic relaxation are obtained from the evolution of the peak to peak linewidth ( ΔHpp) vs the resonant frequency ( fres): 𝛥𝐻pp=𝛥𝐻0+2 √3𝛼𝑓res 𝜇0𝛾 (2) The first term is frequency independent and often attributed magnetic inhomogeneity’s (anisotropy, magnetization). 8 Supplementary Figure 3 - µ0ΔHpp vs fres on 18 nm thick Bi15.Y1.5IG//sGGG The l inewidth frequency dependence from 5 to 19 GHz for in plane magnetized Bi 1.5Y1.5IG//sGGG sample allow to ex tract the damping and the inhomogeneous linewidth parameter using the Equation (2) . 0 5 10 150.00.51.01.5 Hpp(mT) fres (GHz)H0=0.5 mT =1.9*10-39 Supplementary References: 1. Hansen, P., Klages, C. ‐P. & Witter, K. Magnetic and magneto ‐optic properties of praseodymium ‐ and bismuth ‐substituted yttrium iron garnet films. J. Appl. Phys. 60, 721–727 (1986). 2. Ben Youssef, J., , Legall, H. & . U. P. et M. C. Characterisation and physical study of bismuth substituted thin garnet films grown by liquid phase epitaxy (LPE). (1989). 3. Fratello, V. J., Slusky, S. E. G., Brandle, C. D. & Norelli, M. P. Growth -induced anisotropy in bismuth: Rare -earth iron garnets. J. Appl. Phys. 60, 2488 –2497 (1986). 4. Kaplan, B. & Gehring, G. A. The domain structure in ultrathin magnetic films. J. Magn. Magn. Mater. 128, 111–116 (1993).

2018-06-02

A magnetic material combining low losses and large Perpendicular Magnetic Anisotropy (PMA) is still a missing brick in the magnonic and spintronic fields. We report here on the growth of ultrathin Bismuth doped Y$_{3}$Fe$_{5}$O$_{12}$ (BiYIG) films on Gd$_{3}$Ga$_{5}$O$_{12}$ (GGG) and substituted GGG (sGGG) (111) oriented substrates. A fine tuning of the PMA is obtained using both epitaxial strain and growth induced anisotropies. Both spontaneously in-plane and out-of-plane magnetized thin films can be elaborated. Ferromagnetic Resonance (FMR) measurements demonstrate the high dynamic quality of these BiYIG ultrathin films, PMA films with Gilbert damping values as low as 3 10$^{-4}$ and FMR linewidth of 0.3 mT at 8 GHz are achieved even for films that do not exceed 30 nm in thickness. Moreover, we measure Inverse Spin Hall Effect (ISHE) on Pt/BiYIG stacks showing that the magnetic insulator$'$s surface is transparent to spin current making it appealing for spintronic applications.

Ultra-low damping insulating magnetic thin films get perpendicular

1806.00658v1

arXiv:1602.06673v3 [cond-mat.stat-mech] 1 Dec 2016Effects of Landau-Lifshitz-Gilbert damping on domain growt h Kazue Kudo Department of Computer Science, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan (Dated: May 25, 2021) Domain patterns are simulated by the Landau-Lifshitz-Gilb ert (LLG) equation with an easy-axis anisotropy. If the Gilbert damping is removed from the LLG eq uation, it merely describes the precession of magnetization with a ferromagnetic interact ion. However, even without the damping, domains that look similar to those of scalar fields are formed , and they grow with time. It is demon- strated that the damping has no significant effects on domain g rowth laws and large-scale domain structure. In contrast, small-scale domain structure is aff ected by the damping. The difference in small-scale structure arises from energy dissipation due t o the damping. PACS numbers: 89.75.Kd,89.75.Da,75.10.Hk I. INTRODUCTION Coarseningorphase-orderingdynamicsisobservedina widevarietyofsystems. Whenasystemisquenchedfrom a disordered phase to an ordered phase, many small do- mainsareformed, andtheygrowwithtime. Forexample, in the case of an Ising ferromagnet, up-spin and down- spin domains are formed, and the characteristic length scale increases with time. The Ising spins can be inter- preted as two different kinds of atoms in the case of a binary alloy. At the late stage of domain growth in these systems, characteristic length L(t) follows a power-law growth law, L(t)∼tn, (1) wherenis the growth exponent. The growth laws in scalarfieldshavebeenderivedbyseveralgroups: n= 1/2 fornon-conservedscalarfields, and n= 1/3forconserved scalar fields [1–8]. Similar coarsening dynamics and domain growth have been observed alsoin Bose-Einstein condensates (BECs). The characteristic length grows as L(t)∼t2/3in two- dimensional (2D) binary BECs and ferromagnetic BECs with an easy-axis anisotropy [9–11]. The same growth exponent n= 2/3 is found in classical binary fluids in the inertial hydrodynamic regime [1, 12]. It is remark- able that the same growth law is found in both quan- tum and classical systems. It should be also noted that domain formation and coarsening in BECs occur even without energy dissipation. The dynamics in a ferro- magnetic BEC can be described not only by the so- called Gross-Pitaevskii equation, which is a nonlinear Schr¨ odinger equation, but also approximately by a mod- ified Landau-Lifshitz equation in which the interaction between superfluid flow and local magnetization is incor- porated[13–15]. Ifenergydissipationexists, theequation changes to an extended Landau-Lifshitz-Gilbert (LLG) equation [9, 15, 16]. The normal LLG equation is usu- ally used to describe spin dynamics in a ferromagnet. The LLG equation includes a damping term which is called the Gilbert damping. When the system has an easy-axis anisotropy, the damping has the effect to directa spin to the easy-axis direction. The Gilbert damping in the LLG equation corresponds to energy dissipation in a BEC. In other words, domain formation without en- ergy dissipation in a BEC implies that domains can be formed without the damping in a ferromagnet. However, the LLG equation without the damping describes merely the precession of magnetization with a ferromagnetic in- teraction. Inthispaper, wefocusonwhateffectsthedampinghas ondomainformationanddomaingrowth. UsingtheLLG equation (without flow terms), we investigate the mag- netic domain growth in a 2D system with an easy-axis anisotropy. Since our system is simpler than a BEC, we can also give simpler discussions on what causes domain formation. When the easy axis is perpendicular to the x-yplane, the system is an Ising-like ferromagnetic film, and domains in which the zcomponent of each spin has almostthesamevalueareformed. Inordertoobservedo- main formation both in damping and no-damping cases, we limit the initial condition to almost uniform in-plane spins. Actually, without the damping, domain formation does not occur from an initial configuration of spins with totally random directions. Without the damping, the z component is conserved. The damping breaks the con- servation of the zcomponent as well as energy. Here, we should note that the growth laws for conserved and nonconserved scalar fields cannot simply be applied to the no-damping and damping cases, respectively, in our system. Although the zcomponent corresponds to the orderparameterofascalarfield, oursystemhastheother two components. It is uncertain whether the difference in the number of degrees of freedom can be neglected in domain formation. The restofthe paperis organizedas follows. In Sec. II, we describe the model and numerical procedures. Ener- gies and the characteristic length scale are also intro- duced in this section. Results of numerical simulations are shown in Sec. III. Domain patterns at different times and the time evolution of energies and the average do- main size are demonstrated. Scaling behavior is con- firmed in correlation functions and structure factors at late times. In Sec. IV, we discuss why domain formation2 can occur even in the no-damping case, focusing on an almost uniform initial condition. Finally, conclusions are given in Sec. V. II. MODEL AND METHOD The model we use in numerical simulations is the LLG equation, which is widely used to describe the spin dy- namics in ferromagnets. The dimensionless normalized form of the LLG equation is written as ∂m ∂t=−m×heff+αm×∂m ∂t, (2) wheremis the unit vector of spin, αis the dimensionless Gilbert damping parameter. We here consider the 2D systemlyinginthe x-yplane,andassumethatthesystem has a uniaxial anisotropy in the zdirection and that no long-range interaction exists. Then, the dimensionless effective field is given by heff=∇2m+Canimzˆz, (3) whereCaniis the anisotropy parameter, and ˆzis the unit vector in the zdirection. Equation (2) is mathematically equivalent to ∂m ∂t=−1 1+α2m×heff+α 1+α2m×(m×heff). (4) In numerical simulations, we use a Crank-Nicolson method to solve Eq. (4). The initial condition is given as spins that are aligned in the xdirection with a little ran- dom noises: mx≃1 andmy≃mz≃0. Simulations are performed in the 512 ×512 lattice with periodic bound- ary conditions. Averages are taken over 20 independent runs. The energy in this system is written as E=Eint+Eani =1 2/integraldisplay dr(∇m(r))2−1 2Cani/integraldisplay drmz(r)2,(5) which gives the effective field as heff=−δE/δm. The first and second terms are the interfacial and anisotropy energies, respectively. When Cani>0, thezcomponent becomes dominant since a large m2 zlowers the energy. We take Cani= 0.2 in the simulations. The damping parameter αexpresses the rate of energy dissipation. If α= 0, the spatial average of mzas well as the energy E is conserved. Considering mzas the order parameter of this system, we here define the characteristic length scale Lof a do- main pattern from the correlation function G(r) =1 A/integraldisplay d2x/angb∇acketleftmz(x+r)mz(x)/angb∇acket∇ight,(6) whereAis the area of the system and /angb∇acketleft···/angb∇acket∇ightdenotes an ensemble average. The average domain size Lis defined by the distance where G(r), i.e., the azimuth average of G(r), first drops to zero, and thus, G(L) = 0. FIG. 1. (Color online) Snapshots of z-component mzat time t= 102((a) and (b)), 103((c) and (d)), and 104((e) and (f)). Snapshots (g) and (h) are enlarged parts of (e) and (f), respectively. Profiles (i) and (j) of mzare taken along the bottom lines of snapshots (g) and (h), respectively. Left an d right columns are for the no-damping ( α= 0) and damping (α= 0.03) cases, respectively. III. SIMULATIONS Domain patterns appear, regardless of the damping parameter α. The snapshots of the no-damping ( α= 0) and damping ( α= 0.03) cases are demonstrated in the left and right columns of Fig. 1, respectively. Domain patterns at early times have no remarkable difference be- tweenthe twocases. Thecharacteristiclengthscalelooks almostthesamealsoatlatertimes. However,asshownin the enlarged snapshots at late times, difference appears especially around domain walls. Domain walls, where mz≃0, are smooth in the damping case. However, in the no-damping case, they look fuzzy. The difference ap- pears more clearly in profiles of mz(Figs. 1(i) and 1(j)). While the profile in the damping case is smooth, that in the no-damping case is not smooth. Such an uneven profile makes domain walls look fuzzy. The difference in domain structure is closely connected with energydissipation, which is shownin Fig. 2. The in- terfacial energy, which is the first term of Eq. (5), decays forα= 0.03 but increases for α= 0 in Fig. 2 (a). In con-3 0 2000 4000 6000 8000 10000t00.020.040.060.080.1Eint α = 0 α = 0.03(a) 0 2000 4000 6000 8000 10000t-0.1-0.08-0.06-0.04-0.020Eaniα = 0 α = 0.03(b) FIG. 2. (Color online) Time dependence of (a) the inter- facial energy Eintand (b) the anisotropy energy Eani. The interfacial energy increases with time in the no-damping ca se (α= 0) and decreases in the damping case ( α= 0.03). The anisotropy energy decreases with time in both cases. trast, the anisotropy energy, which comes from the total ofm2 z, decreases with time for both α= 0 and α= 0.03. In other words, the energy dissipation relating to the in- terfacial energy mainly causes the difference between the damping and no-damping cases. In the damping case, the interfacial energy decreases with time after a shot- time increase as domain-wall structure becomes smooth. However, in the no-damping case, the interfacial energy increases with time to conserve the total energy that is given by Eq. (5). This corresponds to the result that the domain structure does not become smooth in the no- damping case. Beforediscussinggrowthlaws, we shouldexaminescal- ing laws. Scaled correlation functions of mzat different times are shown in Fig. 3. The functions look pretty similar in both damping and no-damping cases, which reflects the fact that the characteristic length scales in both cases looks almost the same in snapshots. At late times, the correlation functions that are rescaled by the average domain size L(t) collapse to a single function. However, the scaled correlation functions at early times (t= 100 and 1000) do not agree with the scaling func- tion especially in the short range. The disagreement at early times is related with the unsaturation of mz. How mzsaturates is reflected in the time dependence of the0 0.5 1 1.5 2 r/L(t)-0.200.20.40.60.8G(r)t = 100 t = 1000 t = 6000 t = 8000 t = 10000(a) 0 0.5 1 1.5 2 r/L(t)-0.200.20.40.60.8G(r)t = 100 t = 1000 t = 6000 t = 8000 t = 10000(b) FIG.3. (Color online) Scaledcorrelation functions atdiffe rent times in (a) no-damping ( α= 0) and (b) damping ( α= 0.03) cases. The correlation functions at late times collapse to a single function, however, the ones at early times do not. anisotropy energy which is shown in Fig. 2(b). At early times (t/lessorsimilar1000),Eanidecays rapidly. This implies that mzis not saturated enough in this time regime. The de- creasein theanisotropyenergyslowsatlatetimes. In the late-time regime, mzis sufficiently saturated except for domain walls, and the decrease in the anisotropy energy is purely caused by domain growth. This corresponds to the scaling behavior at late times. In Fig. 4, the average domain size Lis plotted for the damping and no-damping cases. In both cases, the average domain size grows as L(t)∼t1/2at late times, although growth exponents at early times look liken= 1/3. Since scaling behavior is confirmed only at late times, the domain growth law is considered to be L(t)∼t1/2rather than t1/3in this system. In our pre- vious work, we saw domain growth as L(t)∼t1/3in a BEC without superfluid flow [9], which was essentially the same system as the present one. However, the time region shown in Ref. [9] corresponds to the early stage (t/lessorsimilar1830) in the present system. Although the growth exponent is supposed to be n= 1/3 for conserved scalar fields, the average domain size grows as L(t)∼t1/2, in our system, at late times even in the no-damping case. This implies that our system without damping cannot be categorized as a model of a4 100 1000 10000t10100 Lα = 0 α = 0.03 t1/2 t1/3 FIG. 4. (Color online) Time dependence of the average do- main size Lforα= 0 and 0 .03. In both damping and no- damping cases, domain size grows as L(t)∼t1/2at late times. Before the scaling regime, early-time behavior look s as ifL(t)∼t1/3. 1 10 100 kL(t)10-1010-910-810-710-610-510-410-3S(k)/L(t)2 t = 6000 t = 8000 t = 10000 1 10 100 kL(t)10-1010-910-810-710-610-510-410-3 t = 6000 t = 8000 t = 10000(a) (b) k-3k-3 FIG. 5. (Color online) Scaling plots of the structure factor scaled with L(t) at different times in (a) no-damping ( α= 0) and (b) damping ( α= 0.03) cases. In both cases, S(k)∼k−3 in the high- kregime. However, they gave different tails in the ultrahigh- kregime. conserved scalar field. Although we consider mzas the order parameter to define the characteristic length scale, the LLG equation is described in terms of a vector field m. Scaling behavior also appears in the structure factor S(k,t), which is given by the Fourier transformation of the correlation function G(r). According to the Porod law, the structure factor has a power-law tail, S(k,t)∼1 L(t)kd+1, (7) in the high- kregime [1]. Here, dis the dimension of the system. Since d= 2 in our system, Eq. (7) leads toS(k,t)/L(t)2∼[kL(t)]−3. In Fig. 5, S(k,t)/L(t)2is plotted as a function of kL(t). The data at different late times collapse to one curve, and they show S(k)∼k−3in the high- kregime (kL∼10) in both the damping and no-damping cases. In the ultrahigh- kregime (kL∼100), tails are different between the two cases, which reflects the difference in domain structure. Since domain walls are fuzzy in the no-damping case, S(k) remains finite. However, in the damping case, S(k) decays faster in the ultrahigh- kregime, which is related with smooth domain walls. IV. DISCUSSION We here have a naive question: Why does domain pattern formation occur even in the no-damping case? Whenα= 0, Eq. (2) is just the equation of the pre- cession of spin, and the energy Eas well as mzis con- served. We here discuss why similar domain patterns are formed from our initial condition in both damping and no-damping cases. Using the stereographic projection of the unit sphere of spin onto a complex plane [17], we rewrite Eq. (4) as ∂ω ∂t=−i+α 1+α2/bracketleftbigg ∇2ω−2ω∗(∇ω)2 1+ωω∗−Caniω(1−ωω∗) 1+ωω∗/bracketrightbigg , (8) whereωis a complex variable defined by ω=mx+imy 1+mz. (9) Equation (8) implies that the effect of the Gilbert damp- ing is just a rescaling of time by a complex constant [17]. The fixed points of Eq. (8) are |ω|2= 1 and ω= 0. Thelinearstabilityanalysisaboutthesefixedpointsgives some clues about domain formation. At the fixed point ω= 1,mx= 1 and my=mz= 0, which corresponds to the initial condition of the numer- ical simulation. Substituting ω= 1 +δωinto Eq. (8), we obtain linearized equations of δωandδω∗. Perform- ing Fourier expansions δω=/summationtext kδ˜ωkeik·randδω∗=/summationtext kδ˜ω∗ −keik·r, we have d dt/parenleftbiggδ˜ωk δ˜ω∗ −k/parenrightbigg =/parenleftbigg ˜α1(Cani−k2) ˜α1Cani ˜α2Cani˜α2(Cani−k2)/parenrightbigg/parenleftbiggδ˜ωk δ˜ω∗ −k/parenrightbigg , (10) where ˜α1=1 2(−i+α)/(1+α2), ˜α2=1 2(i+α)/(1+α2), k= (kx,ky), andk=|k|. The eigenvalues of the 2 ×2 matrix of Eq. (10) are λ(k) =α 2(1+α2)(Cani−2k2)±/radicalbig 4k2(Cani−k2)+α2C2 ani 2(1+α2). (11) Even when α= 0,λ(k) has a positive real part for k <√Cani. Thus, the uniform pattern with mx= 1 is unstable, and inhomogeneous patterns can appear. The positive real parts of Eq. (11) for α= 0 and α= 0.03 have close values, as shown in Fig. 6. This cor- responds to the result that domain formation in the early5 0 0.1 0.2 0.3 0.4 0.5 k00.020.040.060.080.1λ(k)α = 0 α = 0.03 FIG. 6. (Color online) Positive real parts of λ(k) that is given by Eq. (11), which has a positive real value for k <√Cani. The difference between α= 0 and α= 0.03 is small. stage has no remarkable difference between the damping (α= 0.03) and no-damping ( α= 0) cases (See Fig. 1). From the view point of energy, the anisotropy energy does not necessarily keep decaying when α= 0. For con- servation of energy, it should be also possible that both anisotropy and interfacial energies change only a little. Because of the instability of the initial state, mzgrows, and thus, the anisotropy energy decreases. The initial condition, which is given as spins aligned in onedirection with somenoisesin the x-yplane, is the key to observe domain pattern formation in the no-damping case. Actually, if spins have totally random directions, no large domains are formed in the no-damping case, although domains are formed in damping cases ( α >0) from such an initial state. Whenω= 0,mx=my= 0 and mz= 1, which is also one of the fixed points. Substituting ω= 0 +δωinto Eq. (8) and performing Fourier expansions, we have the linearized equation of δ˜ωk, d dtδ˜ωk=i−α 1+α2(k2+Cani)δ˜ωk. (12) This implies that the fixed point is stable for α >0 andneutrally stable for α= 0. Although mz=−1 corre- sponds to ω→ ∞, the same stability is expected for mz=−1 by symmetry. Sincetheinitialconditionisunstable, the z-component of spin grows. Moreover, linear instability is similar for α= 0 and α= 0.03. Since mz=±1 are not unstable, mzcan keep its value at around mz=±1. This is why similar domain patters are formed in both damping and no-damping cases. The main difference between the two cases is that mz=±1 are attracting for α >0 and neu- trally stable for α= 0. Since mz=±1 are stable and at- tractingin the dampingcase, homogeneousdomainswith mz=±1 are preferable, which leads to a smooth profile ofmzsuch as Fig. 1(j). In the damping case, mz=±1 are neutrally stable (not attracting) fixed points, which does not necessarily make domains smooth. V. CONCLUSIONS We have investigated the domain formation in 2D vec- tor fields with an easy-axis anisotropy, using the LLG equation. When the initial configuration is given as al- most uniform spins aligned in an in-plane direction, sim- ilar domain patterns appear in the damping ( α/negationslash= 0) and no-damping ( α= 0) cases. The average domain size grows as L(t)∼t1/2in late times which are in a scal- ing regime. The damping gives no remarkable effects on domain growth and large-scale properties of domain pattern. In contrast, small-scale structures are different between the two cases, which is shown quantitatively in the structure factor. This difference is induced by the re- duction of the interfacial energy due to the damping. It should be noted that the result and analysis especially in the no-damping case are valid for a limited initial condition. Although domains grow in a damping case even from spins with totally random directions, domain growth cannot occur from such a random configuration in the no-damping case. ACKNOWLEDGMENTS This work was supported by MEXT KAKENHI (No. 26103514, “Fluctuation & Structure”). [1] A. Bray, Adv. Phys. 43, 357 (1994) [2] I. M. Lifshitz and V. V. Slyozov, J. Phys. Chem. Solids 19, 35 (1961) [3] C. Wagner, Z. Elektrochem 65, 581 (1961) [4] T. Ohta, D. Jasnow, and K. Kawasaki, Phys. Rev. Lett. 49, 1223 (1982) [5] D. A. Huse, Phys. Rev. B 34, 7845 (1986) [6] A. J. Bray, Phys. Rev. Lett. 62, 2841 (1989) [7] A. J. Bray, Phys. Rev. B 41, 6724 (1990) [8] A. J. Bray and A. D. Rutenberg, Phys. Rev. E 49, R27 (1994)[9] K. Kudo and Y. Kawaguchi, Phys. Rev. A 88, 013630 (2013) [10] J. Hofmann, S. S. Natu, and S. Das Sarma, Phys. Rev. Lett.113, 095702 (2014) [11] L. A. Williamson and P. B. Blakie, Phys. Rev. Lett. 116, 025301 (2016) [12] H. Furukawa, Phys. Rev. A 31, 1103 (1985) [13] A. Lamacraft, Phys. Rev. A 77, 063622 (2008) [14] D. M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85, 1191 (2013) [15] Y. Kawaguchi and M. Ueda, Phys. Rep. 520, 253 (2012)6 [16] K. Kudo and Y. Kawaguchi, Phys. Rev. A 84, 043607 (2011) [17] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53, 2497 (1984)

2016-02-22

Domain patterns are simulated by the Landau-Lifshitz-Gilbert (LLG) equation with an easy-axis anisotropy. If the Gilbert damping is removed from the LLG equation, it merely describes the precession of magnetization with a ferromagnetic interaction. However, even without the damping, domains that look similar to those of scalar fields are formed, and they grow with time. It is demonstrated that the damping has no significant effects on domain growth laws and large-scale domain structure. In contrast, small-scale domain structure is affected by the damping. The difference in small-scale structure arises from energy dissipation due to the damping.

Effects of Landau-Lifshitz-Gilbert damping on domain growth

1602.06673v3

arXiv:1412.1988v1 [cond-mat.mtrl-sci] 5 Dec 2014Calculating linear response functions for finite temperatu res on the basis of the alloy analogy model H. Ebert, S. Mankovsky, K. Chadova, S. Polesya, J. Min´ ar, and D . K¨ odderitzsch Department Chemie/Phys. Chemie, Ludwig-Maximilians-Uni versit¨ at M¨ unchen, Butenandtstrasse 5-13, D-81377 M¨ unchen, Germany (Dated: 8th December 2014) A scheme is presented that is based on the alloy analogy model and allows to account for thermal lattice vibrations as well as spin fluctuations when calcula ting response quantities in solids. Various models to deal with spin fluctuations are discussed concerni ng their impact on the resulting tem- perature dependent magnetic moment, longitudinal conduct ivity and Gilbert damping parameter. It is demonstrated that using the Monte Carlo (MC) spin config uration as an input, the alloy ana- logy model is capable to reproduce results of MC simulations on the average magnetic moment within all spin fluctuation models under discussion. On the o ther hand, response quantities are much more sensitive to the spin fluctuation model. Separate c alculations accounting for either the thermal effect due to lattice vibrations or spin fluctuations show their comparable contributions to the electrical conductivity and Gilbert damping. Howeve r, comparison to results accounting for both thermal effects demonstrate violation of Matthiessen’ s rule, showing the non-additive effect of lattice vibrations and spin fluctuations. The results obtai ned for bcc Fe and fcc Ni are compared with theexperimental data, showing rather good agreement f or thetemperature dependentelectrical conductivity and Gilbert damping parameter. I. INTRODUCTION Finite temperature has often a very crucial influence on the response properties of a solid. A prominent ex- ample for this is the electrical resistivity of perfect non- magnetic metals and ordered compounds that only take a non-zero value with a characteristic temperature ( T) dependence due to thermal lattice vibrations. While the Holstein transport equation1,2provides a sound basis for corresponding calculations numerical work in this field has been done so far either on a model level or for sim- plified situations.3–6In practice often the Boltzmann- formalism is adopted using the constant relaxation time (τ) approximation. This is a very popular approach in particular when dealing with the Seebeck effect, as in this case τdrops out.7,8The constant relaxation time approximation has also been used extensively when deal- ing with the Gilbert damping parameter α.9–11Within the description of Kambersky10,12the conductivity- and resistivity-like intra- and inter-band contributions to α show a different dependency on τleading typically to a minimum for α(τ) or equivalently for α(T).10,11A scheme to deal with the temperature dependent resistiv- ity that is formally much more satisfying than the con- stant relaxation time approximation is achieved by com- bining the Boltzmann-formalism with a detailed calcula- tion of the phonon properties. As was shown by various authors,13–16this parameter-free approach leads for non- magneticmetalsingeneraltoaverygoodagreementwith experimental data. As an alternative to this approach, thermal lattice vibrations have also been accounted for within various studies by quasi-static lattice displacements leading to thermallyinducedstructuraldisorderinthesystem. This point of view provides the basis for the use of the al- loy analogy, i.e. for the use of techniques to deal withsubstitutional chemical disorder also when dealing with temperature dependent quasi-static random lattice dis- placements. An example for this are investigations on the temperature dependence of the resistivity and the Gilbert parameter αbased on the scattering matrix ap- proach applied to layered systems.17The necessary aver- ageovermanyconfigurationsoflatticedisplacementswas takenbymeansofthe supercelltechnique. Incontrastto thistheconfigurationalaveragewasdeterminedusingthe Coherent Potential Approximation (CPA) within invest- igations using a Kubo-Greenwood-like linear expression forα.18The same approach to deal with the lattice dis- placements was also used recently within calculations of angle-resolved photo emission spectra (ARPES) on the basis of the one-step model of photo emission.19 Another important contribution to the resistivity in the case of magnetically ordered solids are thermally in- duced spin fluctuations.20Again, the alloy analogy has been exploited extensively in the past when dealing with theimpactofspinfluctuationsonvariousresponsequant- ities. Representing a frozen spin configuration by means of super cell calculations has been applied for calcula- tions of the Gilbert parameter for α17as well as the resistivity or conductivity, respectively.17,21,22Also, the CPA has been used for calculations of α23as well as the resistivity.20,24A crucial point in this context is obvi- ously the modeling of the temperature dependent spin configurations. Concerning this, rather simple models have been used,23but also quite sophisticated schemes. Here one should mention the transfer of data from Monte Carlo simulations based on exchange parameters calcu- lated in an ab-initio way25as well as work based on the disordered local moment (DLM) method.24,26Although, the standard DLM does not account for transversal spin components it nevertheless allows to represent the para- magnetic regime with no net magnetization in a rigor-2 ous way.Also, for the magnetically ordered regime below the Curie-temperature it could be demonstrated that the uncompensated DLM (uDLM) leads for many situations still to goodagreementwith experimentaldata on the so- called spin disorder contribution to the resistivity.20,24 In the following we present technical details and exten- sionsofaschemethatwasalreadyused beforewhendeal- ing with the temperature dependence of response quant- ities on the basis of Kubo’s response formalism. Various applications will be presented for the conductivity and Gilbert damping parameter accounting simultaneously for various types of disorder. II. THEORETICAL FRAMEWORK A. Configurational average for linear response functions Many important quantities in spintronics can be formulated by making use of linear response formal- ism. Important examples for this are the electrical conductivity,27,28the spin conductivity29or the Gilbert damping parameter.18,30Restricting here for the sake of brevity to the symmetric part of the corresponding re- sponse tensor χµνthis can be expressed by a correlation function of the form: χµν∝Tr/angbracketleftbigˆAµℑG+ˆAνℑG+/angbracketrightbig c. (1) It should be stressed that this not a real restriction as the scheme described below has been used successfully when dealing with the impact of finite temperatures on the anomalous Hall conductivity of Ni.31In this case the more complex Kubo-Stˇ reda- or Kubo-Bastin formulation for the full response tensor has to be used.32 The vector operator ˆAµin Eq. (1) stands for example in case of the electrical conductivity σµνfor the cur- rent density operator ˆjµ28while in case of the Gilbert damping parameter αµνit stands for the torque oper- atorˆTµ.9,18Within the Kubo-Greenwood-like equation (1) the electronic structure of the investigated system is represented in terms of its retarded Green function G+(r,r′,E). Within multiple scattering theory or the KKR (Korringa-Kohn-Rostoker)formalism, G+(r,r′,E) can be written as:33–35 G+(r,r′,E) =/summationdisplay ΛΛ′Zm Λ(r,E)τmn ΛΛ′(E)Zn× Λ′(r′,E)(2) −δmn/summationdisplay ΛZn Λ(r,E)Jn× Λ′(r′,E)Θ(r′ n−rn) +Jn Λ(r,E)Zn× Λ′(r′,E)Θ(rn−r′ n). Herer,r′refer to points within atomic volumes around sitesRm,Rn, respectively, with Zn Λ(r,E) =ZΛ(rn,E) = ZΛ(r−Rn,E) being a function centered at site Rn. Ad- opting a fully relativistic formulation34,35for Eq. (2) one gets in a natural way access to all spin-orbit inducedproperties as for example the anomalous and spin Hall conductivity29,32,36or Gilbert damping parameter.18In this case, the functions Zn ΛandJn Λstand for the reg- ular and irregular, respectively, solutions to the single- site Dirac equation for site nwith the associated single- site scattering t-matrix tn ΛΛ′. The corresponding scat- tering path operator τnn′ ΛΛ′accounts for all scattering events connecting the sites nandn′. Using a suitable spinor representation for the basis functions the com- bined quantum number Λ = ( κ,µ) stands for the relativ- istic spin-orbit and magnetic quantum numbers κandµ, respectively.34,35,37 As was demonstrated by various authors27,28,38rep- resenting the electronic structure in terms of the Green function G+(r,r′,E) allows to account for chemical dis- order in a random alloy by making use of a suitable al- loy theory. In this case ∝an}bracketle{t...∝an}bracketri}htcstands for the configura- tional average for a substitutional alloy concerning the site occupation. Corresponding expressions for the con- ductivity tensor have been worked out by Velick´ y27and Butler28usingthe single-siteCoherentPotentialApprox- imation (CPA) that include in particular the so-called vertex corrections. The CPA can be used to deal with chemical but also with any other type of disorder. In fact, making use of the different time scales connected with the electronic propagation and spin fluctuations the alloy analogy is exploited when dealing with finite temperature magnet- ism on the basis of the disordered local moment (DLM) model.26,39Obviously, the same approach can be used when dealing with response tensors at finite temperat- ures. In connection with the conductivity this is often called adiabatic approximation.40Following this philo- sophy, the CPA has been used recently also when calcu- lating response tensors using Eq. ( 1) with disorder in the system caused by thermal lattice vibrations18,31as well as spin fluctuations.20,41 B. Treatment of thermal lattice displacement A way to account for the impact of the thermal dis- placement of atoms from their equilibrium positions, i.e. for thermal lattice vibrations, on the electronic struc- ture is to set up a representative displacement configura- tion for the atoms within an enlargedunit cell (super-cell technique). In this case one has to use either a very large super-cell or to take the average over a set of super-cells. Alternatively, one may make use of the alloy analogy for the averaging problem. This allows in particular to re- strict to the standard unit cell. Neglecting the correla- tion between the thermal displacements of neighboring atoms from their equilibrium positions the properties of the thermal averaged system can be deduced by making use of the single-site CPA. This basic idea is illustrated by Fig.1. To make use of this scheme a discrete set ofNvdisplacement vectors ∆ Rq v(T) with probability xq v (v= 1,..,Nv) is constructed for each basis atom qwithin3 Figure 1. Configurational averaging for thermal lattice dis - placements: the continuous distribution P(∆Rn(T)) for the atomic displacement vectors is replaced by a discrete set of vectors ∆ Rv(T) occurring with the probability xv. The con- figurational average for this discrete set of displacements is made using the CPA leading to a periodic effective medium. the standard unit cell that is conform with the local sym- metry and the temperature dependent root mean square displacement ( ∝an}bracketle{tu2∝an}bracketri}htT)1/2according to: 1 NvNv/summationdisplay v=1|∆Rq v(T)|2=∝an}bracketle{tu2 q∝an}bracketri}htT. (3) In the general case, the mean square displacement along the direction µ(µ=x,y,z) of the atom ican be either taken from experimental data or represented by the ex- pression based on the phonon calculations42 ∝an}bracketle{tu2 i,µ∝an}bracketri}htT=3/planckover2pi1 2Mi/integraldisplay∞ 0dωgi,µ(ω)1 ωcoth/planckover2pi1ω 2kBT,(4) whereh= 2π/planckover2pi1the Planck constant, kBthe Boltzmann constant, gi,µ(ω) is a partial phonon density of states.42 On the other hand, a rather good estimate for the root mean square displacement can be obtained using Debye’s theory. In this case, for systems with one atom per unit cell, Eq. ( 4) can be reduced to the expression: ∝an}bracketle{tu2∝an}bracketri}htT=1 43h2 π2MkBΘD/bracketleftbiggΦ(ΘD/T) ΘD/T+1 4/bracketrightbigg (5) with Φ(Θ D/T) the Debye function and Θ Dthe Debye temperature43. Ignoring the zero temperature term 1 /4 and assuming a frozen potential for the atoms, the situ- ationcanbe dealt with in full analogytothe treatmentof disorderedalloysonthebasisoftheCPA.Theprobability xvfor a specific displacement vmay normally be chosen as 1/Nv. The Debye temperature Θ Dused in Eq. ( 5) can be either taken fromexperimental data orcalculated rep- resenting it in terms of the elastic constants44. In general the latter approach should give more reliable results in the case of multicomponent systems. To simplify notation we restrict in the following to sys- tems with one atom per unit cell. The index qnumbering sites in the unit cell can therefore be dropped, while the indexnnumbers the lattice sites. Assuming a rigid displacement of the atomic potential in the spirit of the rigid muffin-tin approximation45,46 the correspondingsingle-site t-matrix tlocwith respect to the local frame of reference connected with the displaced atomic position is unchanged. With respect to the globalframe of reference connected with the equilibrium atomic positions Rn, however, the corresponding t-matrix tis given by the transformation: t=U(∆R)tlocU(∆R)−1. (6) The so-called U-transformation matrix U(s) is given in its non-relativistic form by:45,46 ULL′(s) = 4π/summationdisplay L′′il+l′′−l′CLL′L′′jl′′(|s|k)YL′′(ˆs).(7) HereL= (l,m) represents the non-relativistic angu- lar momentum quantum numbers, jl(x) is a spherical Besselfunction, YL(ˆr) a realsphericalharmonics, CLL′L′′ a corresponding Gaunt number and k=√ Eis the electronic wave vector. The relativistic version of the U-matrix is obtained by a standard Clebsch-Gordan transformation.37 The various displacement vectors ∆ Rv(T) can be used to determine the properties of a pseudo-component of a pseudo alloy. Each of the Nvpseudo-components with |∆Rv(T)|=∝an}bracketle{tu2∝an}bracketri}ht1/2 Tis characterized by a corresponding U-matrix Uvand t-matrix tv. As for a substitutional alloy the configurational average can be determined by solving the multi-component CPA equations within the global frame of reference: τnn CPA=Nv/summationdisplay v=1xvτnn v (8) τnn v=/bracketleftbig (tv)−1−(tCPA)−1+(τnn CPA)−1/bracketrightbig−1(9) τnn CPA=1 ΩBZ/integraldisplay ΩBZd3k/bracketleftbig (tCPA)−1−G(k,E)/bracketrightbig−1,(10) where the underline indicates matrices with respect to the combined index Λ. As it was pointed out in the pre- vious work41, the cutoff for the angular momentum ex- pansionin these calculations should be taken l≥lmax+1 with the lmaxvalue used in the calculations for the non- distorted lattice. The first of these CPA equations represents the re- quirement for the mean-field CPA medium that embed- ding of a component vshould lead in the average to no additional scattering. Eq. ( 9) gives the scattering path operator for the embedding of the component vinto the CPA medium while Eq. ( 10) gives the CPA scattering path operator in terms of a Brillouin zone integral with G(k,E) the so-called KKR structure constants. Having solved the CPA equations the linear response quantity of interest may be calculated using Eq. ( 1) as for an ordinary substitutional alloy.27,28This im- plies that one also have to deal with the so-called ver- tex corrections27,28that take into account that one has to deal with a configuration average of the type ∝an}bracketle{tˆAµℑG+ˆAνℑG+∝an}bracketri}htcthat in general will differ from the simpler product ∝an}bracketle{tˆAµℑG+∝an}bracketri}htc∝an}bracketle{tˆAνℑG+∝an}bracketri}htc.4 C. Treatment of thermal spin fluctuations As for the disorder connected with thermal displace- ments the impact of disorder due to thermal spin fluc- tuations may be accounted for by use of the super-cell technique. Alternatively one may again use the alloy analogy and determine the necessary configurational av- erage by means of the CPA as indicated in Fig. 2. As Figure 2. Configurational averaging for thermal spin fluc- tuations: the continuous distribution P(ˆen) for the orienta- tion of the magnetic moments is replaced by a discrete set of orientation vectors ˆ efoccurring with a probability xf. The configurational average for this discrete set of orientatio ns is made using the CPA leading to a periodic effective medium. for the thermal displacements in a first step a set of rep- resentative orientation vectors ˆ ef(withf= 1,...,Nf) for thelocalmagneticmomentisintroduced(seebelow). Us- ing the rigid spin approximation the spin-dependent part Bxcoftheexchange-correlationpotentialdoesnotchange for the local frame of reference fixed to the magnetic mo- ment when the moment is oriented along an orientation vector ˆef. This implies that the single-site t-matrix tloc f in the local frame is the same for all orientation vectors. With respect to the common global frame that is used to deal with the multiple scattering (see Eq. ( 10)) the t-matrix for a given orientation vector is determined by: t=R(ˆe)tlocR(ˆe)−1. (11) Here the transformation from the local to the global frame of reference is expressed by the rotation matrices R(ˆe) that are determined by the vectors ˆ eor correspond- ing Euler angles.37 Again the configurational average for the pseudo-alloy can be obtained by setting up and solvingCPAequations in analogy to Eqs. ( 8) to (10). D. Models of spin disorder The central problem with the scheme described above is obviously to construct a realistic and representative set of orientation vectors ˆ efand probabilities xffor each temperature T. A rather appealing approach is to cal- culate the exchange-coupling parameters Jijof a sys- tem in an ab-initio way25,47,48and to use them in sub- sequent Monte Carlo simulations. Fig. 3(top) shows results for the temperature dependent average reduced magnetic moment of corresponding simulations for bcc- Fe obtained for a periodic cell with 4096 atom sites. The0 0.2 0.4 0.6 0.8 1 1.2 T/TC00.20.40.60.81M(T)MC* KKR (MC*) 0 0.2 0.4 0.6 0.8 1 1.2 T/TC00.20.40.60.81M(T) MC MF-fit to MC (wMC(T)) MF-fit to MC (w=const) Expt MF-fit to Expt (wExpt(T)) 0 0.2 0.4 0.6 0.8 1 1.2 T/TC00.20.40.60.81M(T)MC KKR (MC) KKR (DLM) Figure 3. Averaged reduced magnetic moment M(T) = /angbracketleftmz/angbracketrightT/|/angbracketleftm/angbracketrightT=0|along the z-axis as a function of the tem- perature T. Top: results of Monte Carlo simulations using scheme MC* (full squares) compared with results of sub- sequent KKR-calculations (open squares). Middle: results of Monte Carlo simulations using scheme MC (full squares) compared with results using a mean-field fit with a constant Weiss field wMC(TC) (open diamonds) and a temperature de- pendent Weiss field wMC(T) (open squares). In addition ex- perimental data (full circles) together with a correspondi ng mean-field fit obtained for a temperature dependent Weiss fieldwexp(T). Bottom: results of Monte Carlo simulations using scheme MC (full squares) compared with results sub- sequent KKR-calculations using the MC (triangles up) and a corresponding DLM (triangle down) spin configuration, re- spectively. full line gives the value for the reduced magnetic mo- mentMMC∗(T) =∝an}bracketle{tmz∝an}bracketri}htT/m0projected on the z-axis for the lastMonteCarlostep (ˆ zis the orientationofthetotal moment, i.e. ∝an}bracketle{tm∝an}bracketri}htT∝bardblˆz; the saturated magnetic moment at T= 0 K is m0=|∝an}bracketle{tm∝an}bracketri}htT=0|). This scheme is called MC∗ in the following. In spite of the rather large number of sites (4096) the curve is rather noisy in particular when approaching the Curie temperature. Nevertheless, the5 spin configuration of the last MC step was used as an input for subsequent SPR-KKR-CPA calculations using theorientationvectors ˆ efwiththeprobability xf= 1/Nf withNf= 4096. As Fig. 3(top) shows, the temperature dependent reducedmagnetic moment MKKR(MC∗)(T) de- duced from the electronic structure calculations follows one-to-one the Monte Carlo data MMC∗(T). This is a very encouraging result for further applications (see be- low) as it demonstrates that the CPA although being a mean-field method and used here in its single-site formu- lation is nevertheless capable to reproduce results of MC simulations that go well beyond the mean-field level. However, using the set of vectors ˆ efof scheme MC* also for calculations of the Gilbert damping parameters αas a function of temperature led to extremely noisy and unreliable curves for α(T). For that reason an av- erage has been taken over many MC steps (scheme MC) leading to a much smoother curve for MMC(T) as can be seen from Fig. 3(middle) with a Curie temperature TMC C= 1082 K. As this enlarged set of vectors ˆ efgot too large to be used directly in subsequent SPR-KKR- CPA calculations, a scheme was worked out to get a set of vectors ˆ efand probabilities xfthat is not too large but nevertheless leads to smooth curves for M(T). The first attempt was to use the Curie temperature TMC Ctodeduceacorrespondingtemperatureindependent Weiss-field w(TC) on the basis of the standard mean-field relation: w(TC) =3kBTC m2 0. (12) This leads to a reduced magnetic moment curve MMF(T) that shows by construction the same Curie temperature as the MC simulations. For temperatures between T= 0 K and TC, however, the mean-field reduced magnetic moment MMF(T) is well below the MC curve (see Fig. 3 (middle) ). As an alternative to this simple approach we intro- duced a temperature dependent Weiss field w(T). This allows to describe the temperature dependent magnetic properties using the results obtained beyond the mean- field approximation. At the same time the calculation of the statistical average can be performed treating the model Hamiltonian in termsofthe mean field theory. For this reason the reduced magnetic moment M(T), being a solution of equation (see e.g.49) M(T) =L/parenleftbiggwm2 0M(T) kBT/parenrightbigg , (13) was fitted to that obtained from MC simulations MMC(T)withtheWeissfield w(T)asafittingparameter, such that lim w→w(T)M(T) =MMC(T), (14) withL(x) the Langevin function. The corresponding temperature dependent probability x(ˆe) for an atomic magnetic moment to be oriented alongˆeis proportionalto exp( −w(T)ˆz·ˆe/kBT) (see, e.g.49). To calculate this value we used NθandNφpoints for a reg- ular grid for the spherical angles θandφcorresponding to the vector ˆ ef: xf=exp(−w(T)ˆz·ˆef/kBT)/summationtext f′exp(−w(T)ˆz·ˆef′/kBT).(15) Fig.4shows for three different temperatures the θ- dependent behavior of x(ˆe). As one notes, the MF-fit 0 30 60 90 120 150 180 θ00.050.10.150.20.250.3P(θ)MC MF-fit to MC (wMC(T))T = 200 K 0 30 60 90 120 150 180 θ00.050.10.150.2P(θ)MC MF-fit to MC (wMC(T))T = 400 K 0 30 60 90 120 150 180 θ00.050.1P(θ)MC MF-fit to MC (wMC(T))T = 800 K Figure 4. Angular distribution P(θ) of the atomic magnetic moment mobtained from Monte Carlo simulations (MC) for the temperature T= 200, 400, and 800 K compared with field mean-field (MF) data, xf, (full line) obtained by fitting using a temperature dependent Weiss field w(T) (Eq.13). to the MC-results perfectly reproduces these data for all temperatures. This applies of course not only for the angular resolved distribution of the magnetic moments shown in Fig. 4but also for the average reduced mag- netic moment recalculated using Eq.( 13), shown in Fig. 3. Obviously, the MF-curve MMF(MC)(T) obtained using the temperature dependent Weiss field parameter w(T) perfectly reproduces the original MMC(T) curve. The great advantage of this fitting procedure is that it al- lows to replace the MC data set with a large number6 NMC fof orientation vectors ˆ ef(pointing in principle into any direction) with equal probability xf= 1/NMC fby a much smaller data set with Nf=NθNφwithxfgiven by Eq. (15). Accordingly, the reduced data set can straight for- wardly be used for subsequent electronic structure cal- culations. Fig. 3(bottom) shows that the calcu- lated temperature dependent reduced magnetic moment MKKR−MF(MC)(T) agrees perfectly with the reduced magnetic moment MMC(T) given by the underlying MC simulations. The DLM method has the appealing feature that it combines ab-initio calculations and thermodynamics in a coherent way. Using a non-relativistic formulation, it was shown that the corresponding averaging over all ori- entations of the individual atomic reduced magnetic mo- ments can be mapped onto a binary pseudo-alloy with one pseudo-component having up- and downward orient- ation of the spin moment with concentrations x↑and x↓, respectively.24,50For a fully relativistic formulation, with spin-orbitcoupling included, this simplificationcan- not be justified anymore and a proper average has to be taken over all orientations.51As we do not perform DLM calculationsbut use hereonly the DLM picture to repres- ent MC data, this complication is ignored in the follow- ing. Having the set of orientation vectors ˆ efdetermined by MC simulations the corresponding concentrations x↑ andx↓can straight forwardly be fixed for each temper- ature by the requirement: 1 NfNf/summationdisplay f=1ˆef=x↑ˆz+x↓(−ˆz), (16) withx↑+x↓= 1. Using this simple scheme electronic structure calculations have been performed for a binary alloy having collinear magnetization. The resulting re- duced magnetic moment MKKR−DLM(MC) (T) is shown in Fig.3(bottom). As one notes, again the original MC results are perfectly reproduced. This implies that when calculating the projected reduced magnetic moment Mz that is determined by the averaged Green function ∝an}bracketle{tG∝an}bracketri}ht the transversal magnetization has hardly any impact. Fig.3(middle) gives also experimental data for theM(T).52While the experimental Curie-temperature Texp C= 1044 K52is rather well reproduced by the MC simulations TMC C= 1082 K one notes that the MC-curve MMC(T) is well below the experimental curve. In partic- ular,MMC(T) drops too fast with increasing Tin the low temperature regime and does not show the T3/2- behavior. The reason for this is that the MC simulations do not properly account for the low-energy long-ranged spinwaveexcitationsresponsibleforthelow-temperature magnetization variation. Performing ab-initio calcula- tions for the spin wave energies and using these data for the calculation of M(T) much better agreement with ex- periment can indeed be obtained in the low-temperature regime than with MC simulations.53 As the fitting scheme sketched above needs only thetemperature reduced magnetic moment M(T) as input it can be applied not only to MC data but also to ex- perimental data. Fig. 3shows that the mean field fit MMF(exp)(T) again perfectly fits the experimental re- duced magnetic moment curve Mexp(T). Based on this good agreement this corresponding data set {ˆef,xf}has also been used for the calculation of responsetensors (see below). An additional much simpler scheme to simulate the experimental Mexp(T) curve is to assume the individual atomic moments to be distributed on a cone, i.e. with Nθ= 1 and Nφ>>1.23In this case the opening angle θ(T) of the cone is chosen such to reproduce M(T). In contrasttothestandardDLMpicture,thissimplescheme allows already to account for transversal components of the magnetization. Corresponding results for response tensor calculations will be shown below. Finally, it should be stressed here that the various spin configuration models discussed above assume a rigid spin moment, i.e. its magnitude does not change with temper- ature nor with orientation. In contrast to this Ruban et al.54usealongitudinalspinfluctuation Hamiltonianwith the corresponding parameters derived from ab-initio cal- culations. As a consequence, subsequent Monte Carlo simulations based on this Hamiltonian account in par- ticular for longitudinal fluctuations of the spin moments. A similar approach has been used by Drchal et al.55,56 leading to good agreement with the results of Ruban et al. However, the scheme used in these calculations does not supply in a straightforward manner the necessary input for temperature dependent transport calculations. This is different from the work of Staunton et al.57who performed self-consistent relativistic DLM calculations without the restriction to a collinear spin configuration. This approach in particular accounts in a self-consistent way for longitudinal spin fluctuations. E. Combined chemical and thermally induced disorder The various types of disorder discussed above may be combined with each other as well as with chemical i.e. substitution disorder. In the most general case a pseudo- component ( vft) is characterized by its chemical atomic typet, the spin fluctuation fand lattice displacement v. Using the rigid muffin-tin and rigid spin approxim- ations, the single-site t-matrix tloc tin the local frame is independent from the orientation vector ˆ efand displace- ment vector ∆ Rv, and coincides with ttfor the atomic typet. With respect to the common global frame one has accordingly the t-matrix: tvft=U(∆Rv)R(ˆef)ttR(ˆef)−1U(∆Rv)−1.(17) With this the corresponding CPA equations are identical to Eqs. ( 8) to (10) with the index vreplaced by the combined index ( vft). The corresponding pseudo- concentration xvftcombines the concentration xtof the7 atomic type twith the probability for the orientation vector ˆefand displacement vector ∆ Rv. III. COMPUTATIONAL DETAILS The electronic structure of the investigated ferro- magnets bcc-Fe and fcc-Ni, has been calculated self- consistently using the spin-polarized relativistic KKR (SPR-KKR) band structure method.58,59For the ex- changecorrelationpotential the parametrizationas given by Vosko et al.60has been used. The angular-momentum cutoff of lmax= 3 was used in the KKR multiple scatter- ing expansion. The lattice parameters have been set to the experimental values. In a second step the exchange-coupling parameters Jijhave been calculated using the so-called Lichten- stein formula.25Although the SCF-calculations have been done on a fully-relativistic level the anisotropy of the exchange coupling due to the spin-orbit coupling has been neglected here. Also, the small influence of the magneto-crystallineanisotropyfor the subsequent Monte Carlo (MC) simulations has been ignored, i.e. these have been based on a classical Heisenberg Hamiltonian. The MC simulations were done in a standard way using the Metropolis algorithm and periodic boundary conditions. The theoretical Curie temperature TMC Chas been de- duced from the maximum of the magnetic susceptibility. The temperature dependent spin configuration ob- tained during a MC simulation has been used to con- struct a set of orientations ˆ efand probabilities xfac- cording to the schemes MC* and MC described in sec- tionIIDto be used within subsequent SPR-KKR-CPA calculations (see above). For the corresponding calcu- lation of the reduced magnetic moment the potential obtained from the SCF-calculation for the perfect fer- romagnetic state ( T= 0K) has been used. The calcu- lation for the electrical conductivity as well as the Gil- bertdampingparameterhasbeenperformedasdescribed elsewhere.41,61 IV. RESULTS AND DISCUSSION A. Temperature dependent conductivity Eq. (1) has been used together with the various schemes described above to calculate the temperature dependent longitudinal resistivity ρ(T) of the pure fer- romagnets Fe, Co and Ni. In this case obviously disorder due to thermal displacements of the atoms as well as spin fluctuations contribute to the resistivity. To give an impression on the impact of the thermal displacementsaloneFig. 5givesthe temperaturedepend- ent resistivity ρ(T) of pure Cu (Θ Debye= 315 K) that is found in very good agreement with corresponding ex- perimental data.62This implies that the alloy analogy model that ignores any inelastic scattering events should0 100 200 300 400 500 Temperature (K)01234ρxx (10-6Ω⋅cm)Expt Theory - alloy analogy Theory - LOVA Cu Figure 5. Temperature dependent longitudinal resistivity of fcc-Cuρ(T) obtained by accounted for thermal vibrations as described in section IIBcompared with corresponding ex- perimental data.62In addition results are shown based on the LOVA (lowest order variational approximation) to the Boltzmann formalism.14 in general lead to rather reliable results for the resistivity induced by thermal displacements. Accordingly, com- parison with experiment should allow for magnetically ordered systems to find out the most appropriate model for spin fluctuations. Fig.6(top) shows theoretical results for ρ(T) of bcc- Fe due to thermal displacements ρv(T), spin fluctuations described by the scheme MC ρMC(T) as well as the com- bination of the two influences ( ρv,MC(T)). First of all one notes that ρv(T) is not influenced within the adop- tedmodelbytheCurietemperature TCbutisdetermined only by the Debye temperature. ρMC(T), on the other hand, reaches saturation for TCas the spin disorder does not increase anymore with increasing temperature in the paramagnetic regime. Fig. 6also shows that ρv(T) and ρMC(T)arecomparableforlowtemperaturesbut ρMC(T) exceedsρv(T) more and more for higher temperatures. Most interestingly, however, the resistivity for the com- bined influence of thermal displacements and spin fluctu- ationsρv,MC(T) does not coincide with the sum of ρv(T) andρMC(T) but exceeds the sum for low temperatures and lies below the sum when approaching TC. Fig.6(bottom) shows the results of three differ- ent calculations including the effect of spin fluctuations as a function of the temperature. The curve ρMC(T) is identical with that given in Fig. 6(top) based on Monte Carlo simulations. The curves ρDLM(MC) (T) and ρcone(MC)(T) are based on a DLM- and cone-like repres- entation of the MC-results, respectively. For all three cases results are given including as well as ignoring the vertex corrections. As one notes the vertex corrections play a negligible role for all three spin disorder models. This is fully in line with the experience for the longitud- inal resistivity of disordered transition metal alloys: as long as the the states at the Fermi level have domin- antly d-character the vertex corrections can be neglected in general. On the other hand, if the sp-character dom-8 0 0.2 0.4 0.6 0.8 1 1.2 T/TC020406080100120ρxx (10-6Ω⋅cm)vib fluct (MC) vib + fluct (MC) 0 0.2 0.4 0.6 0.8 1 1.2 T/TC020406080100120ρxx (10-6Ω⋅cm)MC (VC) MC (NVC) DLM (VC) DLM (NVC) cone (VC) cone (NVC) Figure 6. Temperature dependent longitudinal resistivity of bcc-Feρ(T) obtained by accounted for thermal vibrations and spin fluctuations as described in section IIB. Top: ac- counting for vibrations (vib, diamonds), spin fluctuations us- ing scheme MC (fluct, squares) and both (vib+fluct, circles). Bottom: accounting for spin fluctuations ˆ ef= ˆe(θf,φf) us- ing the schemes: MC (squares) with 0 ≤θf≤π;0≤φf≤ 2π, DLM(MC) (triangles up) with θf1= 0,θf2=π, and cone(MC) (triangles down) θf=/angbracketleftθf/angbracketrightT;0≤φf≤2π. The full and open symbols represent the results obtained with th e vertex corrections included (VC) and excluded (NV), respec t- ively. inates inclusion of vertex corrections may alter the result in the order of 10 %.63,64 Comparing the DLM-result ρDLM(MC) (T) with ρMC(T) one notes in contrast to the results for M(T) shown above (see Fig. 3(bottom)) quite an appreciable deviation. This implies that the restricted collinear representation of the spin configuration implied by the DLM-model introduces errors for the configurational average that seem in general to be unacceptable, For the Curie temperature and beyond in the paramagnetic regimeρDLM(MC) (T) andρMC(T) coincide, as it was shown formally before.20 Comparing finally ρcone(MC)(T) based on the conical representationofthe MCspin configurationwith ρMC(T) one notes that also this simplification leads to quite strong deviations from the more reliable result. Never- theless, one notes that ρDLM(MC) (T) agrees with ρMC(T) for the Curie temperature and also accounts to some ex- tent for the impact of the transversal components of themagnetization. The theoretical results for bcc-Fe (Θ Debye= 420 K) based on the combined inclusion of the effects of thermal displacementsandspinfluctuationsusingtheMCscheme (ρv,MC(T)) are compared in Fig. 7(top) with experi- mental data ( ρexp(T)). For the Curie temperature ob- 0 0.2 0.4 0.6 0.8 1 1.2 1.4 T/TC020406080100120ρxx (10-6Ω⋅cm)Expt: J. Bass and K.H. Fischer vib + fluct (MC) vib + fluct (exp) 0 0.2 0.4 0.60.8 1 1.2 1.4 1.61.8 T/TC01020304050ρxx (10-6Ω⋅cm)Expt.: C.Y. Ho et al. (1983) vib vib (PM) fluct vib + fluct Figure 7. Top: Temperature dependent longitudinal res- istivity of bcc-Fe ρ(T) obtained by accounted for thermal vibrations and spin fluctuations using the scheme MC (vib+fluct(MC), squares) and a mean-field fit to the experi- mental temperature magnetic moment Mexp(vib+fluct(exp), diamonds) compared with experimental data (circles).62Bot- tom: corresponding results for fcc-Ni. In addition results are shown accounting for thermal displacements (vib) only for the ferromagnetic (FM) as well paramagnetic (PM) regime. Experimental data have been taken from Ref. 65. viously a very good agreement with experiment is found while for lower temperatures ρv,MC(T) exceeds ρexp(T). This behavior correlates well with that of the temperat- ure dependent reduced magnetic moment M(T) shown in Fig.3(middle). The too rapid decrease of MMC(T) compared with experiment implies an essentially overes- timated spin disorder at any temperature leading in turn to a too large resistivity ρv,MC(T). On the other hand, using the temperature dependence of the experimental reducedmagneticmoment Mexp(T)tosetup thetemper- ature dependent spin configuration as described above a very satisfying agreement is found with the experimental resistivity data ρexp(T). Note also that above TCthe calculated resistivity riches the saturation in contrast to the experimental data where the continuing increase of9 ρexp(T) can be attributed to the longitudinal spin fluctu- ations leading to a temperature dependent distribution of local magnetic moments on Fe atoms.54However, this contribution was not taken into account because of re- striction in present calculations using fixed value for the local reduced magnetic moments. Fig.7(bottom) shows corresponding results for the temperature dependent resistivity of fcc-Ni (Θ Debye= 375 K). For the ferromagnetic (FM) regime that the theoretical results are comparable in magnitude when only thermal displacements ( ρv(T)) or spin fluctuations (ρMF(T)) are accounted for. In the later case the mean fieldw(T) has been fitted to the experimental M(T)- curve. Taking both into account leads to a resistivity (ρv,MF(T)) that are well above the sum of the individual termsρv(T) andρMF(T). Comparing ρv,MF(T) with ex- perimentaldata ρexp(T)ourfindingshowsthatthetheor- etical results overshoots the experimental one the closer one comes to the critical temperature. This is a clear indication that the assumption of a rigid spin moment is quite questionable as the resulting contribution to the resistivity due to spin fluctuations as much too small. In fact the simulations of Ruban et al.54on the basis of a longitudinal spin fluctuation Hamiltonian led on the case of fcc-Ni to a strong diminishing of the averagelocal magnetic moment when the critical temperature is ap- proachedfrom below (about 20% comparedto T= 0K). For bcc-Fe, the change is much smaller (about 3 %) justi- fying on the case the assumption of a rigid spin moment. Taking the extreme point of view that the spin moment vanishescompletely abovethe criticaltemperature orthe paramagnetic (PM) regime only thermal displacements have to be considered as a source for a finite resistivity. Corresponding results are shown in Fig. 7(bottom) to- gether with corresponding experimental data. The very good agreementbetween both obviouslysuggeststhat re- maining spin fluctuations above the critical temperature are of minor importance for the resistivity of fcc-Ni. B. Temperature dependent Gilbert damping parameter Fig.8shows results for Gilbert damping parameter α of bcc-Fe obtained using different models for the spin fluctuations. All curves show the typical conductivity- like behaviorfor low temperatures and the resistivity-like behavior at high temperatures reflecting the change from dominating intra- to inter-band transitions.66The curve denoted expt isbasedon aspin configurationtoted tothe experimental Mexpt(T) data. Using the conical model to fitMexpt(T) as basis for the calculation of α(T) leads obviously to a rather good agreement with αM(expt)(T). Having instead a DLM-like representation of Mexpt(T), on the other hand, transverse spin components are sup- pressed and noteworthy deviations from αM(expt)(T) are found for the low temperature regime. Nevertheless, the deviations are less pronounced than in the case of the0 200 400 600 800 Temperature (K)02468α × 103fluct (MC) fluct (Expt) fluct (DLM) fluct (cone) Figure 8. Temperature dependent Gilbert damping α(T) for bcc-Fe, obtainedbyaccountedfor thermal vibrations andsp in fluctuations accounting for spin fluctuations using scheme MC (squares), DLM(MC) (triangles up), cone(MC) (triangles down) and a MF fit to the experimental temperature reduced magnetic moment (circles). longitudinal resistivity (see Fig. 6(bottom)), where cor- responding results are shown based on MMC(T) as a ref- erence. Obviously, the damping parameter αseems to be less sensitive to the specific spin fluctuation model used than the resistivity. Finally, using the spin con- figuration deduced from Monte Carlo simulations, i.e. based on MMC(T) quite strong deviations for the result- ingαM(MC)(T) fromαM(expt)(T) are found. As for the resistivity (see Fig. 6(bottom)) this seems to reflect the too fast drop of the reduced magnetic moment MMC(T) with temperature in the low temperature regime com- pared with temperature (see Fig. 3). As found before18 accountingonly for thermal vibrations α(T) (Fig.6(bot- tom)) is found comparableto the casewhen only thermal span fluctuations are allowed. Combing both thermal ef- fects does not lead to a curve that is just the sum of the twoα(T) curves. As found for the conductivity (Fig. 6 (top)) obviously the two thermal effects are not simply additive. As Fig. 9(top) shows, the resulting damping parameter α(T) for bcc-Fe that accounts for thermal vi- brationsaswellasspinfluctuationsisfoundinreasonable good agreement with experimental data.18 Fig.9shows also corresponding results for the Gilbert dampingoffcc-Niasafunctionoftemperature. Account- ing only for thermal spin fluctuations on the basis of the experimental M(T)-curveleadsinthis casetocompletely unrealistic results while accounting only for thermal dis- placements leads to results already in rather good agree- ment with experiment. Taking finally both sources of disorder into account again no simple additive behavior is found but the results are nearly unchanged compared to those based on the thermal displacements alone. This implies that results for the Gilbert damping parameter of fcc-Ni hardly depend on the specific spin configura- tion model used but are much more governed by thermal displacements.10 0 200 400 600 800 Temperature (K)0246810α × 103vib vib + fluct (Expt) Expt 1 Expt 2 0 100 200 300 400 500 Temperature (K)00.050.10.150.2αvib fluct (Expt) vib + fluct (Expt) Expt Figure 9. Top: Temperature dependent Gilbert damping α(T) for bcc-Fe, obtained byaccounted for thermal vibrations and spin fluctuations accounting for lattice vibrations onl y (circles) and lattice vibrations and spin fluctuations base d on mean-field fit to the experimental temperature reduced mag- netic moment Mexpt(diamonds) compared with experimental data (dashed and full lines).67,68Bottom: corresponding res- ults for fcc-Ni. Experimental data have been taken from Ref. 67. V. SUMMARY Various schemes based on the alloy analogy that al- low to include thermal effects when calculating responseproperties relevant in spintronics have been presented and discussed. Technical details of an implementation within the framework of the spin-polarized relativistic KKR-CPA band structure method have been outlined that allow to deal with thermal vibrations as well as spin fluctuations. Various models to represent spin fluctu- ations have been compared with each other concerning corresponding results for the temperature dependence of the reduced magnetic moment M(T) as well as re- sponse quantities. It was found that response quantities are much more sensitive to the spin fluctuation model as the reduced magnetic moment M(T). Furthermore, it was found that the influence of thermal vibrations and spin fluctuations is not additive when calculating elec- trical conductivity or the Gilbert damping parameter α. Using experimental data for the reduced magnetic mo- mentM(T) to set up realistic temperature dependent spin configurations satisfying agreement for the electrical conductivity as well as the Gilbert damping parameter could be obtained for elemental ferromagnets bcc-Fe and fcc-Ni. ACKNOWLEDGMENTS This work was supported financially by the Deutsche Forschungsgemeinschaft (DFG) within the projects EB154/20-1, EB154/21-1 and EB154/23-1 as well as the priority program SPP 1538 (Spin Caloric Transport) and the SFB 689 (Spinph¨ anomene in reduzierten Dimen- sionen). Helpful discussions with Josef Kudrnovsk´ y and Ilja Turek are gratefully acknowledged. 1T. Holstein, Annals of Physics 29, 410 (1964) . 2G. D. Mahan, Many-particle physics , Physics of Solids and Liquids (Springer, New York, 2000). 3P. B. Allen, Phys. Rev. B 3, 305 (1971) . 4K. Takegahara and S. Wang, J. Phys. F: Met. Phys. 7, L293 (1977) . 5G. Grimvall, Physica Scripta 14, 63 (1976). 6G. D. Mahan and W. Hansch, J. Phys. 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2014-12-05

A scheme is presented that is based on the alloy analogy model and allows to account for thermal lattice vibrations as well as spin fluctuations when calculating response quantities in solids. Various models to deal with spin fluctuations are discussed concerning their impact on the resulting temperature dependent magnetic moment, longitudinal conductivity and Gilbert damping parameter. It is demonstrated that using the Monte Carlo (MC) spin configuration as an input, the alloy analogy model is capable to reproduce results of MC simulations on the average magnetic moment within all spin fluctuation models under discussion. On the other hand, response quantities are much more sensitive to the spin fluctuation model. Separate calculations accounting for either the thermal effect due to lattice vibrations or spin fluctuations show their comparable contributions to the electrical conductivity and Gilbert damping. However, comparison to results accounting for both thermal effects demonstrate violation of Matthiessen's rule, showing the non-additive effect of lattice vibrations and spin fluctuations. The results obtained for bcc Fe and fcc Ni are compared with the experimental data, showing rather good agreement for the temperature dependent electrical conductivity and Gilbert damping parameter.

Calculating linear response functions for finite temperatures on the basis of the alloy analogy model

1412.1988v1

arXiv:0905.0112v2 [cond-mat.other] 3 Aug 2009Spin excitations in a monolayer scanned by a magnetic tip M.P. Magiera1, L. Brendel1, D.E. Wolf1andU. Nowak2 1Department of Physics and CeNIDE, University of Duisburg-E ssen, D-47048 Duisburg, Germany, EU 2Theoretical Physics, University of Konstanz, D-78457 Kons tanz, Germany, EU PACS68.35.Af – Atomic scale friction PACS75.10.Hk – Classical spin models PACS75.70.Rf – Surface Magnetism Abstract. - Energy dissipation via spin excitations is investigated f or a hard ferromagnetic tip scanning a soft magnetic monolayer. We use the classical Hei senberg model with Landau-Lifshitz- Gilbert (LLG) dynamics including a stochastic field represe nting finite temperatures. The friction force depends linearly on the velocity (provided it is small enough) for all temperatures. For low temperatures, the corresponding friction coefficient is proportional to the phenomenological damping constant of the LLG equation. This dependence is los t at high temperatures, where the friction coefficient decreases exponentially. These finding s can be explained by properties of the spin polarisation cloud dragged along with the tip. Introduction. – While on the macroscopic scale the phenomenology of friction is well known, several new as- pects are currently being investigated on the micron and nanometer scale [1,2]. During the last two decades, the research on microscopic friction phenomena has advanced enormously, thanks to the development of Atomic Force Microscopy (AFM, [3]), which allows to measure energy dissipation caused by relative motion of a tip with respect to a substrate. Recently the contribution of magnetic degrees of free- dom to energy dissipation processes has attracted increas- ing interest [4–8]. Today, magnetic materials can be con- trolleddowntothenanometerscale. Newdevelopmentsin the data storage industry, spintronics and quantum com- puting require a better understanding of tribological phe- nomena in magnetic systems. For example, the reduction of heat generation in reading heads of hard disks which workat nanometerdistancesis animportant issue, as heat can cause data loss. Magnetic Force Microscopy(MFM), where both tip and surface are magnetic, is used to investigate surface mag- netism and to visualise domain walls. Although recent studies have attempted to measure energy dissipation be- tween an oscillating tip and a magnetic sample [9,10], the dependenceofthefrictionforceonthetip’sslidingvelocity hasnot been consideredyet apartfroma workby C. Fusco et al.[8] which is extended by the present work to temper- aturesT∝negationslash=0. The relative motion of the tip with respect to the surface can lead to the creation of spin waves whichpropagate inside the sample and dissipate energy, giving rise to magnetic friction. We will firstpresent asimulationmodel anddefine mag- netic friction. The model contains classical Heisenberg spins located on a rigid lattice which interact by exchange interaction with each other. Analogous to the reading head of a hard disc or a MFM tip, an external fixed magnetic moment is moved across the substrate. Using Langevin dynamics and damping, it is possible to simu- late systems at finite temperatures. The main new results concern the temperature dependence of magnetic friction. Simulation model and friction definition. – To simulate a solid magnetic monolayer (on a nonmagnetic substrate), we consider a two-dimensional rigid Lx×Ly lattice of classical normalised dipole moments (“spins”) Si=µi/µs, where µsdenotes the material-dependent magnetic saturationmoment(typically afew Bohrmagne- tons). These spins, located at z= 0 and with lattice spac- inga, represent the magnetic moments of single atoms. They can change their orientation but not their absolute value, so that there are two degrees of freedom per spin. Weuseopenboundaryconditions. Aconstantpointdipole Stippointing in the z-direction and located at z= 2arep- resentsthemagnetictip. Itismovedparalleltothesurface with constant velocity v. This model has only magnetic degrees of freedom and thus focusses on their contributions to friction. For a real tip one could expect that magnetic, just like nonmag- p-1M.P. Magiera 1 L. Brendel 1 D.E. Wolf 1 U. Nowak 2 netic [11,12] interactions might also lead to atomic stick- slip behaviour, and hence to phononic dissipation with a velocity-independent friction contribution as described by the Prandtl-Tomlinson model [13,14]. However, this requires a periodic potential between tip and substrate, that is strong enough compared to the elastic deformation energytoallowformultiple localpotentialenergyminima. The magnetic tip-substrate interactions are unlikely to be strong enough. The Hamiltionan consists of two parts: H=Hsub+Hsub−tip. (1) The first one represents the internal ferromagnetic short- range interaction within the substrate. The second one describes the long-range coupling between the substrate and the tip. The interaction between the substrate moments is mod- eled by the anisotropic classical Heisenberg model, Hsub=−J/summationdisplay /angbracketlefti,j/angbracketrightSi·Sj−dzN/summationdisplay i=1S2 i,z. (2) J >0 describes the ferromagnetic exchange interaction between two nearest neighbours, expressed by the an- gular brackets ∝angbracketlefti,j∝angbracketright.dz<0 quantifies the anisotropy, which prefers in-plane orientations of the spins. The dipole-dipole-interaction between the substrate spins is neglected, because it is much weaker than the exchange interaction. A quantitative comparison of our simulation results with the ones obtained in [8], where the dipole- dipole-interaction inside the substrate was taken into ac- count, justifies this approximation, which reduces simula- tion time enormously. The long-range interaction between substrate and tip is described by a dipole-dipole interaction term Hsub−tip=−wN/summationdisplay i=13 (Si·ei)(Stip·ei)−Si·Stip R3 i,(3) whereRi=|Ri|denotes the norm of the distance vector Ri=ri−rtip, andeiits unit vector ei=Ri/Ri.riand rtipdenote the position vectors of the substrate spins and the tip respectively. wquantifies the dipole-dipole cou- pling of the substrate and the tip. Note, however, that the results of the present study only depend on the com- bination w|Stip|, which is the true control parameter for the tip-substrate coupling. Theproperequationofmotionofthemagneticmoments is theLandau-Lifshitz-Gilbert (LLG, [15]) equation, ∂ ∂tSi=−γ (1+α2)µs[Si×hi+αSi×(Si×hi)],(4) which is equivalent to the Gilbert equation of motion [16]: ∂ ∂tSi=−γ µsSi×/bracketleftbigg hi−αµs γ∂Si ∂t/bracketrightbigg . (5)The first term on the right-hand side of eq. (4) describes the dissipationless precession of each spin in the effective fieldhi(to be specified below). The second term de- scribes the relaxation of the spin towards the direction ofhi.γdenotes the gyromagnetic ratio (for free electrons γ= 1.76086×1011s−1T−1), andαis a phenomenological, dimensionless damping parameter. The effective field contains contributions from the tip and from the exchange interaction, as well as a thermally fluctuating term ζi[17,18], hi=−∂H ∂Si+ζi(t). (6) The stochastic, local and time-dependent vector ζi(t) ex- presses a “Brownian rotation”, which is caused by the heat-bath connected to each magnetic moment. In our simulations this vector is realised by uncorrelated random numbers with a Gaussian distribution, which satisfy the relations ∝angbracketleftζi(t)∝angbracketright= 0 and (7) ∝angbracketleftζκ i(t)ζλ j(t′)∝angbracketright= 2αµs γkBTδi,jδκ,λδ(t−t′),(8) whereTis the temperature, δi,jexpresses that the noise at different lattice sites is uncorrelated, and δκ,λrefers to the absence of correlations among different coordinates. Tofind aquantitywhichexpressesthe friction occurring in the system, it is helpful to discuss energy transfers be- tween tip, substrate and heat-bath first. It is straightfor- ward to separate the time derivative of the system energy, eq. (1), into an explicit and an implicit one, dH dt=∂H ∂t+N/summationdisplay i=1∂H ∂Si·∂Si ∂t. (9) The explicit time dependence is exclusively due to the tip motion. The energy transfer between the tip and the sub- strate is expressed by the first term of eq. (9), which jus- tifies to call it the “pumping power” Ppump: Ppump=∂H ∂t=∂Hsub−tip ∂rtip·v =w/summationdisplay αvαN/summationdisplay i=13 R4 i/braceleftbigg (Si·eiei,α−Si,α)(Stip·ei) (10) +(Stip·eiei,α−Stip,α)(Si·ei) −Si·Stipei,α+3ei,α(Si·ei)(Stip·ei)/bracerightbigg At any instance, the substrate exerts a force −∂Hsub−tip ∂rtip on the tip. Due to Newton’s third law, Ppumpis the work per unit time done by the tip on the substrate. Its time and thermal average ∝angbracketleftPpump∝angbracketrightis the average rate at which energy is pumped into the spin system. In a steady state it must be equal to the average dissipation rate, i.e.to p-2Spin excitations in a monolayer scanned by a magnetic tip h hSω δθδϕ Figure1: Asingle spininamagnetic fieldrotatingwithangul ar velocity ω, is dragged along with a phase shift δϕand aquires an out of plane component δθ. the net energy transferred to the heat bath per unit time due to spin relaxation. The magnetic friction force can therefore be calculated by F=∝angbracketleftPpump∝angbracketright v. (11) The second term of eq. (9) describes the energy transfer between the spin system and the heat bath. Inserting eq. (5) intoPdiss=−/summationtextN i=1∂H ∂Si·∂Si ∂tleads to Pdiss=N/summationdisplay i=1∂H ∂Si·/bracketleftbiggγ µsSi×ζi−αSi×∂Si ∂t/bracketrightbigg =−Ptherm+Prelax. (12) The first term, Pthermcontaining ζi, describes, how much energy is transferred into the spin system due to the ther- mal perturbation by the heat bath. The second term, Prelaxproportional to the damping constant α, describes the rate of energy transfer into the heat bath due to the relaxation of the spins. AtT= 0,Pthermis zero. The spins are only perturbed by the external pumping at v∝negationslash= 0. Then Prelax=Pdiss=γα µs(1+α2)N/summationdisplay i=1(Si×hi)2,(T= 0), (13) where for the last transformation we used eq. (4) in order toshowexplicitlytherelationshipbetweendissipationrate and misalignment between spins and local fields. It will be instructive to compare the magnetic substrate scanned by a dipolar tip with a much simpler system, inwhich the substrate is replaced by a single spin Ssub- jected to an external field h(t) that rotates in the plane perpendicular to a constant angular velocity ω(replac- ing the tip velocity). It is straight forward to obtain the steady state solution for T= 0, where in the co-rotating frameSis at rest. Slags behind h/hby an angle δϕand gets a component δθinω-direction (cf. fig. 1), which are in first order given by δϕ α=δθ=ωµs hγ+O/parenleftBigg/parenleftbiggωµs hγ/parenrightbigg3/parenrightBigg .(14) Inserting this into the ( N=1)-case of eq. (13) yields a dis- sipation rate of Pdiss=αω2µs/γ, which corresponds to a “viscous” friction F=Pdiss/ω∝αω. It is instructive to give a simple physical explanation for eq. (14), instead of presenting the general solution, which can be found in [19]. Two timescales exist in the system, which can be readily obtained from eq. (4); first, the inverse Lamor frequency τLamor= (1+α2)µs/γh, and second, the relaxation time τrelax=τLamor/α. They govern the time evolution of δϕandδθ. In leading order, δ˙θ=δϕ τLamor−δθ τrelax, (15) δ˙ϕ=ω−δθ τLamor−δϕ τrelax. (16) Thefirstequationdescribes, how δθwouldincreasebypre- cession of the spin around the direction of the field, which is counteracted by relaxation back towards the equator. The second equation describes that without relaxation into the field direction, δϕwould increase with velocity ωminus the azimuthal component of the precession veloc- ity, which is in leading order proportional to δθ. Setting the left hand sides to zeroin the steady state, immediately gives the solution (14). Inthe (T= 0)-study[8], the timeaverage ∝angbracketleftPrelax∝angbracketright/vwas used to calculate the friction force. As pointed out above, this quantity agrees with (11) in the steady state. For finite temperatures, however, (11) is numerically better behaved than ∝angbracketleftPrelax∝angbracketright/v. The reason is the following: ForT∝negationslash= 0, the spins are also thermally agitated, even without external pumping, when the dissipation rate Pdiss vanishes. This shows that the two terms PthermandPrelax largely compensate each other, and only their difference is the dissipation rate we are interested in. This fact makes it difficult to evaluate (12) and is the reasonwhy we prefer to work with (11) as the definition of the friction force. We have also analyzed the fluctuations of the friction force (11). The power spectrum has a distinct peak at frequency v/a, which means that the dominant temporal fluctuations are due to the lattice periodicity with lattice constant a. A more complete investigation of the fluctu- ations, which should also take into account, how thermal positional fluctuations influence the friction force, remains to be done. p-3M.P. Magiera 1 L. Brendel 1 D.E. Wolf 1 U. Nowak 2 Technical remarks. – Because of the vector pro- duct in eq. (4), the noise ζienters in a multiplicative way, calling for special attention to the interpretation of this stochastic differential equation (Stratonovich vs. Itˆ o sense). The physical origin of the noise renders it generi- callycoloured and thus selects the Stratonovich interpre- tation as the appropriate one (Wong-Zakai theorem [20]), in which its appearance as white noise is an idealisation. Accordingly we employ the Heun integration scheme [21]. After each time step the spins are rescaled so that their length remains unchanged. To get meaningful results, it is of prime importance to reach a steady state. The initial configuration turned out to be a crucial factor for achieving this within acceptable computing time. Therefore a long initialisation run is per- formed, before the tip motion starts. Moreover, the system size is another limiting factor. In order to avoid that the tip reaches the system boundary before the steady state is reached, we use a “conveyorbelt method” allowing to do the simulation in the comoving frame of the tip. The tip is placed in a central point above the substrate plane, e.g.at ((Lx+1)/2,(Ly+1)/2). After an equilibration time, the tip starts to move with fixed velocity in x-direction. When it passed exactly one lattice constant a, the front line (at x=Lx) is duplicated and added to the lattice at x=Lx+ 1. Simultaneously, the line at the opposite boundary of the system (at x= 1) is deleted, so that the simulation cell is of fixed size and con- sists of the Lx×Lyspins centered around the tip position, with open boundaries. Note that this is different from pe- riodicboundaryconditions, becausethespinconfiguration deleted at one side is different from the one added at the opposite side. We compared the results obtained for a small system in the co-moving frame of the tip with some runs for a system that was long enough in x-direction that the steady state could be reached in the rest frame of the sample, and confirmed that the same friction results and steady state properties could be obtained with drastically reduced computation time. It is convenient to rewrite the equations of motion in natural units. An energy unit is prescribed by the ex- change energy Jof two magnetic moments. It rescales the energy related parameters dzandwas well as the simu- lated temperature, kBT′=kBT J. (17) The rescaled time further depends on the material con- stantsµsandγ, t′=Jγ µst. (18) A length scale is given by the lattice constant a, so a nat- ural velocity for the system can be defined, v′=µs γJav. (19)-1-0.5 0 0.5 1 -4-3-2-1 0 1 2 3 4mx(x) x-x0α = 0.3, v = 0.05 α = 0.3, v = 0.10 α = 0.5, v = 0.10 tanh(0.75 x)-0.2 0 0 0.2 vx0/αα = 1.0 α = 0.7 α = 0.5 Figure 2: Local magnetisation ( x-component) at T=0 along the lattice axes in x-direction which are closest to the tip ( i.e. aty=±0.5). Analogous to a domain wall one finds a tanh x- profile. Depending on the damping constant αand the velocity v, its zero is shifted backwards from the tip position by a valu e x0≈−0.88αv, as shown in the inset. From the natural length and energy, the natural force re- sults to: F′=a JF (20) From now on, all variables are given in natural units, and we will drop the primes for simplicity. The typical exten- sion of the simulated lattices is 50 ×30,which was checked to be big enough to exclude finite size-effects. For the tip coupling, we chose large values ( e.g.wStip= (0,0,−10)), to get a large effective field on the substrate. Usually it is assumed that the dipole-dipole coupling constant has a value of about w= 0.01, which means that the magnetic moment of the tip is chosen a factor of 1000 times larger than the individual substrate moments. The anisotropy constant is set to dz=−0.1 in all simulations. The damp- ing constant αis varied from 0 .1 to the quite large value 1. At finite temperatures typically 50 simulation runs with different random number seeds are performed to get reli- able ensemble averages. Simulation results. – In [8] it was found that the magnetic friction force depends linearly on the scanning velocityvand the damping constant αfor small velocities (v≤0.3). For higher velocities the friction force reaches a maximum and then decreases. In this work we focus on the low-velocity regime with the intention to shed more light on the friction mechanism and its temperature de- pendence.1 Adiabatic approximation at T=0.If we assume the fieldhfor each spin to vary slowly enough to allow the so- lution (14) to be attained as adiabatic approximation, the linear dependence F∝αvfrom [8] follows immediately: At every point, the temporal change of the direction of h, defining a local ωfor (14), is proportional to v. We 1It should be noted that the smallest tip velocities we simula ted, are of the order of 10−2(aJγ/µ s), which is still fast compared to typical velocities in friction force microscopy experimen ts. p-4Spin excitations in a monolayer scanned by a magnetic tip -1-0.5 0 0.5 1 -20-15-10-5 0 5 10 15 20mx x0.11.0 2345678910|m| r0.11.0 2345678910|m| rkBT=0.1 kBT=0.5 kBT=0.9 kBT=1.5 Figure 3: Left: Magnetisation profiles as in fig. 2 for several temperatures with wStip= 10,α= 0.5 andv= 0.01. Middle and right: Absolute value of the average magnetisation as a func tion of the distance rfrom (x,y) = (0,0), directly underneath the tip. For small temperatures (upper two curves) it decreases with a power law (cf. double-logarithmic plot, middle), for high temperatures (lower two curves) it decreases exponentiall y (cf. semi-logarithmic plot, right). confirmed the validity of the adiabatic approximation nu- merically by decomposing S−h/hwith respect to the local basis vectors ∂t(h/h),h, and their cross-product, all of them appropriately normalized. In other words, we ex- tractedδϕandδθdirectly and found them in excellent agreement with (14). The lag of Swith respect to hmanifests itself also macroscopically in the magnetisation field as we will show now. Thetip-dipoleisstrongenoughtoalignthesubstrate spins to nearly cylindrical symmetry. Since the anisotropy is chosen to generate an easy plane ( dz<0), spins far away from the tip try to lie in the xy-plane, while close to the tip they tilt into the z-direction. This is displayed in fig. 2 where the x-component of the local magnetisation is shown along a line in x-direction for a fixed y-coordinate. Remarkably, these magnetisation profiles for different val- ues ofvandαcollapse onto a unique curve, if they are shifted by corresponding offsets x0with respect to the tip position. As expected from (14), the magnetisation profile stays behind the tip by a ( y-dependent) shift x0∝αv(cf. inset of fig. 2). Friction at T>0.With increasing temperature the magnetisation induced by the tip becomes smaller, as shown in fig. 3. One can distinguish a low tempera- ture regime, where the local magnetisation decreases alge- braically with the distance from the tip, and a high tem- perature regime, where it decreases exponentially. The transitionbetween these regimeshappens around T≈0.7. For all temperatures the friction force Fturns out to be proportional to the velocity (up to v≈0.3), as for T=0, with a temperature dependent friction coefficient F/v. The two temperature regimes manifest themselves also here, as shown in fig. 4: For low temperatures the fric- tion coefficients depend nearly linearly on α, reflecting the T=0 behaviour. Towards the high temperature regime, however, the α-dependence vanishes, and all friction coef- ficients merge into a single exponentially decreasing func- 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6F/v kBTwStip = 10, α=0.5 wStip = 10, α=0.7 wStip = 5, α=0.5 wStip = 5, α=0.7 Figure 4: Friction coefficients for different α,wStipandkBT. One can distinguish between a low temperature regime, where the friction coefficient depends on αbut not on wStip, and a high temperature regime, where it depends on wStipbut not onα. tion. The low temperature behaviour can be understood es- sentially along the lines worked out for T=0, as result of a delayed, deterministic response (precession and relax- ation) to the time dependent tip field. At high tempera- tures, however,frictionresultsfromtheabilityofthetipto propagate partial order through the thermally disorderd medium. The magnetisationpattern in the wakeofthe tip no longer adapts adiabatically to the dwindling influence ofthe tip, but decaysdue to thermal disorder. Then, aris- ingtemperatureletstheorderedareaaroundthetipshrink whichleadstotheexponentialdecreaseofthefrictioncoef- ficient. However, it increases with the tip strength, wStip, as stronger order can be temporarily forced upon the re- gion around the tip. By contrast, the tip strength looses its influence on friction in the limit T→0, because the substrate spins are maximally polarised in the tip field . This picture of the two temperature regimes is sup- p-5M.P. Magiera 1 L. Brendel 1 D.E. Wolf 1 U. Nowak 2 -3-2.5-2-1.5-1-0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6x0/v kBTwStip = 10, α=0.5 wStip = 10, α=0.7 wStip = 5, α=0.5 wStip = 5, α=0.7 Figure 5: Distance x0by which the magnetisation pattern lags behind the tip is proportional to vfor all temperatures. The proportionality constant depends on αonly in the low temper- ature regime. ported by the distance x0, by which the magnetisation pattern lags behind the tip. It is proportional to vfor all temperatures, but α-dependent only in the low tempera- ture regime, cf. fig. 5. Conclusion and outlook. – In this work, we could explain the low-velocity, zero-temperature findings from [8], namely that the magnetic friction force in the Heisen- berg model has a linear velocity dependence with a coeffi- cient proportional to the damping constant α. In the spin polarisation cloud dragged along with the tip, each sub- stratespin followsthe localfield with a lagproportionalto the frequency of the field change and to α. Moreover, the magnetisationpattern aroundthe tip getsdistorted due to precession. These effects directly give rise to the observed magneticfriction andcould be evaluated quantitativelyby means of a single spin model. Second, for the first time the temperature dependence of magnetic friction in the Heisenberg model was investi- gated in the framework of Landau-Lifshitz-Gilbert (LLG) dynamics with a stochastic contribution to the magnetic field. Two regimes were found, which can be charac- terised by their different relaxation behaviour. While in the low-temperature regime the response of the system on the perturbation due to the moving tip is dominated by the deterministic precession and relaxation terms in the LLG equation, thermal perturbations competing with the one caused by the moving tip are essential in the high- temperature regime. This explains, why magnetic friction depends on αbut noton wStipforlowtemperatures, while it depends on wStipbut not on αfor high temperatures where it decreases exponentially with T. Important extensions of the present investigation in- clude the effects ofa tip magnetisationpointing in a differ- ent than the z-direction, of the strength and sign of spin anisotropy, dz, or of the thickness of the magnetic layer. Both, spin anisotropyand lattice dimension will be crucial for the critical behaviour, as well as for the critical tem-perature itself. Studies dealing with these quantities are already in progress and will be reported in a future work. ∗∗∗ This work was supported by the German Research Foundation (DFG) via SFB 616 “Energy dissipation at surfaces”. Computation time was granted in J¨ ulich by the John-von-Neumann Institute of Computing (NIC). References [1]Persson B. ,Sliding Friction (Springer, Berlin, Heidel- berg, New York) 1998. [2]Urbakh M., Klafter J., Gourdon D. andIs- raelachvili J. ,Nature,430(2004) 525. [3]Gnecco E., Bennewitz R., Gyalog T. andMeyer E. , J. Phys.: Condens. Matter ,13(2001) R619. [4]Acharyya M. andChakrabarti B. K. ,Phys. Rev. B , 52(1995) 6550. [5]Ort´ın J.andGoicoechea J. ,Phys. Rev. B ,58(1998) 5628. [6]Corberi F., Gonnella G. andLamura A. ,Phys. Rev. Lett.,81(1998) 3852. [7]Kadau D., Hucht A. andWolf D. E. ,Phys. Rev. Lett. , 101(2008) 137205. [8]Fusco C., Wolf D. E. andNowak U. ,Phys. Rev. B , 77(2008) 174426. [9]Grutter P., Liu Y., LeBlanc P. andDurig U. ,Appl. Phys. Lett. ,71(1997) 279. [10]Schmidt R., Lazo C., Holscher H., Pi U. H., Caciuc V., Schwarz A., Wiesendanger R. andHeinze S. , Nano Letters ,9(2009) 200. [11]Zw¨orner O., H ¨olscher H., Schwarz U. D. and Wiesendanger R. ,Appl. Phys. A ,66(1998) S263. [12]Gnecco E., Bennewitz R., Gyalog T., Loppacher C., Bammerlin M., Meyer E. andG¨untherodt H.- J.,Phys. Rev. Lett. ,84(2000) 1172. [13]Prandtl L. ,Zs. f. angew. Math. u. Mech. ,8(1928) 85. [14]Tomlinson G. A. ,Philos. Mag. ,7(1929) 905. [15]Landau L. D. andLifshitz E. M. ,Phys. Z. Sowjetunion , 8(1935) 153. [16]Gilbert T. L. ,IEEE Trans. Magn. ,40(2004) 3443. [17]N´eel L.,C. R. Acad. Sc. Paris ,228(1949) 664. [18]Brown W. F. ,Phys. Rev. ,130(1963) 1677. [19]Magiera M. P. ,Computer simulation of magnetic fric- tionDiploma Thesis, Univ. of Duisburg-Essen (2008). [20]Horsthemke W. andLefever R. ,Noise-Induced Tran- sitions(Springer) 1983. [21]Garc´ıa-Palacios J. L. andL´azaro F. J. ,Phys. Rev. B,58(1998) 14937. p-6

2009-05-01

Energy dissipation via spin excitations is investigated for a hard ferromagnetic tip scanning a soft magnetic monolayer. We use the classical Heisenberg model with Landau-Lifshitz-Gilbert (LLG)-dynamics including a stochastic field representing finite temperatures. The friction force depends linearly on the velocity (provided it is small enough) for all temperatures. For low temperatures, the corresponding friction coefficient is proportional to the phenomenological damping constant of the LLG equation. This dependence is lost at high temperatures, where the friction coefficient decreases exponentially. These findings can be explained by properties of the spin polarization cloud dragged along with the tip.

Spin excitations in a monolayer scanned by a magnetic tip

0905.0112v2

arXiv:1310.7657v1 [astro-ph.SR] 29 Oct 2013Nature of Prominences and their role in Space Weather Proceedings IAU Symposium No. 300, 2014 B. Schmieder, JM. Malherbe & S. Wu, eds.c/circlecopyrt2014 International Astronomical Union DOI: 00.0000/X000000000000000X Observational Study of Large Amplitude Longitudinal Oscillations in a Solar Filament Kalman Knizhnik1,2,Manuel Luna3, Karin Muglach2,4 Holly Gilbert2, Therese Kucera2, Judith Karpen2 1Department of Physics and Astronomy The Johns Hopkins University, Baltimore, MD 21218 email: kalman.knizhnik@nasa.gov 2NASA/GSFC, Greenbelt, MD 20771, USA 3Instituto de Astrof´ ısica de Canarias, E-38200 La Laguna, T enerife, Spain 4ARTEP, Inc., Maryland, USA Abstract. On 20 August 2010 an energetic disturbance triggered damped large-amplitude lon- gitudinal (LAL) oscillations in almost an entire filament. I n the present work we analyze this periodic motion in the filament to characterize the damping a nd restoring mechanism of the oscillation. Our method involves placing slits along the ax is of the filament at different angles with respect to the spine of the filament, finding the angle at w hich the oscillation is clearest, and fitting the resulting oscillation pattern to decaying si nusoidal and Bessel functions. These functions represent the equations of motion of a pendulum da mped by mass accretion. With this method we determine the period and the decaying time of t he oscillation. Our preliminary results support the theory presented by Luna and Karpen (201 2) that the restoring force of LAL oscillations is solar gravity in the tubes where the threads oscillate, and the damping mechanism is the ongoing accumulation of mass onto the oscillating thr eads. Following an earlier paper, we have determined the magnitude and radius of curvature of the dipped magnetic flux tubes host- ing a thread along the filament, as well as the mass accretion r ate of the filament threads, via the fitted parameters. Keywords. solar prominences, oscillations, magnetic structures 1. Procedure LAL oscillations consist of periodic motions of the prominence thread s along the mag- netic field that are disturbed by a small energetic event close to the filament (see Luna et al. paper in this volume). Luna and Karpen (2012) argue that pro minence oscillations can be modeled as a damped oscillating pendulum, whose equation of mo tion satis- fies a zeroth-order Bessel function. In their model, a nearby trig ger event causes quasi- stationary preexisting prominence threads sitting in the dips of the magnetic structure to oscillate back and forth, with the restoring force being the proj ected gravity in the tubes where the threads oscillate (e.g. Luna et al. (2012)). In this paper, we report pre- liminary results of comparisonsof observations of prominence oscilla tions with the model presented by Luna and Karpen (2012). More details will be available in the forthcoming paper by Luna et al. (2013). In this analysis, we place slits along the filament spine and measure the intensity along each slit as a function of time. Fig. 1 (left) shows the filament in the AI A 171˚A filter with the slits overlaid. Each slit is then rotated in increments of 0.5◦from 0◦to 60◦ with respect to the filament spine. We select the best slit according t o the following criteria: (a) continuity of oscillations, (b) amplitude of the oscillation is maximized, (c) clear transition from dark to bright regions, (d) maximum number of cycles. The oscillation for a representative slit is shown in Figure 1 (right), wh ich corresponds to the grey slit in Figure 1 (left). We identify the position of the cente r of mass of the 12 K. Knizhnik, M. Luna, K. Muglach, H. Gilbert, T. Kucera, J. Karpen Figure 1. Left: Filament seen in AIA 171 with best slits overlaid. Right: An intensity distance– time slit, showing an oscillation with the Bessel fit (white c urve) to equation (2.1) in Luna et al. (this volume). The sinusoidal fit was not as good as the Bes sel fit and is not shown. thread by finding the intensity minimum along the slit, indicated by black crosses in Figure 1 (right). These points are then fit to equation (2.1) of Luna et al. (this volume), and the resulting fit is shown in white. 2. Results Fitting our data to equation (2.1) of Luna et al. (this volume) yields va lues ofχ2 ranging between 1-13. Using equation (2.2) of Luna et al. (this volum e), we find the averageradiusofcurvatureofthe magneticfield dips that suppor t the oscillatingthreads. We find it to be approximately 60 Mm. We also calculate a threshold value for the field itself that would allow it to support the observedthreads. Using equ ation (3.1) of Luna et al.(this volume), wefind anaveragemagneticfield of ∼20G, assumingatypicalfilament number density of 1011cm−3, in good agreement with measurements (e.g. Mackay et al. 2010). On average, the oscillations form an angle of ∼25owith respect to the filament spine, and have a period of ∼0.8 hours. To explain the very strong damping mass must accrete onto the threads at a rate of about 60 ×106kg/hr. 3. Conclusions We conclude that the observedoscillationsarealongthe magneticfie ld, which formsan angle of∼25owith respect to the filament spine (Tandberg-Hanssen & Anzer, 19 70). We find that both the curvature and the magnitude of the magnetic fie ld are approximately uniform on different threads. Both the Bessel and sinusoidal func tions are well fitted, indicating that mass accretion is a likely damping mechanism of LAL oscilla tions, and that the restoring force is the projected gravity in the dips where the threads oscillate. The mass accretion rate agrees with the theoretical value (Karpe n et al., 2006, Luna, Karpen, & DeVore, 2012). References Karpen, J. T., Antiochos, S. K., Klimchuk, J. A. 2006, ApJ, 63 7, 531 Luna, M., Karpen, J. T., & Devore, C. R. 2012a, ApJ, 746, 30 Luna, M., & Karpen, J. 2012, ApJ, 750, L1 Luna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., this volume , 2014 Luna, M., Knizhnik, K., Muglach, K., Gilbert, H, Kucera, T. & Karpen, J., ApJ, 2013,in prep. Mackay, D., Karpen, J., Ballester, J., Schmieder, B., Aulan ier, G. 2010, Sp. Sci. Rev. , 151, 333 Tandberg-Hanssen, E. and Anzer, U. 1970, Solar Physics 15, 158T

2013-10-29

On 20 August 2010 an energetic disturbance triggered damped large-amplitude longitudinal (LAL) oscillations in almost an entire filament. In the present work we analyze this periodic motion in the filament to characterize the damping and restoring mechanism of the oscillation. Our method involves placing slits along the axis of the filament at different angles with respect to the spine of the filament, finding the angle at which the oscillation is clearest, and fitting the resulting oscillation pattern to decaying sinusoidal and Bessel functions. These functions represent the equations of motion of a pendulum damped by mass accretion. With this method we determine the period and the decaying time of the oscillation. Our preliminary results support the theory presented by Luna and Karpen (2012) that the restoring force of LAL oscillations is solar gravity in the tubes where the threads oscillate, and the damping mechanism is the ongoing accumulation of mass onto the oscillating threads. Following an earlier paper, we have determined the magnitude and radius of curvature of the dipped magnetic flux tubes hosting a thread along the filament, as well as the mass accretion rate of the filament threads, via the fitted parameters.

Observational Study of Large Amplitude Longitudinal Oscillations in a Solar Filament

1310.7657v1

POLYNOMIAL STABILIZATION OF NON-SMOOTH DIRECT/INDIRECT ELASTIC/VISCOELASTIC DAMPING PROBLEM INVOLVING BRESSE SYSTEM STEPHANE GERBI, CHIRAZ KASSEM, AND ALI WEHBE Abstract. We consider an elastic/viscoelastic problem for the Bresse system with fully Dirichlet or Dirichlet- Neumann-Neumann boundary conditions. The physical model consists of three wave equations coupled in certain pattern. The system is damped directly or indirectly by global or local Kelvin-Voigt damping. Ac- tually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness of the damping coecient at the interface play a crucial role in the type of the stabilization of the corre- sponding semigroup. Indeed, using frequency domain approach combined with multiplier techniques and the construction of a new multiplier function, we establish dierent types of energy decay rate (see the table of stability results at the end). Our results generalize and improve many earlier ones in the literature (see [7]) and in particular some studies done on the Timoshenko system with Kelvin-Voigt damping (see for instance [9], [23] and [25]). Contents 1. Introduction 1 2. Well-posedness of the problem 4 3. Strong stability of the system 6 4. Polynomial stability for non smooth damping coecients at the interface 10 5. The case of only one local viscoelastic damping with non smooth coecient at the interface 17 6. Lack of exponential stability 24 7. Additional results and summary 25 Acknowledgments 26 References 26 1.Introduction 1.1.The Bresse system with Kelvin-Voigt damping. Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. There are several mathemati- cal models representing physical damping. The most often encountered type of damping in vibration studies are linear viscous damping and Kelvin-Voigt damping which are special cases of proportional damping. Vis- cous damping usually models external friction forces such as air resistance acting on the vibrating structures and is thus called \external damping", while Kelvin-Voigt damping originates from the internal friction of the material of the vibrating structures and thus called \internal damping". The stabilization of conser- vative evolution systems (wave equation, coupled wave equations, Timoshenko system ...) by viscoelastic Kelvin-Voigt type damping has attracted the attention of many authors. In particular, it was proved that the stabilization of wave equation with local Kelvin-Voigt damping is greatly in uenced by the smoothness of the damping coecient and the region where the damping is localized (near or faraway from the boundary) even in the one-dimensional case, see [6, 16]. This surprising result initiated the study of an elastic system with local Kelvin-Voigt damping There are a few number of publications concerning the stabilization of Bresse or Timoshenko systems with viscoelastic Kelvin-Voigt damping. (see Subsection 1.2 below). 2010 Mathematics Subject Classication. 35B37, 35D05, 93C20, 73K50. Key words and phrases. Bresse system, Kelvin-Voigt damping, polynomial stability, non uniform stability, frequency domain approach. 1arXiv:2006.16595v2 [math.AP] 22 Feb 2022Figure 1. After deformation the particle M0of the beam is at the position M. In this paper, we study the stability of Bresse system with localized non-smooth Kelvin-Voigt damping coecient at the interface and we brie y state results when the Kelvin-Voigt damping coecients are either global or localized but smooth at the interface since the tools used for the study of non-smooth coecient are used in the same, but much simpler, way when the coecients act on the totality of the domain or are smooth enough at the interface. These results generalize and improve many earlier ones in the literature. The Bresse system is usually considered in studying elastic structures of the arcs type (see [14]). It can be expressed by the equations of motion: 1'tt=Qx+`N 2 tt=MxQ 1wtt=Nx`Q where N=k3(wx`') +F3; Q =k1('x+ +`w) +F1; M =k2 x+F2 F1=D1('xt+ t+`wt); F 2=D2 xt; F 3=D3(wxt`'t) and whereF1,F2andF3are the Kelvin-Voigt dampings. When F1=F2=F3= 0,N,QandMdenote the axial force, the shear force and the bending moment. The functions '; ; andwmodel the vertical, shear angle, and longitudinal displacements of the lament. Here 1=A;  2=I; k 1=k0GA; k 3=EA; k 2= EI; ` =R1whereis the density of the material, Eis the modulus of elasticity, Gis the shear modulus, k0is the shear factor, Ais the cross-sectional area, Iis the second moment of area of the cross-section, and Ris the radius of curvature. see gure 1 reproduces from [7]. The damping coecients D1,D2andD3are bounded non negative functions over (0 ;L). So we will consider the system of partial dierential equations given on (0 ;L)(0;+1) by the following form: (1.1)8 >>>>< >>>>:1'tt[k1('x+ +`w) +D1('xt+ t+`wt)]x`k3(wx`')`D3(wxt`'t) = 0; 2 tt[k2 x+D2 xt]x+k1('x+ +`w) +D1('xt+ t+`wt) = 0; 1wtt[k3(wx`') +D3(wxt`'t)]x+`k1('x+ +`w) +`D1('xt+ t+`wt) = 0; with fully Dirichlet boundary conditions: (1.2) '(0;
) ='(L;
) = (0;
) = (L;
) =w(0;
) =w(L;
) = 0 in R+; or with Dirichlet-Neumann-Neumann boundary conditions: (1.3) '(0;
) ='(L;
) = x(0;
) = x(L;
) =wx(0;
) =wx(L;
) = 0 in R+; 2in addition to the following initial conditions: (1.4)'(
;0) ='0(
); (
;0) = 0(
); w(
;0) =w0(
); 't(
;0) ='1(
); t(
;0) = 1(
); wt(
;0) =w1(
);in (0;L): We dene the three wave speeds as: c1=s k1 1; c 2=s k2 2; c 3=s k3 1: In the absence of the three Kelvin-Voigt damping terms, the system (1.1) is a system of three coupled wave equations. This system is conservative whereas when at least one of the three Kelvin-Voigt damping is present, the system is dissipative. The combination of direct damping, that is damping that acts in the equation involving the unknown itself and indirect damping that acts on another unknown than the one concerns by the equation, makes this study much delicate. We note that when R!1 , then`!0 and the Bresse model reduces, by neglecting w, to the well-known Timoshenko beam equations: (1.5)8 < :1'tt[k1('x+ ) +D1('xt+ t)]x= 0; 2 tt[k2 x+D2 xt]x+k1('x+ ) +D1('xt+ t) = 0 with dierent types of boundary conditions and with initial data. 1.2.Motivation, aims and main results. The stability of elastic Bresse system with dierent types of damping (frictional, thermoelastic, Cattaneo, ...) has been intensively studied (see Subsection 1.3), but there are a few number of papers concerning the stability of Bresse or Timoshenko systems with local viscoelastic Kelvin-Voigt damping. In fact, in [7], El Arwadi and Youssef studied the theoretical and numerical stability on a Bresse system with Kelvin-Voigt damping under fully Dirichlet boundary conditions. Using multiplier techniques, they established an exponential energy decay rate provided that the system is subject to three global Kelvin-Voigt damping. Later, a numerical scheme based on the nite element method was introduced to approximate the solution. Zhao et al. in [25], considered a Timoshenko system with Dirichlet-Neumann boundary conditions. They obtained the exponential stability under certain hypotheses of the smoothness and structural condition of the coecients of the system, and obtain the strong asymptotic stability under weaker hypotheses of the coecients. Tian and Zhang in [23] considered a Timoshenko system under fully Dirichlet boundary conditions and with two locally or globally Kelvin-Voigt dampings. First, in the case when the two Kelvin-Voigt dampings are globally distributed, they showed that the corresponding semigroup is analytic. On the contrary, they proved that the energy of the system decays exponentially or polynomially and the decay rate depends on properties of material coecient function. In [9], Ghader and Wehbe generalized the results of [25] and [23]. Indeed, they considered the Timoshenko system with only one locally or globally distributed Kelvin-Voigt damping and subject to fully Dirichlet or to Dirichlet-Neumann boundary conditions. They established a polynomial energy decay rate of type t1for smooth initial data. Moreover, they proved that the obtained energy decay rate is in some sense optimal. In [19], Maryati et al. considered the transmission problem of a Timoshenko beam composed by Ncomponents, each of them being either purely elastic, or a Kelvin-Voigt viscoelastic material, or an elastic material inserted with a frictional damping mechanism. They proved that the energy decay rate depends on the position of each component. In particular, they proved that the model is exponentially stable if and only if all the elastic components are connected with one component with frictional damping. Otherwise, only a polynomial energy decay rate is established. So, the stability of the Bresse system with local viscoelastic Kelvin-Voigt damping is still an open problem. The purpose of this paper is to study the Bresse system in the presence of local non-smooth dampings coecient at interface and under fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The system is given by (1.1)-(1.2) or (1.1)-(1.3) with initial data (1.4). WhenD1,D2,D32L1(0;L), using frequency domain approach combined with multiplier techniques and the construction of new multiplier functions, we establish a polynomial stability of type1 t(see Theorem 34.2). Moreover, in the presence of only one local damping D2acting on the shear angle displacement (D1=D3= 0), we establish a polynomial energy decay estimate of type1p t(see Theorem 5.1). Finally, in the absence of at least one damping, we prove the lack of uniform stability for the system (1.1)-(1.3) even with smoothness of damping coecients. In these cases, we conjecture the optimality of the obtained decay rate. For clarity, let ;6=!= (;)(0;L): Here and thereafter, andwill be considered as interfaces. 1.3.Literature concerning the Bresse system. In [17], Liu and Rao considered the Bresse system with two thermal dissipation laws. They proved an exponential decay rate when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, they showed polynomial decays depending on the boundary conditions. These results are im- proved by Fatori and Rivera in [8] where they considered the case of one thermal dissipation law globally distributed on the displacement equation. Wehbe and Najdi in [20] extended and improved the results of [8], when the thermal dissipation is locally distributed. Wehbe and Youssef in [24] considered an elastic Bresse system subject to two locally internal dissipation laws. They proved that the system is exponentially stable if and only if the waves propagate at the same speed. Otherwise, a polynomial decay holds. Alabau et al. in [2] considered the same system with one globally distributed dissipation law. The authors proved the existence of polynomial decays with rates that depend on some particular relation between the coecients. In [10], Guesmia et al. considered Bresse system with innite memories acting in the three equations of the system. They established asymptotic stability results under some conditions on the relaxation functions regardless the speeds of propagation. These results are improved by Abdallah et al. in [1] where they considered the Bresse system with innite memory type control and/or with heat conduction given by Cattaneo's law acting in the shear angle displacement. The authors established an exponential energy decay rate when the waves propagate at same speed. Otherwise, they showed polynomial decays. In [4], Benaissa and Kasmi, considered the Bresse system with three control of fractional derivative type acting on the boundary conditions. They established a polynomial decay estimate. 1.4.Organization of the paper. This paper is organized as follows: In Section 2, we prove the well- posedness of system (1.1) with either the boundary conditions (1.2) or (1.3). Next, in Section 3, we prove the strong stability of the system in the lack of the compactness of the resolvent of the generator. In Section 4 when the coecient functions D1,D2, andD3are not smooth, we prove the polynomial stability of type1 t. In section 5, we prove the polynomial energy decay rate of type1p tfor the system in the case of only one local non-smooth damping D2acting on the shear angle displacement. In Section 6, under boundary conditions (1.3), we prove the lack of uniform (exponential) stability of the system in the absence of at least one damping. Finally in Section 7, we will brie y state the analytic stabilization of the system (1.1) when the three damping coecient act on the whole spatial domain (0 ;L) and the exponential stability when the three damping coecient are localized on ( ;) and are smooth at the interfaces. 2.Well-posedness of the problem In this part, using a semigroup approach, we establish the well-posedness result for the systems (1.1)-(1.2) and (1.1)-(1.3). Let ( '; ;w ) be a regular solution of system (1.1)-(1.2), its associated energy is given by: (2.1)E(t) =1 2ZL 0 1j'tj2+2j tj2+1jwtj2+k1j'x+ +`wj2 dx +ZL 0 k2j xj2+k3jwx`'j2 dx ; and it is dissipated according to the following law: (2.2) E0(t) =ZL 0 D1j'xt+ t+`wtj2+D2j xtj2+D3jwxt`'tj2
dx0: 4Now, we dene the following energy spaces: H1= H1 0(0;L)L2(0;L)
3andH2=H1 0(0;L)L2(0;L) H1 (0;L)L2 (0;L)
2; where L2 (0;L) =ff2L2(0;L) :ZL 0f(x)dx= 0gandH1 (0;L) =ff2H1(0;L) :ZL 0f(x)dx= 0g: Both spacesH1andH2are equipped with the inner product which induces the energy norm: (2.3)kUk2 Hj=k(v1;v2;v3;v4;v5;v6)k2 Hj =1 v2 2+2 v4 2+1 v6 2+k1 v1 x+v3+`v5 2 +k2 v3 x 2+k3 v5 x`v1 2; j= 1;2 here and afterk
kdenotes the norm of L2(0;L) . Remark 2.1. In the case of boundary condition (1.2), it is easy to see that expression (2.3) denes a norm on the energy space H1. But in the case of boundary condition (1.3) the expression (2.3) dene a norm on the energy space H2ifL6=n `for all positive integer n. Then, here and after, we assume that there does not exist any n2Nsuch thatL=n `whenj= 2. Next, we dene the linear operator AjinHjby: D(A1) = U2H 1jv2;v4;v62H1 0(0;L); k1 v1 x+v3+`v5
+D1 v2 x+v4+`v6
 x2L2(0;L);  k2v3 x+D2v4 x x2L2(0;L); k3(v5 x`v1) +D3(v6 x`v2) x2L2(0;L) ; D(A2) = U2H 2jv22H1 0(0;L);v4;v62H1 (0;L);v3 xj0;L=v5 xj0;L= 0;  k1 v1 x+v3+`v5
+D1 v2 x+v4+`v6
 x2L2(0;L);  k2v3 x+D2v4 x x2L2 (0;L); k3(v5 x`v1) +D3(v6 x`v2) x2L2 (0;L) and (2.4) Aj0 BBBBBBBB@v1 v2 v3 v4 v5 v61 CCCCCCCCA=0 BBBBBBBBB@v2 1 1 k1(v1 x+v3+`v5) +D1(v2 x+v4+`v6) x+`k3(v5 x`v1) +`D3(v6 x`v2)
v4 1 2 (k2v3 x+D2v4 x)xk1 v1 x+v3+`v5
D1(v2 x+v4+`v6)
v6 1 1 k3(v5 x`v1) +D3(v6 x`v2) x`k1 v1 x+v3+`v5
`D1(v2 x+v4+`v6)
1 CCCCCCCCCA for allU= v1;v2;v3;v4;v5;v6
T2D(Aj). So, ifU= (';'t; ; t;w;wt)Tis the state of (1.1)-(1.2) or (1.1)-(1.3), then the Bresse beam system is transformed into a rst order evolution equation on the Hilbert spaceHj: (2.5)( Ut=AjU; j = 1;2 U(0) =U0(x); where U0(x) = ('0(x);'1(x); 0(x); 1(x);w0(x);w1(x))T: Remark 2.2. It is easy to see that there exists a positive constant c0such that for any ('; ;w )2 H1 0(0;L)
3forj= 1and for any ('; ;w )2H1 0(0;L) H1 (0;L)
2forj= 2, (2.6) k1k'x+ +`wk2+k2k xk2+k3kwx`'k2c0 k'xk2+k xk2+kwxk2 : 5On the other hand, we can show by a contradiction argument the existence of a positive constant c1such that, for any ('; ;w )2 H1 0(0;L)
3forj= 1and for any ('; ;w )2H1 0(0;L) H1 (0;L)
2forj= 2, (2.7) c1 k'xk2+k xk2+kwxk2 k1k'x+ +`wk2+k2k xk2+k3kwx`'k2: Therefore the norm on the energy space Hjgiven in (2.3) is equivalent to the usual norm on Hj. Proposition 2.3. Assume that coecients functions D1,D2andD3are non negative. Then, the operator Ajis m-dissipative in the energy space Hj, forj= 1;2. Proof. LetU= v1;v2;v3;v4;v5;v6
T2D(Aj). By a straightforward calculation, we have: (2.8) Re ( AjU;U)Hj=ZL 0 D1v2 x+v4+`v62+D2v4 x2+D3v6 x`v22 dx: AsD10; D 20 andD30 , we get thatAjis dissipative. Now, we will check the maximality of Aj. For this purpose, let F= f1;f2;f3;f4;f5;f6
T2Hj;we have to prove the existence of U= v1;v2;v3;v4;v5;v6
T2D(Aj) unique solution of the equation AjU=F. Let '1;'3;'5
2 H1 0(0;L)
3forj= 1 and '1;'3;'5
2 H1 0(0;L)(H1 (0;L))2
forj= 2 be a test function. Writing AjUand replacing the rst, third and th component of Ubyf1;f3;f5and now multiplying the second, the fourth and the sixth equation by respectively '1;'3;'5, after integrating by parts, we obtain the following form: (2.9)8 >< >:k1 v1 x+v3+`v5
'1 x`k3 v5 x`v1
'1=h1; k2v3 x'3 x+k1 v1 x+v3+`v5
'3=h3; k3 v5 x`v1
'5 x+`k1 v1 x+v3+`v5
'5=h5; where h1=1f2'1+D1 f1 x+f3+`f5
'1 x`D3(f5 x`f1)'1; h3=2f4'3+D2f3 x'3 x+D1 f1 x+f3+`f5
'3;and h5=1f6'5+D3 f5`f1
'5 x+`D1 f1 x+f3+`f5
'5: Using Lax-Milgram Theorem (see [21]), we deduce that (2.9) admits a unique solution in H1 0(0;L)
3for j= 1 and in H1 0(0;L)(H1 (0;L))2
forj= 2. Thus,AjU=Fadmits an unique solution U2D(Aj) and consequently 0 2(Aj). Then,Ajis closed and consequently (Aj) is open set of C(see Theorem 6.7 in [13]). Hence, we easily get R(IAj) =Hjfor suciently small  > 0. This, together with the dissipativeness of Aj, imply that D(Aj) is dense inHjand thatAjis m-dissipative in Hj(see Theorems 4.5, 4.6 in [21]). The proof is thus complete.  Thanks to Lumer-Phillips Theorem (see [18, 21]), we deduce that Ajgenerates a C0-semigroup of contraction etAjinHjand therefore problem (2.5) is well-posed. We have thus the following result. Theorem 2.4. For anyU02Hj, problem (2.5) admits a unique weak solution U2C(R+;Hj): Moreover, if U02D(Aj);then U2C(R+;D(Aj))\C1(R+;Hj): 3.Strong stability of the system In this part, we use a general criteria of Arendt-Batty in [3] to show the strong stability of the C0- semigroupetAjassociated to the Bresse system (1.1) in the absence of the compactness of the resolvent of Aj. Before, we state our main result, we need the following stability condition: (SSC) There exist i2f1;2;3g; d0>0 and<2[0;L] such thatDid0>0 on (;): 6Theorem 3.1. Assume that condition (SSC) holds. Then the C0semigroupetAjis strongly stable in Hj, j= 1;2, i.e., for all U02Hj, the solution of (2.5) satises lim t!+1 etAjU0 Hj= 0: For the proof of Theorem 3.1, we need the following two lemmas. Lemma 3.2. Under the same condition of Theorem 3.1, we have (3.1) ker ( iAj) =f0g; j= 1;2;for all2R: Proof. We will prove Lemma 3.2 in the case D1=D3= 0 on (0;L) andD2d0>0 on (;)(0;L). The other cases are similar to prove. First, from Proposition 2.3, we claim that 0 2(Aj):We still have to show the result for 2R. Suppose that there exist a real number 6= 0 and 06=U= v1;v2;v3;v4;v5;v6
T2D(Aj) such that: (3.2) AjU=iU: Our goal is to nd a contradiction by proving that U= 0. Taking the real part of the inner product in Hj ofAjUandU, we get: (3.3) Re ( AjU;U)Hj=ZL 0D2v4 x2dx= 0: Since by assumption D2d0>0 on (;), it follows from equality (3.3) that: (3.4) D2v4 x= 0 in (0 ;L) andv4 x= 0 in (;): Detailing (3.2) we get: v2=iv1; (3.5) k1 v1 x+v3+`v5
x+`k3 v5 x`v1
=i1v2; (3.6) v4=iv3; (3.7) k2v3 x+D2v4 x
xk1 v1 x+v3+`v5
=i2v4; (3.8) v6=iv5; (3.9) k3 v5 x`v1
x`k1 v1 x+v3+`v5
=i1v6: (3.10) Next, inserting (3.4) in (3.7) and using the fact that 6= 0, we get: (3.11) v3 x= 0 in (;): Moreover, substituting equations (3.5), (3.7) and (3.9) into equations (3.6), (3.8) and (3.10), we get: (3.12)8 >< >:12v1+k1 v1 x+v3+`v5
x+`k3 v5 x`v1
= 0; 22v3+ k2v3 x+iD2v3 x
xk1 v1 x+v3+`v5
= 0; 12v5+k3 v5 x`v1
x`k1 v1 x+v3+`v5
= 0: Now, we introduce the functions bvi, fori= 1;::;6 bybvi=vi x:It is easy to see that bvi2H1(0;L). It follows from equations (3.4) and (3.11) that: (3.13) bv3=bv4= 0 in (;) and consequently system (3.12) will be, after dierentiating it with respect to x, given by: 12bv1+k1 bv1 x+`bv5
x+`k3 bv5 x`bv1
= 0 in ( ;); (3.14) bv1 x+`bv5= 0 in ( ;); (3.15) 12bv5+k3 bv5 x`bv1
x`k1 bv1 x+`bv5
= 0 in ( ;): (3.16) 7Furthermore, substituting equation (3.15) into (3.14) and (3.16), we get: 12bv1+`k3 bv5 x`bv1
= 0 in ( ;); (3.17) bv1 x+`bv5= 0 in ( ;); (3.18) 12bv5+k3 bv5 x`bv1
x= 0 in ( ;): (3.19) Dierentiating equation (3.17) with respect to x, a straightforward computation with equation (3.19) yields: 12 bv1 x`bv5
= 0 in (;): Equivalently (3.20) bv1 x`bv5= 0 in (;): Hence, from equations (3.18) and (3.20), we get: (3.21) bv5= 0 andbv1 x= 0 in (;): Pluggingbv5= 0 in (3.17), we get: (3.22) 12`2k3
bv1= 0: In order to nish our proof, we have to distinguish two cases: Case 1:6=`rk3 1. Using equation (3.22) , we deduce that: bv1= 0 in (;): SettingV= bv1;bv1 x;bv3;bv3 x;bv5;bv5 x
T. By continuity of bvion (0;L), we deduce that V() = 0. Then system (3.12) could be given as: (3.23)Vx=BV; in (0;) V() = 0 ; where (3.24) B=0 BBBBBBBBBBB@0 1 0 0 0 0 21+`2k3 k10 0 1`(k1+k3) k10 0 0 0 1 0 0 0k1 k2+iD 2k122 k2+iD 20`k1 k2+iD 20 0 0 0 0 0 1 0`(k3+k1) k3`k1 k30`2k121 k301 CCCCCCCCCCCA: Using ordinary dierential equation theory, we deduce that system (3.23) has the unique trivial solution V= 0 in (0;). The same argument as above leads us to prove that V= 0 on (;L). Consequently, we obtainbv1=bv3=bv5= 0 on (0;L). It follows that bv2=bv4=bv6= 0 on (0;L), thusbU= 0. This gives that U=C, whereCis a constant. Finally, from the boundary condition (1.2) or (1.3), we deduce that U= 0. Case 2:=`rk3 1. The fact that bv1 x= 0 on (;), we getbv1=con (;), wherecis a constant. By continuity of bv1on (0;L), we deduce that bv1() =c. We know also that bv3=bv5= 0 on (;) from (3.13) and (3.21). Hence, setting V() = (c;0;0;0;0;0)T=V0, we can rewrite system (3.12) on (0 ;) under the form:  Vx=bBV; V() =V0; 8where bB=0 BBBBBBBBBBBBB@0 1 0 0 0 0 0 0 0 1`(k1+k3) k10 0 0 0 1 0 0 0k1 k2+i`q k3 1D2k122 k2+i`q k3 1D 20`k1 k2+i`q k3 1D20 0 0 0 0 0 1 0`(k3+k1) k3`k1 k30`2(k1k3) k301 CCCCCCCCCCCCCA: IntroducingeV= bv1 x;bv3;bv3 x;bv5;bv5 x
Tand eB=0 BBBBBBBBBBB@0 0 1`(k1+k3) k10 0 0 1 0 0 k1 k2+i`q k3 1D2k122 k2+i`q k3 1D 20`k1 k2+i`q k3 1D20 0 0 0 0 1 `(k3+k1) k3`k1 k30`2(k1k3) k301 CCCCCCCCCCCA: Then system (3.12) could be given as: (3.25)( eVx=eBeV;in (0;); eV() = 0 : Using ordinary dierential equation theory, we deduce that system (3.25) has the unique trivial solution eV= 0 in (0;). This implies that on (0 ;), we havebv3=bv5= 0. Consequently, v3=c3andv5=c5where c3andc5are constants. But using the fact that v3(0) =v5(0) = 0, we deduce that v3=v5= 0 on (0;). Substituting v3andv5by their values in the second equation of system (3.12), we get that v1 x= 0. This yieldsv1=c1, wherec1is a constant. But as v1(0) = 0, we get: v1= 0 on (0;). ThusU= 0 on (0;). The same argument as above leads us to prove that U= 0 on (;L) and therefore U= 0 on (0;L). Thus the proof is complete.  Lemma 3.3. Under the same condition of Theorem 3.1, (iIAj);j= 1;2is surjective for all 2R. Proof. We will prove Lemma 3.3 in the case D1=D3= 0 on (0;L) andD2d0>0 on (;)(0;L) and the other cases are similar to prove. Since 02(Aj), we still need to show the result for 2R. For any F= f1;f2;f3;f4;f5;f6
T2Hj; 2R; we prove the existence of U= v1;v2;v3;v4;v5;v6
T2D(Aj) solution of the following equation: (3.26) ( iIAj)U=F: Equivalently, we have the following system: iv1v2=f1; (3.27) 1iv2k1 v1 x+v3+`v5
x`k3 v5 x`v1
=1f2; (3.28) iv3v4=f3; (3.29) 2iv4(k2v3 x+D2v4 x)x+k1 v1 x+v3+`v5
=2f4; (3.30) iv5v6=f5; (3.31) 1iv6k3 v5 x`v1
x+`k1 v1 x+v3+`v5
=1f6: (3.32) 9From (3.27),(3.29) and (3.31), we have: (3.33) v2=iv1f1; v3=iv3f3; v6=iv5f5: Inserting (3.33) in (3.28), (3.30) and (3.32), we get: (3.34)8 >< >:2v1k11 1 v1 x+v3+`v5
x`k31 1 v5 x`v1
=h1; 2v31 2(k2+iD 2)v3 xx+k11 2 v1 x+v3+`v5
=h3; 2v5k31 1 v5 x`v1
x+`k11 1 v1 x+v3+`v5
=h5; where h1=f2+if1; h3=f4+if31 2D2f3 xx; h5=f6+if5: For allv= v1;v3;v5
T2 H1 0(0;L)
3forj= 1 andv= v1;v3;v5
T2H1 0(0;L)H1 (0;L)2forj= 2, we dene the linear operator Lby: Lv=0 B@k11 1 v1 x+v3+`v5
x`k31 1 v5 x`v1
1 2(k2+iD 2)v3 xx+k11 2 v1 x+v3+`v5
k31 1 v5 x`v1
x+`k11 1 v1 x+v3+`v5
1 CA: For clarity, we consider the case j= 1. The proof in the case j= 2 is very similar. Using Lax-Milgram theo- rem, it is easy to show that Lis an isomorphism from ( H1 0(0;L))3onto (H1(0;L))3. Letv= v1;v3;v5
T andh= h1;h3;h5
T, then we transform system (3.34) into the following form: (3.35) ( 2IL )v=h: Since the operator Lis an isomorphism from ( H1 0(0;L))3onto (H1(0;L))3andIis a compact operator from (H1 0(0;L))3onto (H1(0;L))3, then using Fredholm's Alternative theorem, problem (3.35) admits a unique solution in ( H1 0(0;L))3if and only if 2IL is injective. For that purpose, let ~ v= ~v1;~v3;~v5
Tin ker(2IL ). Then, if we set ~ v2=i~v1, ~v4=i~v3and ~v6=i~v5, we deduce that ~V= (~v1;~v2;~v3;~v5;~v6) belongs toD(A1) and it is solution of: (iIA 1)~V= 0: Using Lemma 3.2, we deduce that ~ v1= ~v3= ~v5= 0. This implies that equation (3.35) admits a unique solution in v= (v1;v3;v5)2(H1 0(0;L))3and k11 1 v1 x+v3+`v5
x`k31 1 v5 x`v1
2L2(0;L); 1 2(k2v3 x+iD 2v3 xD2f3 x)x+k11 2 v1 x+v3+`v5
2L2(0;L); k31 1 v5 x`v1
x+`k11 1 v1 x+v3+`v5
2L2(0;L): By settingv2=iv1f1,v3=iv3f3andv6=iv5f5, we deduce that V= (v1;v2;v3;v4;v5;v6) belongs toD(A1) and it is the unique solution of equation (3.26) and the proof is thus complete.  Proof of Theorem 3.1. Following a general criteria of Arendt-Batty in [3], the C0semigroupetAjof con- tractions is strongly stable if Ajhas no pure imaginary eigenvalues and (Aj)\iRis countable. By Lemma 3.2, the operator Ajhas no pure imaginary eigenvalues and by Lemma 3.3, R( iAj) =Hjfor all2R. Therefore the closed graph theorem of Banach implies that (Aj)\iR=;. Thus, the proof is complete.  4.Polynomial stability for non smooth damping coefficients at the interface Before we state our main result, we recall the following results (see [11], [22] for part i), [5] for ii)and [21] for iii). Theorem 4.1. LetA:D(A)H!H be an unbounded operator generating a C0-semigroup of contractions etAonH. Assume that i2(A), for all2R. Then, the C0-semigroup etAis: i) Exponentially stable if and only if lim jj!+1 sup 2Rk(iIA)1kL(H) <+1: 10ii) Polynomially stable of order1 l(l>0)if and only if lim jj!+1 sup 2Rjjlk(iIA)1kL(H) <+1: iii) Analytically stable if and only if lim jj!+1 sup 2Rjjk(iIA)1kL(H) <+1: It was proved that, see [6, 16], the stabilization of wave equation with local Kelvin-Voigt damping is greatly in uenced by the smoothness of the damping coecient and the region where the damping is localized (near or faraway from the boundary) even in the one-dimensional case. So, in this section, we consider the Bresse systems (1.1)-(1.2) and (1.1)-(1.3) subject to three local viscoelastic Kelvin-Voigt dampings with non smooth coecients at the interface. Using frequency domain approach combined with multiplier techniques and the construction of a new multiplier function, we establish the polynomial stability of the C0-semigroup etAj, j= 1;2. For this purpose, let ;6= (i;i)(0;L),i= 1;2;3;be an arbitrary nonempty open subsets of (0;L). We consider the following stability condition: (4.1)9di 0>0 such that Didi 0>0 in (i;i); i= 1;2;3;and3\ i=1(i;i) = (;)6=;: Our main result in this section can be given by the following theorem: Theorem 4.2. Assume that condition (4.1) holds. Then, there exists a positive constant c >0such that for allU02D(Aj),j= 1;2;the energy of the system satises the following decay rate: (4.2) E(t)c tkU0k2 D(Aj): Referring to [5], (4.2) is veried if the following conditions (H1) iR(Aj) and (H3) lim !+1sup 2R1 2 (iIdAj)1 L(Hj) =O(1) hold. Condition iR(Aj) is already proved in Lemma 3.2 and Lemma 3.3. We will establish (H3) by contradiction. Suppose that there exist a sequence of real numbers ( n)n, with jnj!+1and a sequence of vectors (4.3) Un= v1 n;v2 n;v3 n;v4 n;v5 n;v6 n
T2D(Aj) withkUnkHj= 1 such that (4.4) 2 n(inUnAjUn) = f1 n;f2 n;f3 n;f4 n;f5 n;f6 n
T!0 inHj; j = 1;2: We will check the condition (H3) by nding a contradiction with (4.3)-(4.4) such as kUnkHj=o(1). 11Equation (4.4) is detailed as: inv1 nv2 n=f1 n 2n; (4.5) i1nv2 n k1 v1 n
x+v3 n+`v5 n
+D1 v2 n
x+v4 n+`v6 n
 x `k3 v5 n
x`v1 n `D3 v6 n
x`v2 n =1f2 n 2n; (4.6) inv3 nv4 n=f3 n 2n; (4.7) i2nv4 n k2 v3 n
x+D2 v4 n
x x+k1 v1 n
x+v3 n+`v5 n +D1 v2 n
x+v4 n+`v6 n =2f4 n 2n; (4.8) inv5 nv6 n=f5 n 2n; (4.9) i1nv6 n k3 v5 n
x`v1 n
+D3 v6 n
x`v2 n
 x+`k1 v1 n
x+v3 n+`v5 n +`D1 v2 n
x+v4 n+`v6 n =1f6 n 2n: (4.10) From (4.3), (4.5), (4.7) and (4.9), we deduce that: (4.11) kv1 nk=O(1 n);kv3 nk=O(1 n);kv5 nk=O(1 n): For clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n. Lemma 4.3. Under all the above assumptions, we have: (4.12)kD1=2 1 v2 x+v4+`v6
k=o(1) ;kD1=2 2v4 xk=o(1) ;kD1=2 3 v6 x`v2
k=o(1)  and (4.13)kv2 x+v4+`v6k=o(1) ;kv4 xk=o(1) ;kv6 x`v2k=o(1) in (;): Proof. Taking the inner product of (4.4) with UinHj, we get: Re i3kUk22(AjU;U)
Hj=2Re (AjU;U)Hj =2ZL 0 D1jv2 x+v4+`v6j2+D2jv4 xj2+D3jv6 x`v2j2
dx=o(1): (4.14) Thanks to (4.1), we obtain the desired asymptotic equations (4.12) and (4.13). Thus the proof is complete.  Remark 4.4. These estimates are crucial for the rest of the prooof and they will be used to prove each point of the global proof divided in several lemmas. Lemma 4.5. Under all the above assumptions, we have: (4.15)kv1 x+v3+`v5k=o(1) 2;kv3 xk=o(1) 2;kv5 x`v1k=o(1) 2in (;): Proof. First, using equations (4.5), (4.7) and (4.9), we obtain: (4.16)  v1 x+v3+`v5
=i(v2 x+f1 x 2+v4+f3 2+`v6+`f5 2): Consequently, (4.17)Z 2jv1 x+v3+`v5j2dx2Z jv2 x+v4+`v6j2dx+ 2Z jf1 x+f3+`f5j2 4dx: 12Using the rst estimate of (4.13) and the fact that f1,f3,f5converge to zero in H1 0(0;L) (or inH1 ?(0;L)) in (4.17), we deduce: (4.18)Z 2jv1 x+v3+`v5j2dx=o(1) 2: In a similar way, one can prove: (4.19)Z 2jv3 xj2dx=o(1) 2andZ 2jv5 x`v1j2dx=o(1) 2: The proof is thus complete.  Here and after designates a xed positive real number such that 0 <+<<L . Then, we dene the cut-o function 2C1 c(R) by: = 1 on [+;];01; = 0 on (0 ;L)n(;): Lemma 4.6. Under all the above assumptions, we have: (4.20)Z +v12dx=o(1) ;Z +v32dx=o(1) ;Z +v52dx=o(1) : Proof. First, multiplying equation (4.5) by iv1inL2(0;L) and integrating by parts, we get: (4.21) ZL 0v12dxiZL 0v2v1dx=iZL 0f1 2v1dx: Asv1is uniformly bounded in L2(0;L) andf1converges to zero in H1 0(0;L), we get that the term on the right hand side of (4.21) converges to zero and consequently (4.22) ZL 0v12dxiZL 0v2v1dx=o(1) 2: Moreover, multiplying (4.6) by 1 1v1inL2(0;L), then integrating by parts we obtain: iZL 0v2v1dx+1 1ZL 0 k1 v1 x+v3+`v5
+D1 v2 x+v4+`v6

 v1 xdx `k31 1ZL 0 v5 x`v1
v1dx`1 1ZL 0D3 v6 x`v2
v1dx=ZL 0f2 2v1dx: (4.23) Using (4.12), (4.15), the fact that f2converges to zero in L2(0;L) andv1,v1 xare uniformly bounded in L2(0;L) in (4.23), we get: (4.24) iZL 0v2v1dx=o(1) : Finally, using (4.24) in (4.22) and the deniton of , we get: ZL 0v12dx=o(1) ;Z +v12dx=o(1) : In a same way, we show: Z +v32dx=o(1) ;Z +v52dx=o(1) : The proof is thus complete.  Now, we introduce new multiplier functions. For this purpose, let ;6=!= (+;): 13Lemma 4.7. The solution (u;y;z )of the following system (4.25)8 < :12u+k1(ux+y+`z)x+`k3(zx`u)i1!u=v1; 22y+k2yxxk1(ux+y+`z)i1!y =v3; 12z+k3(zx`u)x`k1(ux+y+`z)i1!z=v5 with fully Dirichlet boundary conditions: (4.26) u(0) =u(L) =y(0) =y(L) =z(0) =z(L) = 0 or with Dirichlet-Neumann-Neumann boundary conditions: (4.27) u(0) =u(L) =yx(0) =yx(L) =zx(0) =zx(L) = 0 veries the following inequality: ZL 0 1juj2+2jyj2+1jzj2+k2jyxj2 +k1jux+y+`zj2+k3jzx`uj2 dxCZL 0 jv1j2+jv3j2+jv5j2
dx; (4.28) whereCis a constant independent of n. Proof. We consider the following Bresse system subject to three local viscous dampings: (4.29)8 >>>>< >>>>:1uttk1(ux+y+`z)xlk3(zx`u) +1!ut= 0; 2yttk2yxx+k1(ux+y+`z) +1!yt = 0; 1zttk3(zx`u)x+`k1(ux+y+`z) +1!zt= 0 with fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. Systems (4.29)-(4.26) and (4.29)- (4.27) are well posed in the space H1= H1 0(0;L)L2(0;L)
3and in the space H2= H1 0(0;L)L2(0;L)
 H1 (0;L)L2 (0;L)
2respectively. In addition, both are exponentially stable (see [24]). Therefore, following Huang [11] and Pruss [22], we deduce that the resolvent of the associated operator: Aauxj:D(Aauxj)Hj!Hj dened by D(Aaux 1) = H1 0( )\H2( )
3 H1 0( )
3; D(Aaux 2) = U2H2:u2H1 0\H2;y;z2H1 ?\H2;~u;yx;zx2H1 0;~y;~z2H1 ? and Aauxj0 BBBBBBBB@u ~u y ~y z ~z1 CCCCCCCCA=0 BBBBBBBBB@~u 1 1[k1(ux+y+`z)x+`k3(zx`u)1!~u] ~y 1 2[k2yxxk1(ux+y+`z)1!~y] ~z 1 1[k3(zx`u)x`k1(ux+z+`z)1!~z]1 CCCCCCCCCA is uniformly bounded on the imaginary axis. So, by setting ~ u=iu, ~y=iyand ~z=iz, we deduce that: 0 BBBBBBBB@u ~u y ~y z ~z1 CCCCCCCCA= iAauxj
10 BBBBBBBBB@0 1 1v1 0 1 2v3 0 1 1v11 CCCCCCCCCA: 14This yields: k(u;~u;y;~y;z;~z)k2 Hjk iAauxj
1kL(Hj)k(0;1 1v1;0;1 2v3;0;1 1v5)kHj CZL 0 jv1j2+jv3j2+jv5j2
dx; (4.30) whereCis a constant independent of n. Consequently, (4.28) holds. The proof is thus complete.  Lemma 4.8. Under all the above assumptions, we have: (4.31)ZL 0jv1j2dx=o(1);ZL 0jv3j2dx=o(1);ZL 0jv5j2dx=o(1): Proof. For clarity of the proof, we divide the proof into several steps. Step 1. First, multiplying (4.5) by i1u, whereuis a solution of system (4.25), we get: (4.32) ZL 012uv1dxiZL 01uv2dx=1ZL 0if1 udx: Moreover, multiplying (4.6) by uand integrating by parts, we obtain: iZL 01uv2dxZL 0k1uxxv1dxZL 0`k3(`u)v1dx+ZL 0k1uxv3dx+ZL 0`k1uxv5dx +ZL 0`k3uxv5dx+ZL 0D1(v2 x+v4+`v6)uxdxZL 0`D3(v6 x`v2)udx=1ZL 0f2 2udx: (4.33) Now, combining (4.32) and (4.33), we get: ZL 0 12u+k1uxx+`k3(`u) v1dxZL 0k1uxv3dxZL 0`k1uxv5dxZL 0`k3uxv5dx ZL 0D1(v2 x+v4+`v6)uxdx+ZL 0`D3(v6 x`v2)udx=1ZL 0if1 +f2 2 udx: (4.34) Step 2. Similarly to Step 1, multiplying (4.7) by i2yand (4.8) by y, whereyis a solution of system (4.25), we get: ZL 0 22y+k2yxxk1y v3dx+ZL 0k1yxv1dxZL 0`k1yv5dx ZL 0D2v4 xyxdxZL 0D1(v2 x+v4+`v6)ydx=2ZL 0if3 +f4 2 ydx: (4.35) Step 3. As in Step 1 and Step 2, by multiplying (4.9) by i1zand (4.10) by z, wherezis a solution of system (4.25), we get: ZL 0 12z+k3zxx`k1(`z) v5dx+ZL 0`k3zxv1dx+ZL 0`k1zxv1dx ZL 0`k1zv3dxZL 0D3 v6 x`v2
zxdx`ZL 0D1(v2 x+v4+`v6)zdx=1ZL 0if5 +f6 2 zdx:(4.36) 15Step 4. First, combining (4.34), (4.35) and (4.36), we obtain: ZL 0 12u+k1(ux+y+`z)x+`k3(zx`u) v1dx+ZL 0 22y+k2yxxk1(ux+y+`z) v3dx +ZL 0 12z+k3(zx`u)x`k1(ux+y+`z) v5dxZL 0D1(v2 x+v4+`v6)uxdx +ZL 0`D3(v6 x`v2)udxZL 0D2v4 xyxdxZL 0D1(v2 x+v4+`v6)ydxZL 0D3 v6 x`v2
zxdx (4.37) `ZL 0D1(v2 x+v4+`v6)zdx=1ZL 0if1 +f2 2 udx2ZL 0if3 +f4 2 ydx 1ZL 0if5 +f6 2 zdx: Combining equation (4.25) and (4.37), multiplying by 2, we get: ZL 0jv1j2dx+ZL 0jv3j2dx+ZL 0jv5j2dx=iZ +(2uv1dx+2yv3+2zv5)dx +ZL 0D 1(v2 x+v4+`v6)uxdxZL 0`D3(v6 x`v2)2udx+ZL 0D 2v4 xyxdx +ZL 0D1(v2 x+v4+`v6)2ydx+ZL 0D 3 v6 x`v2
zxdx+`ZL 0D1(v2 x+v4+`v6)2zdx (4.38) 1ZL 0 if1+f2
udx2ZL 0 if3+f4
ydx1ZL 0 if5+f6
zdx: Using estimates (4.20) and the fact that 2u,2yand2zare uniformly bounded in L2(0;L) due to (4.28), we get: (4.39) iZ +(2uv1dx+2yv3+2zv5)dx=o(1) 1=2: In addition, using (4.12) and the fact that ux,yxandzxare uniformly bounded in L2(0;L) due to (4.28). we get: (4.40)ZL 0D 1(v2 x+v4+`v6)uxdx+ZL 0D 2v4 xyxdxZL 0D 3 v6 x`v2
zxdx=o(1): Also, by using (4.12) and the fact that 2u,2yand2zare uniformly bounded in L2(0;L) due to (4.28), we obtain: (4.41)ZL 0`D3(v6 x`v2)2udx+ZL 0D1(v2 x+v4+`v6)2ydx+`ZL 0D1(v2 x+v4+`v6)2zdx=o(1) : Moreover, we have: (4.42)1ZL 0 if1+f2
udx2ZL 0 if3+f4
ydx1ZL 0 if5+f6
zdx=o(1); sincef1,f3,f5converge to zero in H1 0(0;L) (or inH1 ?(0;L)),f2,f4,f6converge to zero in L2(0;L), and 2u,2y,2zare uniformly bounded in L2(0;L). Finally, inserting (4.39) - (4.42) into (4.38), we get the desired estimates in (4.31). Thus the proof is complete.  Lemma 4.9. Under all the above assumptions, we have: (4.43)ZL 0jv1 xj2dx=o(1);ZL 0jv3 xj2dx=o(1);ZL 0jv5 xj2dx=o(1): 16Proof. First, multiplying (4.6) by v1and then integrating by parts, we get: iZL 01v2v1dx+k1ZL 0jv1 xj2dx+k1ZL 0 v3+`v5
v1xdx+ZL 0D1 v2 x+v4+`v6
v1xdx `k3ZL 0 v5 x`v1
v1dx`ZL 0D3 v6 x`v2
v1dx=1ZL 0f2 2v1dx: (4.44) Then, using (4.11), (4.12) and the fact that v1 x, v5 x`v1
are uniformly bounded in L2(0;L) due to (4.3), we obtain: k1ZL 0 v3+`v5
v1xdx+ZL 0D1 v2 x+v4+`v6
v1xdx `k3ZL 0 v5 x`v1
v1dx`ZL 0D3 v6 x`v2
v1dx=o(1): (4.45) Asf2converges to zero in L2(0;L) andv1is uniformly bounded in L2(0;L), we have: (4.46) 1ZL 0f2 2v1dx=o(1): Next, inserting (4.45) and (4.46) into (4.44), we get: iZL 01v2v1dx+k1ZL 0jv1 xj2dx=o(1): (4.47) Using Lemma 4.8 and the fact that v2is uniformly bounded in L2(0;L) due to (4.47), we deduce: ZL 0jv1 xj2dx=o(1): Similarly, one can prove that: ZL 0jv3 xj2dx=o(1);ZL 0jv5 xj2dx=o(1): Thus, the proof is complete.  Proof of Theorem 4.2. Using Lemma 4.8 and Lemma 4.9, we get that kUkHj=o(1). Therefore, we get a contradiction with (4.3) and consequently (H3) holds. Thus the proof is complete  Remark 4.10. It is known that for a single one-dimensional wave equation with damping coecient D1= d0>0on!, the optimal solution decay rate is 1=t2. The new multipliers (one for each equation) we have used here, dened by system (4.25) , do not permit to obtain a decay rate of 1=t2but only 1=t. This may be due to the coupling eects and we do not know if this decay rate of 1=tis optimal. 5.The case of only one local viscoelastic damping with non smooth coefficient at the interface In control theory, it is important to reduce the number of control such as damping terms. So, this section is devoted to show the polynomial stability of systems (1.1)-(1.2) and (1.1)-(1.3) subject to only one viscoelastic Kelvin-Voigt damping with non smooth coecient at the interface. For this purpose, we consider the following condition: (5.1)D1=D3= 0 in (0;L) and9d0>0 such that D2d0>0 in;6= (;)(0;L): The main result of this section is given by the following theorem: Theorem 5.1. Assume that condition (5.1) is satised. Then, there exists a positive constant c >0such that for all U02D(Aj),j= 1;2;the energy of system (1.1) satises the following decay rate: (5.2) E(t)cp tkU0k2 D(Aj): 17Referring to [5], (5.2) is veried if the following conditions (H1) iR(Aj) and (H4) lim jj!+1sup 2R1 4 (iIAj)1 L(Hj) =O(1) hold. Condition iR(Aj) is already proved in Lemma 3.2 and Lemma 3.3. We will establish (H4) by contradiction. Suppose that there exist a sequence of real numbers ( n)n, with jnj!+1and a sequence of vectors (5.3) Un= v1 n;v2 n;v3 n;v4 n;v5 n;v6 n
T2D(Aj) withkUnkHj= 1 such that (5.4) 4 n(inUnAjUn) = f1 n;f2 n;f3 n;f4 n;f5 n;f6 n
T!0 inHj; j = 1;2: We will check the condition (H4) by nding a contradiction with (5.3)-(5.4) such as kUnkHj=o(1). Equation (5.4) is detailed as: inv1 nv2 n=f1 n 4n; (5.5) i1nv2 nk1 v1 n
x+v3 n+`v5 n x`k3 v5 n
x`v1 n =1f2 n 4n; (5.6) inv3 nv4 n=f3 n 4n; (5.7) i2nv4 n k2 v3 n
x+D2 v4 n
x x+k1 v1 n
x+v3 n+`v5 n =2f4 n 4n; (5.8) inv5 nv6 n=f5 n 4n; (5.9) i1nv6 n k3 v5 n
x`v1 n
 x+`k1 v1 n
x+v3 n+`v5 n =1f6 n 4n: (5.10) Inserting (5.5), (5.7), and (5.9) into (5.6),(5.8) and (5.10) respectively, we get 12 nv1 n+k1 v1 n
x+v3 n+`v5 n x+`k3 v5 n
x`v1 n =i1f1 n 3n1f2 n 4n; (5.11) 22 nv3 n+ k2 v3 n
x+D2 v4 n
x xk1 v1 n
x+v3 n+`v5 n =i2f3 n 3n2f4 n 4n; (5.12) 12 nv5 n+ k3 v5 n
x`v1 n
 x`k1 v1 n
x+v3 n+`v5 n =i1f5 n 3n1f6 n 4n: (5.13) From (5.5), (5.7), (5.9) and (5.3), we deduce that: (5.14) kv1 nk=O(1 n);kv3 nk=O(1 n);kv5 nk=O(1 n): For clarity, we divide the proof into several lemmas. From now on, for simplicity, we drop the index n. Lemma 5.2. Under all the above assumptions, we have: (5.15)ZL 0D2jv4 xj2dx=o(1) 4;Z jv4 xj2dx=o(1) 4 and (5.16)ZL 0jv3 xj2dx=o(1) 6;Z jv3 xj2dx=o(1) 6: 18Proof. Taking the inner product of (5.4) with UinHj, we get: Re i5kUk24(AjU;U)
Hj=4Re (AjU;U)Hj=4ZL 0D2jv4 xj2dx=o(1): (5.17) Thanks to (5.1), we obtain the desired asymptotic equation (5.15). Next, dierentiating equation (5.7), we get: iv3 x=v4 x+f3 x 4; and consequentlyZ jv3 xj2dx2Z jv4 xj2dx+ 2Z jf3 xj2 8dx: Using (5.15) and the fact that f3converges to zero in H1 0(0;L) (or inH1 (0;L) ) in the above equation, we get the desired estimate (5.16). Thus the proof is complete.  Remark 5.3. Again, these estimates are crucial for the rest of the proof and they will be used to prove each point of the global proof divided in several lemmas. Lemma 5.4. Under all the above assumptions, we have: (5.18)ZL 0jv3j2dx=O(1) 2andZ +jv3j2dx=O(1) 2: Proof. First, multiplying (5.12) by 1 2v3and integrating by parts, we get: ZL 0jv3j2dx=1 2ZL 0 k2v3 x+D2v4 x
 0v3+v3x dx+1 2ZL 0k1 v1 x+v3+`v5
v3dx ZL 0if3 3v3dxZL 0f4 4v3dx: (5.19) Then, using (5.15), (5.16), kv3k=O(1 ) and the fact that f3,f4converge to zero in H1 0(0;L) (or inH1 ?(0;L)), L2(0;L) respectively, we deduce that: 1 2ZL 0 k2v3 x+D2v4 x
 0v3+v3x dxZL 0if3 3v3dxZL 0f4 4v3dx=o(1) 3: (5.20) Next, inserting (5.20) into (5.19), we obtain: ZL 0jv3j2dx=1 2ZL 0k1 v1 x+v3+`v5
v3dx+o(1) 3: Using Cauchy-Shwartz and Young's inequalities in the above equation, we get: ZL 0jv3j2dx2j1 2j2ZL 0k2 1v1 x+v3+`v52 2dx+1 2ZL 0jv3j2dx+o(1) 3; Consequently, 1 2ZL 0jv3j2dx2j1 2j2ZL 0k2 1v1 x+v3+`v52 2dx+o(1) 3: Finally, using the fact that v1 x+v3+`v5
is uniformly bounded in L2(0;L) and the denition of , we get the desired estimates in (5.18) and the proof is thus complete.  Lemma 5.5. Under all the above assumptions, we have: (5.21)ZL 0jv1 xj2dx=o(1);Z +jv1 xj2dx=o(1) and (5.22)ZL 0jv1j2dx=o(1);Z +jv1j2dx=o(1): 19Proof. Our rst aim here is to proveZL 0jv1 xj2dx=o(1): For this sake, multiplying (5.8) by v1xand integrating by parts, we get: iZL 02v40v1dxiZL 02v4 xv1dx+ZL 0(k2v3 x+D2v4 x)(v1xx)dx+ZL 0(k2v3 x+D2v4 x)(0v1x)dx +ZL 0k1jv1 xj2dx+ZL 0k1v3v1xdx+ZL 0`k1v5v1xdx=ZL 02f4 4v1xdx: (5.23) Now, we need to estimate each term of (5.23): Using (5.14), (5.18) and the fact that f3converges to zero in H1 0(0;L) (orH1 (0;L)), we get: (5.24)iZL 02v40v1dx=iZL 02(iv3f3 4)0v1=ZL 022v30v1+iZL 0f3 30v1=o(1): Using (5.15) and the fact that v1is uniformly bounded in L2(0;L), we obtain: (5.25) iZL 02v4 xv1dx=o(1) 2: From (5.6), we remark that1 v1 xxis uniformly bounded in L2(0;L). This fact combined with (5.15) and (5.16) yields (5.26)ZL 0(k2v3 x+D2v4 x)(v1xx )dx=o(1) : Using (5.15), (5.16) and the fact that v1 xis uniformly bounded in L2(0;L), we get: (5.27)ZL 0(k2v3 x+D2v4 x)(0v1x)dx=o(1) 2: Using (5.14) and the fact that v1 xis uniformly bounded in L2(0;L), we obtain: (5.28)ZL 0k1v3v1xdx+ZL 0`k1v5v1xdx=o(1): Using the fact that f4converges to zero in L2(0;L) andv1 xis uniformly bounded in L2(0;L), we get: (5.29)ZL 02f4 4v1xdx=o(1) 4: Finally, inserting equations (5.24)-(5.29) into (5.23) and using the denition of , we get the desired estimates in (5.21). Next, our second aim is to proveZL 0jv1j2dx=o(1): For this, multiplying (5.11) by 1 1v1and integrating by parts, we get: ZL 0jv1j2dx=1 1ZL 0k1(v1 x+v3+`v5)(0v1+v1x)dx1 1ZL 0`k3(v5 x`v1)v1 (5.30) ZL 0f2 4+if1 3 v1dx: So, using (5.14), (5.21), the fact that ( v1 x+v3+`v5), (v5 x`v1) are uniformly bounded in L2(0;L) andf1, f2converge respectively to zero in H1 0(0;L),L2(0;L) in the right hand side of the above equation and using the denition of , we get the desird estimates in (5.22).  20Lemma 5.6. Under all the above assumptions, we have: (5.31)ZL 0jv1 xj2dx=o(1) 2;Z +jv1 xj2dx=o(1) 2 and (5.32)ZL 0jv1j2dx=o(1) 2Z +jv1j2dx=o(1) 2: Proof. For the clarity of the proof, we divide the proof into several steps: Step 1. In this step, we will prove (5.33) 1ZL 0jv1j2dxk1ZL 0jv1 xj2dx+`(k1+k3)Re(ZL 0v5 xv1dx) =o(1) 2: For this sake, multiplying (5.11) by v1and integrating by parts, we get: 1ZL 0jv1j2dxk1ZL 0jv1 xj2dxk1ReZL 00v1 xv1dx +k1Re(ZL 0v3 xv1dx) +`(k1+k3)Re(ZL 0v5 xv1dx) `2k3ZL 0jv1j2dx=o(1) 4: (5.34) Now, we need to estimate some terms of (5.34) as follows: We get after integrating by parts k1ReZL 00v1 xv1dx =k1 2ZL 00(jv1j2)xdx=k1 2ZL 000jv1j2dx: Using (5.22) in the previous equation, we obtain: (5.35) k1ReZL 00v1 xv1dx =o(1) 2: Using (5.16) and (5.22), we deduce that (5.36) k1Re(ZL 0v3 xv1dx) `2k3ZL 0jv1j2dx=o(1) 2: Finally, inserting (5.35) and (5.36) in (5.34), we get the desired estimate (5.33). Step 2. In this step, we will prove k1+k3 k1 ReZL 0(k2v3 x+D2v4 x)v1xx)dx + (k1+k3)ZL 0jv1 xj2dx `(k1+k3)ReZL 0v5 xv1dx =o(1) 2: (5.37) In order to prove (5.37), multiplying (5.12) by the multiplier k1+k3 k1 v1xand integrating by parts, we get: 2k1+k3 k1 ReZL 02v3v1xdx | {z } I1k1+k3 k1 ReZL 0(k2v3 x+D2v4 x)xv1xdx | {z } I2 + (k1+k3)ZL 0jv1 xj2dx+ (k1+k3)ReZL 0v3v1xdx |{z } I3`(k1+k3)ReZL 0v5 xv1dx =o(1) 2: (5.38) 21Next, we need to estimate I1;I2andI3. Integrating by parts I1and then using (5.16), (5.18) and (5.22), we deduce that: (5.39) I1=2k1+k3 k1 ReZL 0 0v3v1+v3 xv1 dx =o(1) 2: Integrating by parts I2and then using (5.15), (5.16) and (5.21), we get: I2=k1+k3 k1 ReZL 0 k2v3 x+D2v4 x
0v1xdx dx+k1+k3 k1 ReZL 0(k2v3 x+D2v4 x)v1xxdx dx =k1+k3 k1 ReZL 0(k2v3 x+D2v4 x)v1xxdx dx+o(1) 2: (5.40) By using (5.18) and (5.21), we deduce that (5.41) I3= (k1+k3)ReZL 0v3v1xdx =o(1) 2: Finally, inserting (5.39), (5.40), and (5.41) into (5.38), we get the desired estimate (5.37). Step3. Combining (5.33) and (5.37), we get (5.42) 1ZL 0jv1j2dx+k3ZL 0jv1 xj2dx+k1+k3 k1 ReZL 0 k2v3 x+D2v4 x
v1xxdx =o(1) 2: Step 4. In this step, we conclude the proof of the main estimates (5.31) and (5.32). For this aim, multiplying (5.11) by k2v3x+D2v4x , we get: Re(ZL 01v1 k2v3x+D2v4x dx) +k1Re(ZL 0 k2v3x+D2v4x v1 xxdx) +k1Re(ZL 0 v3 x+`v5 x
 k2v3x+D2v4x dx) +`k3Re(ZL 0 v5 x`v1
 k2v3x+D2v4x dx) | {z } I=o(1) 2:(5.43) Using the fact that v3 xand (v5 x`v1) are uniformly bounded in L2(0;L), (5.15) and (5.16), we get: I=k1Re(ZL 0 v3 x+`v5 x
 k2v3x+D2v4x dx) +`k3Re(ZL 0 v5 x`v1
 k2v3x+D2v4x dx) =o(1) 2: (5.44) Substitute (5.44) in (5.43), we get: (5.45) k1Re(ZL 0 k2v3x+D2v4x v1 xxdx) =Re(ZL 01v1 k2v3x+D2v4x dx) +o(1) 2: Now, substitute (5.45) in (5.42), we obtain: (5.46)1ZL 0jv1j2dx+k3ZL 0jv1 xj2dx=k1+k3 k2 1 Re(ZL 01v1 k2v3x+D2v4x dx) +o(1) 2: 22We will now apply Young's inequality in (5.46). For this sake , let >0 be given. We get: 1ZL 0jv1j2dx+k3ZL 0jv1 xj2dx1 k1+k3 k2 12ZL 012k2v3x+D2v4x2 dx | {z } =o(1) 2+ZL 01jv1j2dx+o(1) 2 ZL 01jv1j2dx+o(1) 2: Consequently, we have: (1)1ZL 0jv1j2dx+k3ZL 0jv1 xj2dx=o(1) 2: Finally, it is sucient to take =1 2in the previous equation to get the desired estimates in (5.31) and (5.32). The proof is thus complete.  Lemma 5.7. Under all the above assumptions, we have: (5.47)Z +jv5 xj2dx=o(1) Proof. Multiplying (5.11) by v5xand integrating over (0 ;L), we get: (`k1+`k3)ZL 0jv5 xj2=1ZL 02v1v5xdx+k1ZL 0v1 x0v5xdx+k1ZL 0v1 xv5xx dx (5.48) k1ZL 0v3 xv5xdx+`2k3ZL 0v1v5xdx+o(1) 3: Finally, using (5.16), (5.31), (5.32), the fact that v5 xis uniformly bounded in L2(0;L) and1 v5 xxis uniformly bounded in L2(0;L) due to (5.10) in the right hand side of the previous equation, we get the desired estimate (5.47). The proof is thus complete.  Lemma 5.8. Under all the above assumptions, we have: (5.49)Z +jv5j2dx=o(1): Proof. Multiplying (5.13) by 1 1v5, we get: ZL 0jv5j2dx=1 1ZL 0k3(v5 x`v1)(0v5+v5x)dx+1 1ZL 0`k1(v1 x+v3+`v5)v1dx (5.50) ZL 0f6 4+if5 3 v5dx: Using (5.14), (5.47), the fact that ( v1 x+v3+`v5), (v5 x`v1) are uniformly bounded in L2(0;L),f5,f6 converge to zero respectively in H1 0(0;L) (or inH1 (0;L)),L2(0;L) in the right hand side of the above equation, we deduce:ZL 0jv5j2dx=o(1): Finally, using the denition of , we get the desired estimate (5.49). The proof is thus complete.  Proof of Theorem 5.1 It follows from Lemmas 5.2, 5.4, 5.5, 5.7 and 5.8 that kUnkHj=o(1) on (+;):So one can use estimate (4.38) with D1=D3= 0 and Lemma 4.9 to conclude that kUnkHj=o(1) on (0;L) which is a contradiction with (5.3). Consequently, condition (H4) holds and the energy of smooth solutions of system (1.1) decays polynomially as tgoes to innity.  236.Lack of exponential stability It was proved that the Bresse system subject to one or two viscous dampings is exponentially stable if and only if the wave propagate at the same speed (see [24] and [1]). In the case of viscoelastic damping, the situation is more delicate. In this section, we prove that the Bresse system (1.1)-(1.3) subject to two global viscoelastic dampings is not exponentially stable even if the waves propagate at same speed. So, we assume that: (6.1) D1= 0 and D2=D3= 1 in (0 ;L): Theorem 6.1. Under hypothesis (6.1) , the Bresse system (1.1) -(1.3) , is not exponentially stable in the energy spaceH2. Proof. For the proof of Theorem 6.1, it suces to show that there exists a sequence ( n)Rwith lim n!+1jnj= +1, and a sequence ( Vn)D(A2), such that (inIA 2)Vnis bounded inH2and lim n!+1kVnk= +1. For the sake of clarity, we skip the index n. LetF= (0;0;0;f4;0;0)2H 2with f4(x) = cosnx L ; =np2k2 L2; n2N: We solve the following equations: (6.2) iv1v2= 0; (6.3) i 1v2k1 v1 xx+v3 x+`v5 x
`k3 v5 x`v1
` v6 x`v2
= 0; (6.4) iv3v4= 0; (6.5) i 2v4k2v3 xx+k1 v1 x+v3+`v5
=2f4; (6.6) iv5v6= 0; (6.7) i 1v6k3 v5 xx`v1 x
+`v2 x+`k1 v1 x+v3+`v5
= 0: Eliminating v2,v4andv6in (6.3), (6.5) and (6.7) by (6.2), (6.4) and (6.6), we get: (6.8) 21v1+k1 v1 xx+v3 x+`v5 x
+`(k3+i) v5 x`v1
= 0; (6.9) 22v3+k2v3 xxk1 v1 x+v3+`v5
=2f4; (6.10) 21v5+k3 v5 xx`v1 x
i`v1 x`k1 v1 x+v3+`v5
= 0: This can be solved by the ansatz: (6.11) v1=Asinnx L ; v3=Bcosnx L ; v5=Ccosnx L whereA,BandCdepend on are constants to be determined. Notice that k2n L
222= 0, and inserting (6.11) in (6.8)-(6.10) we obtain that: (6.12)n L2 k121+ (k3+i)`2 A+k1n L B+ (k1+k3+i)`n L C= 0; (6.13) k1n L A+k1B+`k1C=2; (6.14) ( k1+k3+i)`n L A+`k1B+ k3n L2 21+`2k1 C= 0: 24Equivalently, (6.15)0 B@n L
2k121+ (k3+i)`2k1n L
(k1+k3+i)`n L
k1n L
k1 `k1 (k1+k3+i)`n L
`k1k3n L
221+`2k11 CA0 @A B C1 A=0 @0 2 01 A: This implies that: (6.16) A=(k212k3)2 2L (k22 1k312+2`2)k2n+O(n2); (6.17) B=2 k1k32 2+ (k1k3)1+`2
k22+k2 22 1
k1((k31+`2)2+k22 1)k2+O(n1); (6.18) C=i`2 2Lp2k2 ((k31+`2)2+k22 1)k2n+O(n2): Now, letVn= v1; iv1;v3; iv3;v5; iv5
, wherev1;v3andv5are given by (6.11) and (6.16)-(6.18). It is easy to check that kVnkH2p2kv3kjBjjnj!+1asn!+1: On the other hand, using (6.2)-(6.7), we deduce that k(iIA 2)Vnk2 H2=k(0;0;0;2f4iD 2v3 xx;0;iD 3v5 xx)k2 H2c: Consequently,k(iIA 2)Vnk2 H2is bounded as ntends to +1. Thus the proof is complete.  Remark 6.2. By a similar way, we can prove that the Bresse system (1.1) -(1.3) subject to only one vis- coelastic damping is also not exponentially stable even if the waves propagate at same speed.  7.Additional results and summary Global Kelvin{Voigt damping : analytic stability . In [12], Huang considered a one-dimensional wave equation with global Kelvin-Voigt damping and he proved that the semigroup associated to the equation is not only exponentially stable, but also is analytic. So, it is logic that in the case of three waves equations with three global dampings, the decay will be also analytic. In this part, we state the analytic stability of the Bresse systems (1.1)-(1.2) and (1.1)-(1.3) provided that there exists a positive constant d0such that: (7.1) D1; D 2D3d0>0 for every x2(0;L): Theorem 7.1. Assume that condition (7.1) holds. Then, the C0-semigroup etAj, forj= 1;2;is analytically stable. The proof relies on the characterization of the analytic stability stated in theorem 4.1 and on the same kind of proof used for the preceding results : we use a contradiction argument and much simpler estimation to obtain the result. This much simpler proof is left to the reader. Localized smooth damping : exponential stability . In [15], K. Liu and Z. Liu considered a one- dimensional wave equation with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam. They proved that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is not exponentially stable. And in [16], K. Liu and Z. Liu reconsidered the one-dimensional linear wave equation with the Kelvin- Voigt damping presented on a subinterval but with smooth transition at the end of the interval. They proved that the smoothness of the damping coecient at the interface leads to an exponential stability. They were the rst researchers to suggest that discontinuity of material properties at the interface and the \type" of the damping can aect the qualitative behavior of the energy decay. The smoothness of the coecient at the interface plays a crucial role in the stabilization of the wave equation. In this part, we generalize these results on Bresse system. 25So we consider the Bresse systems (1.1)-(1.2) and (1.1)-(1.3) subject to three local viscoelastic Kelvin- Voigt dampings with smooth coecients at the interface. We establish uniform (exponential) stability of the C0-semigroup etAj,j= 1;2. For this purpose, let ;6=!= (;)(0;L) be the biggest nonempty open subset of (0 ;L) satisying: (7.2) 9d0>0 such that Did0;for almost every x2!; i= 1;2;3: Theorem 7.2. Assume that condition (7.2) holds. Assume also that D1,D2,D32W1;1(0;L). Then, the C0-semigroup etAjis exponentially stable in Hj,j= 1;2, i.e., for all U02Hj, there exist constants M1 and>0independent of U0such that: etAjU0 HjMetkU0kHj; t0; j= 1;2: Again, the proof relies on the characterization of the exponential stability stated in theorem 4.1 and on the same kind of arguments used for the proof of the preceding results : we use a contradiction argument and simpler estimation to obtain the result. This proof is left to the reader. The following table summarizes the results of this study: Regularity of D1Regularity of D2Regularity of D3 Localization Energy decay rate L1(0;L)L1(0;L)L1(0;L)Did0>0 in (0;L) i= 1;2;3Analytic stability W1;1(0;L)W1;1(0;L)W1;1(0;L)Did0>0 in! i= 1;2;3Exponential stability L1(0;L)L1(0;L)L1(0;L)3\ i=1suppDi=! Polynomial of type1 t 0 L1(0;L) 0 D2d0>0 in! Polynomial of type1p t Acknowledgments The authors thanks professor Kais Ammari for his valuable discussions and comments. Chiraz Kassem would like to thank the AUF agency for its support in the framework of the PCSI project untitled Theoretical and Numerical Study of Some Mathematical Problems and Applications . Ali Wehbe would like to thank the CNRS and the LAMA laboratory of Mathematics of the Universit e Savoie Mont Blanc for their supports. The authors thank also the referees for very useful comments. References [1]F. Abdallah, M. Ghader, and A. 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Control Optim., 26 (1988), pp. 714{724. [13]T. Kato ,Perturbation theory for linear operators , Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. [14]J. E. Lagnese, G. Leugering, and E. J. P. G. Schmidt ,Modeling, analysis and control of dynamic elastic multi-link structures , Systems & Control: Foundations & Applications, Birkh auser Boston, Inc., Boston, MA, 1994. [15]K. Liu and Z. Liu ,Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping , SIAM J. Control Optim., 36 (1998), pp. 1086{1098. [16] ,Exponential decay of energy of vibrating strings with local viscoelasticity , Z. Angew. Math. Phys., 53 (2002), pp. 265{280. [17]Z. Liu and B. Rao ,Energy decay rate of the thermoelastic Bresse system , Z. Angew. Math. Phys., 60 (2009), pp. 54{69. [18]Z. Liu and S. Zheng ,Semigroups associated with dissipative systems , vol. 398 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999. [19]T. K. Maryati, J. E. Mu ~noz Rivera, A. Rambaud, and O. Vera ,Stability of an N-component Timoshenko beam with localized Kelvin-Voigt and frictional dissipation , Electron. J. Dierential Equations, (2018), pp. Paper No. 136, 18. [20]N. Najdi and A. Wehbe ,Weakly locally thermal stabilization of Bresse systems , Electron. J. Dierential Equations, (2014), pp. No. 182, 19. [21]A. Pazy ,Semigroups of linear operators and applications to partial dierential equations , vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. [22]J. Pr uss,On the spectrum of C0-semigroups , Trans. Amer. Math. Soc., 284 (1984), pp. 847{857. [23]X. Tian and Q. Zhang ,Stability of a Timoshenko system with local Kelvin-Voigt damping , Z. Angew. Math. Phys., 68 (2017), pp. Paper No. 20, 15. [24]A. Wehbe and W. Youssef ,Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks , J. Math. Phys., 51 (2010), pp. 103523, 17. [25]H. L. Zhao, K. S. Liu, and C. G. Zhang ,Stability for the Timoshenko beam system with local Kelvin-Voigt damping , Acta Math. Sin. (Engl. Ser.), 21 (2005), pp. 655{666. Laboratoire de Math ematiques UMR 5127 CNRS, Universit e de Savoie Mont Blanc, Campus scientifique, 73376 Le Bourget du Lac Cedex, France Email address :stephane.gerbi@univ-smb.fr Universit e Libanaise, Facult e des Sciences 1, EDST, Equipe EDP-AN, Hadath, Beyrouth, Liban Email address :shiraz.kassem@hotmail.com Universit e Libanaise, Facult e des Sciences 1, EDST, Equipe EDP-AN, Hadath, Beyrouth, Liban Email address :ali.wehbe@ul.edu.lb 27

2020-06-30

We consider an elastic/viscoelastic transmission problem for the Bresse system with fully Dirichlet or Dirichlet-Neumann-Neumann boundary conditions. The physical model consists of three wave equations coupled in certain pattern. The system is damped directly or indirectly by global or local Kelvin-Voigt damping. Actually, the number of the dampings, their nature of distribution (locally or globally) and the smoothness of the damping coefficient at the interface play a crucial role in the type of the stabilization of the corresponding semigroup. Indeed, using frequency domain approach combined with multiplier techniques and the construction of a new multiplier function, we establish different types of energy decay rate (see the table of stability results below). Our results generalize and improve many earlier ones in the literature and in particular some studies done on the Timoshenko system with Kelvin-Voigt damping.

Polynomial stabilization of non-smooth direct/indirect elastic/viscoelastic damping problem involving Bresse system

2006.16595v2

arXiv:2205.13364v1 [math.PR] 26 May 2022Ergodic results for the stochastic nonlinear Schr¨ odinger equation with large damping Zdzis/suppress law Brze´ zniak∗, Benedetta Ferrario†and Margherita Zanella‡ May 27, 2022 Abstract We study the nonlinear Schr¨ odinger equation with linear da mping, i.e. a zero order dissipation, and additive noise. Working in Rdwithd= 2 ord= 3, we prove the uniqueness of the invariant measure when the damping coefficient is sufficiently large. 1 Introduction The nonlinear Schr¨ odinger equation occurs as a basic model in many areas of physics: hydrodynamics, plasma physics, optics, molecular biolog y, chemical reaction,etc. Itdescribesthepropagationofwavesinmediawithb othnonlinear and dispersive responses. Inthisarticle,weinvestigatethelongtimebehaviorofthenonlinearS chr¨ odinger equation with a linear damping term and an additive noise. This is our mod el (1.1)/braceleftBigg du(t)+/bracketleftbig i∆u(t)+iα|u(t)|2σu(t)+λu(t)/bracketrightbig dt= ΦdW(t) u(0) =u0, The unknown is u:Rd→C. We consider σ >0,λ >0 andα∈ {−1,1}; for α= 1 this is called the focusing equation and for α=−1 this is the defocusing one. Many results are known about existence and uniqueness of solution s, in different spatial domains and with different noises; see [ BRZ14,BRZ16,BRZ17, BHW19,CHS19,DBD99,DBD03]. Basically these results are obtained without damping but can be easily extended to the case with λ>0. When there is no damping and no forcing term ( λ= 0 and Φ = 0), the equation is conservative. However, with a noise and a damping term, we expect that the energy injected by the noise is dissipated by the damping te rm; because of this balance it is meaningful to look for stationary solutions or inva riant measures. Ekren, Kukavica and Ziane [ EKZ17] and Kim [ K06] provide the ∗Department of Mathematics, University of York, Heslington , York, YO105DD, U E-mail address: zdzislaw.brzezniak@york.ac.uk †Dipartimento di Scienze Economiche e Aziendali, Universit ` a di Pavia, 27100 Pavia, Italy E-mail address : benedetta.ferrario@unipv.it ‡Dipartimento di Matematica ”Francesco Brioschi”, Politec nico di Milano, Via Bonardi 13, 20133 Milano, Italy E-mail address : margherita.zanella@polimi.it 1existence of invariant measures of the equation ( 1.1) for any damping coefficient λ >0; see also the more general setting of [ BFZ21] for the two dimensional case in a different spatial domain and with multiplicative noise and the bo ok [HW19] for the numerical analysis approach. Notice that the damping λuis weaker than the dissipation given by a Laplacian −λ∆u; for this reason we say thatλuis a zero-order dissipation. This implies that the results of existence or uniqueness of invariant measures for the damped Schr¨ odinger equation are less easy than for the stochastic parabolic equations (see, e.g., [ DPZ96]). A similar issue appears in the stochastic damped 2D Euler equations, fo r which the existence of invariant measures has been recently proven in [ BF20]; there again the difficulty comes from the absence of the strong dissipation , given by the Laplacian in the Navier-Stokes equations. The question of the uniqueness of invariant measures is quite challen ging for the SPDE’s with azero-orderdissipation. Debussche and Odasso[ DO05] proved the uniqueness ofthe invariant measurefor the cubic focusing Sch r¨ odingerequa- tion (1.1), i.e.σ=α= 1, when the spatial domain is a bounded interval; however no uniqueness results are known for larger dimension. For the one- dimensional stochastic damped KdV equation there is a recent resu lt by Glatt- Holtz, Martinez and Richards [ GHMR21 ]. However for nonlinear SPDE’s of parabolic type, i.e. with the stronger dissipation term, the uniquene ss issue has been solved in many cases; see, e.g., the book [ DPZ96] by Da Prato and Zabczyk, and the many examples in the paper [ GHMR17 ] by Glatt-Holtz, Mat- tingly and Richards, dealing with the coupling technique. Let us point o ut that the coupling technique allows for the uniqueness result without rest riction on the damping parameter λbut all the examples solved so far areset in a bounded spatial domain and not in Rd. The aim of our paper is to investigate the uniqueness of the invariant mea- sures for equation ( 1.1) in dimension d= 2 andd= 3, with some restrictions on the nonlinearity when d= 3. However our technique fails for larger dimension. Notice that also the results for the attractor in the deterministic s etting are known for d≤3 (see [L95]). Our main result is Theorem 5.1; it provides a sufficient condition to get the uniqueness of the invariant measure, involvingλ and the intensity of the noise. To optimize this condition ( 5.1) we perform a detailed analysis on how the solution depends on λ. As far as the contents of this paper are concerned, in Section 2we introduce the mathematical setting; in Section 3by means of the Strichartz estimates we prove a regularity result on the solutions; this will allow to prove in Sec tion4 that the support of any invariant measure is contained in V∩L∞(Rd) and some estimates of the moments are given. Finally Section 5presents the uniqueness result. The four appendices contain auxiliary results. 2 Assumptions and basic results Forp≥1,Lp(Rd) is the classical Lebesgue space of complex valued functions, and the inner product in the real Hilbert space L2(Rd) is denoted by /an}b∇acketle{tu,v/an}b∇acket∇i}ht=/integraldisplay Rdu(y)v(y)dy. 2We consider the Laplace operator ∆ as a linear operator in L2(Rd); so A0=−∆, A 1= 1−∆ are non-negative linear operators and {eitA0}t∈Ris a unitary group in L2(Rd). Moreover for s≥0 we consider the power operator As/2 1inL2(Rd) with domain Hs={u∈L2(Rd) :/ba∇dblAs/2 1u/ba∇dblL2(Rd)<∞}. Our two main spaces are H:= L2(Rd) andV:=H1(Rd). We setH−s(Rd) for the dual space of Hs(Rd) and denote again by /an}b∇acketle{t·,·/an}b∇acket∇i}htthe duality bracket. We define the generalized Sobolev spaces Hs,p(Rd) with norm given by /ba∇dblu/ba∇dblHs,p(Rd) =/ba∇dblAs/2 1u/ba∇dblLp(Rd). We recall the Sobolev embedding theorem, see e.g. [BL76][Theorem 6.5.1]: if 1 <q<p< ∞with 1 p=1 q−r−s d, then the following inclusion holds Hr,q(Rd)⊂Hs,p(Rd) and there exists a constant Csuch that /ba∇dblu/ba∇dblHs,p(Rd)≤C/ba∇dblu/ba∇dblHr,q(Rd)for all u∈Hr,q(Rd). Remark 2.1. Ford= 1the spaceVis a subset of L∞(R)and is a multiplicative algebra. This simplifies the analysis of the Schr¨ odinger eq uation(1.1). However ford≥2the analysis is more involved. We write the nonlinearity as (2.1) Fα(u) :=α|u|2σu. LemmaC.1provides a priori estimates on it. As far as the stochastic term is concerned, we considera realHilbe rt spaceU with an orthonormal basis {ej}j∈Nand a complete probability space (Ω ,F,P). LetWbe aU-canonical cylindrical Wiener process adapted to a filtration F satisfying the usual conditions. We can write it as a series W(t) =∞/summationdisplay j=1Wj(t)ej, with{Wj}ja sequence of i.i.d. real Wiener processes. Hence (2.2) Φ W(t) =∞/summationdisplay j=1Wj(t)Φej for a given linear operator Φ : U→V. Now we rewrite the Schr¨ odinger equation ( 1.1) in the abstract form as (2.3)/braceleftBigg du(t)+[−iA0u(t)+iFα(u(t))+λu(t)] dt= ΦdW(t) u(0) =u0 We work under the following assumptions on the noise and the nonlinea rity. The initial data u0is assumed to be V. 3Assumption 2.2 (on the noise) .We assume that Φ :U→Vis a Hilbert- Schmidt operator, i.e. (2.4) /ba∇dblΦ/ba∇dblLHS(U,V):=/parenleftbig∞/summationdisplay j=1/ba∇dblΦej/ba∇dbl2 V/parenrightbig1/2<∞. This means that /ba∇dblΦ/ba∇dbl2 LHS(U;V)=∞/summationdisplay j=1/ba∇dblA1/2 1Φej/ba∇dbl2 H=∞/summationdisplay j=1/ba∇dblΦej/ba∇dbl2 H+∞/summationdisplay j=1/ba∇dbl∇Φej/ba∇dbl2 H. In order to compare our setting with the more general one of our p revious paper [BFZ21] in the two-dimensional setting, we point out that Φ is also a Hilbert-Schmidt operator from UtoH(and we denote /ba∇dblΦ/ba∇dblLHS(U;H):=/parenleftBig/summationtext j∈N/ba∇dblΦej/ba∇dbl2 H/parenrightBig1/2 ) and, ford= 2, aγ-radonifying operator from UtoLp(R2) for any finite p. Assumption 2.3 (on the nonlinearity ( 2.1)). •Ifα= 1(focusing), let 0≤σ<2 d. •Ifα=−1(defocusing), let/braceleftBigg 0≤σ<2 d−2,ford≥3 σ≥0, ford≤2 We recall the continuous embeddings H1(R2)⊂Lp(R2)∀p∈[2,∞) H1(Rd)⊂Lp(Rd)∀p∈[2,2d d−2] ford≥3 Hence forσchosen as in Assumption 2.3there is the continuous embedding (2.5) H1(Rd)⊂L2+2σ(Rd). Moreover if σd<2(σ+1) the following Gagliardo-Niremberg inequality holds (2.6) /ba∇dblu/ba∇dblL2+2σ(Rd)≤C/ba∇dblu/ba∇dbl1−σd 2(1+σ) L2(Rd)/ba∇dbl∇u/ba∇dblσd 2(1+σ) L2(Rd). In particular this holds for the values of σspecified in the Assumption 2.3. In the focusing case, thanks to the Young inequality for any ǫ >0 there exists Cǫ>0 such that (2.7) /ba∇dblu/ba∇dbl2+2σ L2+2σ(Rd)≤ǫ/ba∇dbl∇u/ba∇dbl2 L2(Rd)+Cǫ/ba∇dblu/ba∇dbl2+4σ 2−σd L2(Rd). We recall the classical invariant quantities for the deterministic unf orced Schr¨ odinger equation ( λ= 0, Φ = 0), the mass and the energy (see [ C03]): M(u) =/ba∇dblu/ba∇dbl2 H, (2.8) H(u) =1 2/ba∇dbl∇u/ba∇dbl2 H−α 2(1+σ)/ba∇dblu/ba∇dbl2+2σ L2+2σ(Rd). (2.9) They are both well defined on V, thanks to ( 2.5). 4Remark 2.4. In the defocusing case α=−1, we have (2.10) H(u)≥1 2/ba∇dbl∇u/ba∇dbl2 H≥0∀u∈V. In the focusing case α= 1, the energy has no positive sign but we can modify it by adding a term and recover the sign property. We introduc e the modified energy (2.11) ˜H(u) =1 2/ba∇dbl∇u/ba∇dbl2 H−1 2(1+σ)/ba∇dblu/ba∇dbl2+2σ L2(1+σ)(Rd)+G/ba∇dblu/ba∇dbl2+4σ 2−σd H whereGis the constant appearing in the following particular form o f(2.7) (2.12)1 2(1+σ)/ba∇dblu/ba∇dbl2+2σ L2(1+σ)(Rd)≤1 4/ba∇dbl∇u/ba∇dbl2 H+G/ba∇dblu/ba∇dbl2+4σ 2−σd H. Therefore (2.13) ˜H(u)≥1 4/ba∇dbl∇u/ba∇dbl2 H≥0∀u∈V. Above we denoted by Ca generic positive constant which might vary from one line to the other, except Gwhich is the particular constant in ( 2.12) coming from the Gagliardo-Niremberg inequality. Moreover we shall use this notation: ifa,b≥0 satisfy the inequality a≤CAbwith a constant CA>0 depending on the expression A, we writea/lessorsimilarAb; for a generic constant we put no subscript. If we havea/lessorsimilarAbandb/lessorsimilarAa, we writea≃Ab. Now we recall the known results on solutions and invariant measures ; then we provide the improved estimates for the mass and the energy. 2.1 Basic results We recall from [ DBD03] the basic results on the solutions; for any u0∈V there exists a unique global solution u={u(t;u0)}t≥0, which is a continuous V-valued process. Here uniqueness is meant as pathwise uniqueness . Actually their result is given without damping but one can easily pass from λ= 0 to anyλ>0. The difference consists in getting uniform in time estimates for the damped equationoverthe time interval [0 ,+∞), whereasthey hold on any finite time interval [0 ,T] whenλ= 0. Let us state the result from De Bouard and Debussche [ DBD03]. Theorem 2.5. Under Assumptions 2.2and2.3, for every u0∈Vthere exists a uniqueV-valued and continuous solution of (2.3). This is a Markov process inV. Moreover for any finite T >0and integer m≥1there exist positive constantsC1andC2(depending on T,mand/ba∇dblu0/ba∇dblV) such that Esup 0≤t≤T[M(u(t))m]≤C1 and E/bracketleftbigg sup 0≤t≤TH(u(t))/bracketrightbigg ≤C2. 5We notice that the last estimate can be obtained for any power m>1 for the energy in the defocusing case as well as for the modified energy in th e focusing case. Similar computations can be found in [ EKZ17] for the defocusing case; however their Lemma 5.1 has to be modified in the focusing case. The s trategy is the same as that given in the next Section 2.2. As soon as a unique solution in Vis defined, we can introduce the Markov semigroup. Let us denote by u(t;x) the solution evaluated at time t>0, with initial value x. We define (2.14) Ptf(x) =E[f(u(t;x))] for any Borelian and bounded function f:V→R. A probability measure µon the Borelian subsets of Vis said to be an in- variant measure for ( 2.3) when (2.15)/integraldisplay VPtf dµ=/integraldisplay Vf dµ ∀t≥0,f∈Bb(V). We recall Theorem 3.4 from [ EKZ17]. Theorem 2.6. Under Assumptions 2.2and2.3, there exists an invariant mea- sure supported in V. 2.2 Mean estimates In this section we revise some bounds on the moments of the mass, t he en- ergy and the modified energy, in order to see how these quantities d epend on the damping coefficient λ. This improves the results by [ DBD03][§3] and [EKZ17][Lemma 5.1]. This is the result for the mass M(u) =/ba∇dblu/ba∇dbl2 H. Proposition 2.7. Letu0∈V. Then under assumptions 2.2and2.3, for every m≥1there exists a positive constant C(depending on m) such that (2.16) E[M(u(t))m]≤e−λmtM(u0)m+C/ba∇dblΦ/ba∇dbl2m LHS(U;H)λ−m for anyt≥0. Proof.Let us start by proving the estimate ( 2.16) form= 1. We apply the Itˆ o formula to M(u(t)) dM(u(t))+2λM(u(t))dt=/ba∇dblΦ/ba∇dbl2 LHS(U;H)dt+2Re/an}b∇acketle{tu(t),ΦdW(t)/an}b∇acket∇i}ht. Taking the expected value and using the fact that the stochastic in tegral is a martingale by Theorem 2.5, we obtain, for any t≥0, d dtE[M(u(t))] =−2λE[M(u(t))]+/ba∇dblΦ/ba∇dbl2 LHS(U,H). Solving this ODE we obtain E[M(u(t))] =e−2λtM(u0)+/ba∇dblΦ/ba∇dbl2 LHS(U,H)/integraldisplayt 0e−2λ(t−s)ds ≤e−2λtM(u0)+1 2λ/ba∇dblΦ/ba∇dbl2 LHS(U,H), 6which proves ( 2.16) form= 1. For larger values of m, let us apply the Itˆ o formula to M(u(t))mto obtain M(u(t))m=M(u0)m−2λm/integraldisplayt 0M(u(s))mds +2m/integraldisplayt 0M(u(s))m−1Re/an}b∇acketle{tu(s),ΦdW(s)/an}b∇acket∇i}ht +m/ba∇dblΦ/ba∇dbl2 LHS(U,H)/integraldisplayt 0M(u(s))m−1ds +2(m−1)m/integraldisplayt 0M(u(s))m−2∞/summationdisplay j=1[Re/an}b∇acketle{tu(s),Φej/an}b∇acket∇i}ht]2ds.(2.17) With the Young inequality we get m/ba∇dblΦ/ba∇dbl2 LHS(U,H)M(u)m−1+2(m−1)mM(u)m−2∞/summationdisplay j=1[Re/an}b∇acketle{tu,Φej/an}b∇acket∇i}ht]2 ≤m(2m−1)/ba∇dblΦ/ba∇dbl2 LHS(U,H)M(u)m−1 ≤λmM(u)m+/ba∇dblΦ/ba∇dbl2m HS(U;H)/parenleftbiggm−1 m/parenrightbiggm−1 (2m−1)mλ1−m.(2.18) By Theorem 2.5we know that the stochastic integral in ( 2.17) is a martingale, so taking the expected value on both sides of ( 2.17) we obtain EM(u(t))m≤ M(u0)m−λm/integraldisplayt 0EM(u(s))mds+/ba∇dblΦ/ba∇dbl2m HS(U;H)Cmλ1−mt. By means of Gronwall inequality we get EM(u(t))m≤e−λmtM(u0)m+/ba∇dblΦ/ba∇dbl2m HS(U;H)Cmλ1−m/integraldisplayt 0e−λm(t−s)ds ≤e−λmtM(u0)m+/ba∇dblΦ/ba∇dbl2m HS(U;H)Cm mλ−m for anyt≥0. Notice that the estimates on the mass do not depend on α, whereas this happens in the following result. Indeed, consider the energy H(u) given in ( 2.9) and the modified energy ˜H(u) given in ( 2.11). We deal with d≥2, since the cased= 1 is easier and already analysed in [ DO05]. Therefore the condition σ<2 dimplies that σ<1 whend≥2. We introduce the function (2.19) φ1(σ,λ,Φ) =/ba∇dblΦ/ba∇dbl2 LHS(U;V)+/ba∇dblΦ/ba∇dbl2+2σ LHS(U;V)λ−σ. Notice that the mapping λ/ma√sto→φ1(σ,λ,Φ) is strictly decreasing. The same prop- erty holds for the other function φ2(d,σ,λ,m, Φ) appearing the next statement. The expression of φ2(d,σ,λ,1,Φ) is given in ( 2.34); however for general mthe expression is similar but longer and we avoid to specify it. 7Proposition 2.8. Letd≥2andu0∈V. Under Assumptions 2.2and2.3, we have the following estimates: i)Whenα=−1, for everym≥1there exists a positive constant C=C(d,σ,m) such that (2.20) EH(u(t))m≤e−λmtH(u0)m+Cφm 1λ−m for anyt≥0. ii)Whenα= 1, for every m≥1there exist a smooth positive function φ2= φ2(d,σ,λ,m, Φ)and positive constants a=a(d,σ),C1=C(d,σ,m)andC2= C(d,σ,m)such that (2.21)E˜H(u(t))m ≤e−maλt/parenleftBig ˜H(u0)m+C1(λ−m+λ−m−1 2)M(u0)m(1+2σ 2−σd)/parenrightBig +C2φm 2λ−m for anyt≥0. Moreover the mapping λ/ma√sto→φ2(d,σ,λ,m, Φ)is strictly decreasing. Proof.The Itˆ o formula for H(u(t)) is (2.22)dH(u(t))+2λH(u(t))dt=αλσ σ+1/ba∇dblu(t)/ba∇dbl2σ+2 L2σ+2(Rd)dt −∞/summationdisplay j=1Re/an}b∇acketle{t∆u(t)+α|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t)+1 2/ba∇dbl∇Φ/ba∇dbl2 LHS(U;H)dt −α 2/ba∇dbl|u(t)|σΦ/ba∇dbl2 LHS(U;H)dt−ασ∞/summationdisplay j=1/an}b∇acketle{t|u(t)|2σ−2,[Re(u(t)Φej)]2/an}b∇acket∇i}htdt. Below we repeatedly use the H¨ older and Young inequalities. In partic ular (2.23) Am−1B≤ǫλAm+Cǫλ1−mBm and (2.24) Am−2B≤ǫλAm+Cǫλ1−m 2Bm 2 for positive A,B,λ,ǫ . •In the defocusing case α=−1, we neglect the first term in the r.h.s. in ( 2.22), i.e. (2.25)dH(u(t))+2λH(u(t))dt≤ −∞/summationdisplay j=1Re/an}b∇acketle{t∆u(t)−|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t) +/bracketleftBig1 2/ba∇dbl∇Φ/ba∇dbl2 LHS(U;H)+1 2/ba∇dbl|u(t)|σΦ/ba∇dbl2 LHS(U;H)+σ∞/summationdisplay j=1/an}b∇acketle{t|u(t)|2σ−2,[Re(u(t)Φej)]2/an}b∇acket∇i}ht/bracketrightBig dt. Moreover thanks to the Assumption 2.3we use the H¨ older and Young inequali- 8ties to get 1 2/ba∇dbl|u|σΦ/ba∇dbl2 LHS(U;H)+σ∞/summationdisplay j=1/an}b∇acketle{t|u|2σ−2,[Re(u(t)Φej)]2/an}b∇acket∇i}ht ≤1 2/ba∇dbl|u|σ/ba∇dbl2 L2σ+2 σ(Rd)∞/summationdisplay j=1/ba∇dblΦej/ba∇dbl2 L2σ+2(Rd)+σ/ba∇dbl|u|2σ/ba∇dbl L2σ+2 2σ(Rd)∞/summationdisplay j=1/ba∇dbl|Φej|2/ba∇dblLσ+1(Rd) ≤2σ+1 2/ba∇dblu/ba∇dbl2σ L2σ+2(Rd)∞/summationdisplay j=1/ba∇dblΦej/ba∇dbl2 L2σ+2(Rd) ≤2σ+1 2/ba∇dblu/ba∇dbl2σ L2σ+2(Rd)/ba∇dblΦ/ba∇dbl2 LHS(U;V)by (2.5) ≤λ 2+2σ/ba∇dblu/ba∇dbl2+2σ L2+2σ(Rd)+Cσ/ba∇dblΦ/ba∇dbl2+2σ LHS(U;V)λ−σ ≤λH(u)+C/ba∇dblΦ/ba∇dbl2+2σ LHS(U;V)λ−σ(2.26) Now we insert this estimate in ( 2.25) and take the mathematical expectation to get rid of the stochastic integral d dtEH(u(t))+2λEH(u(t))≤1 2/ba∇dblΦ/ba∇dbl2 LHS(U;V)+λEH(u(t))+C/ba∇dblΦ/ba∇dbl2+2σ LHS(U;V)λ−σ, i.e. d dtEH(u(t))+λEH(u(t))≤1 2/ba∇dblΦ/ba∇dbl2 LHS(U;V)+C/ba∇dblΦ/ba∇dbl2+2σ LHS(U;V)λ−σ. By Gronwall lemma we get EH(u(t))≤e−λtH(u0)+1 2/ba∇dblΦ/ba∇dbl2 LHS(U;V)λ−1+C/ba∇dblΦ/ba∇dbl2+2σ LHS(U;V)λ−σ−1 for anyt≥0. This proves ( 2.20) form= 1. For higher powers m>1, by means of Itˆ o formula we get (2.27)dH(u(t))m=mH(u(t))m−1dH(u(t)) +m(m−1) 2H(u(t))m−2∞/summationdisplay j=1[Re/an}b∇acketle{t∆u(t)−|u(t)|2σu(t),Φej/an}b∇acket∇i}ht]2dt. We estimate the latter term using H¨ older and Young inequality: 1 2/summationdisplay j[Re/an}b∇acketle{t∆u−|u|2σu,Φej/an}b∇acket∇i}ht]2 ≤/summationdisplay j[Re/an}b∇acketle{t∆u,Φej/an}b∇acket∇i}ht]2+/summationdisplay j[Re/an}b∇acketle{t|u|2σu,Φej/an}b∇acket∇i}ht]2 ≤ /ba∇dbl∇u/ba∇dbl2 H/summationdisplay j/ba∇dbl∇Φej/ba∇dbl2 H+/ba∇dbl|u|2σu/ba∇dbl2 L2σ+2 2σ+1(Rd)/summationdisplay j/ba∇dblΦej/ba∇dbl2 L2+2σ(Rd) ≤ /ba∇dbl∇u/ba∇dbl2 H/ba∇dblΦ/ba∇dbl2 LHS(U;V)+/ba∇dblu/ba∇dbl2(2σ+1) L2+2σ(Rd)/ba∇dblΦ/ba∇dbl2 LHS(U;V) ≤ǫλH(u)2+Cǫ,σ/parenleftBig /ba∇dblΦ/ba∇dbl4 LHS(U;V)+/ba∇dblΦ/ba∇dbl4(1+σ) LHS(U;V)λ−2σ/parenrightBig λ−1 9for anyǫ>0. Inserting in ( 2.27) and using the Young inequality ( 2.24) we get (2.28)dH(u(t))m≤mH(u(t))m−1dH(u(t)) +1 2mλH(u(t))mdt+C/parenleftBig /ba∇dblΦ/ba∇dbl4 LHS(U;V)+/ba∇dblΦ/ba∇dbl4(σ+1) LHS(U;V)λ−2σ/parenrightBigm/2 λ−m+1dt We estimate H(u(t))m−1dH(u(t)) using ( 2.25), (2.26), and the Young in- equality ( 2.23). Then we take the mathematical expectation in ( 2.28) and ob- tain (2.29)d dtEH(u(t))m+mλEH(u(t))m ≤Cσ,m/parenleftBig /ba∇dblΦ/ba∇dbl2 LHS(U;V)+/ba∇dblΦ/ba∇dbl2(1+σ) LHS(U;V)λ−σ/parenrightBigm λ−m+1. By Gronwall lemma we get ( 2.20). •In the focusing case α= 1, we neglect the last two terms in the r.h.s. in ( 2.22) and get (2.30)dH(u(t))+2λH(u(t))dt≤λσ σ+1/ba∇dblu(t)/ba∇dbl2+2σ L2+2σ(Rd)dt −∞/summationdisplay j=1Re/an}b∇acketle{t∆u(t)+|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t)+1 2/ba∇dbl∇Φ/ba∇dbl2 LHS(U;H)dt. We write the Itˆ o formula for the modified energy ˜H(u) =H(u)+GM(u)1+2σ 2−σd. Keeping in mind ( 2.17) and (2.18) for the power m= 1+2σ 2−σdof the mass, we have (2.31)d˜H(u(t))+2λ˜H(u(t))dt ≤λσ σ+1/ba∇dblu(t)/ba∇dbl2σ+2 L2σ+2(Rd)dt−2λ2σ 2−σdGM(u(t))1+2σ 2−σddt +CM(u(t))1+2σ 2−σddt+C/ba∇dblΦ/ba∇dbl2+4σ 2−σd LHS(U;H)dt+1 2/ba∇dbl∇Φ/ba∇dbl2 LHS(U;H)dt −/summationdisplay jRe/an}b∇acketle{t∆u(t)+|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t) +2(1+2σ 2−σd)GM(u(s))2σ 2−σdRe/an}b∇acketle{tu(t),ΦdW(t)/an}b∇acket∇i}ht. Since (1−2 2−σd)≤0 by Assumption 2.3, we get σ σ+1/ba∇dblu/ba∇dbl2σ+2 L2σ+2(Rd)−4σ 2−σdGM(u)1+2σ 2−σd ≤ (2.12)σ 2/ba∇dbl∇u/ba∇dbl2 H+2σ(1−2 2−σd)GM(u)1+2σ 2−σd ≤σ 2/ba∇dbl∇u/ba∇dbl2 H≤ (2.13)2σ˜H(u). 10Then (2.32)d˜H(u(t))+2(1−σ)λ˜H(u(t))dt ≤/parenleftBig CM(u(t))1+2σ 2−σd+C/ba∇dblΦ/ba∇dbl2+4σ 2−σd LHS(U;H)+1 2/ba∇dbl∇Φ/ba∇dbl2 LHS(U;H)/parenrightBig dt −/summationdisplay jRe/an}b∇acketle{t∆u(t)+|u(t)|2σu(t),Φej/an}b∇acket∇i}htdWj(t) +2(1+2σ 2−σd)GM(u(s))2σ 2−σdRe/an}b∇acketle{tu(t),ΦdW(t)/an}b∇acket∇i}ht. So considering the mathematical expectation we obtain d dtE˜H(u(t))+2(1−σ)λE˜H(u(t)) ≤CE[M(u(t))1+2σ 2−σd]+C/ba∇dblΦ/ba∇dbl2+4σ 2−σd LHS(U;H)+1 2/ba∇dbl∇Φ/ba∇dbl2 LHS(U;H) ≤Ce−λ(1+2σ 2−σd)tM(u0)1+2σ 2−σd+C/ba∇dblΦ/ba∇dbl2(1+2σ 2−σd) LHS(U;H)λ−(1+2σ 2−σd) +C/ba∇dblΦ/ba∇dbl2+4σ 2−σd LHS(U;H)+1 2/ba∇dbl∇Φ/ba∇dbl2 LHS(U;H)(2.33) thanks to ( 2.16). By means of the Gronwall lemma, setting a= min(2 −2σ,1+ 2σ 2−σd)>0 we obtain E˜H(u(t))≤e−2(1−σ)λt˜H(u0)+e−aλtλ−1CM(u0)1+2σ 2−σd+Cφ2λ−1 whereφ2=φ2(d,σ,λ,1,Φ) is equal to (2.34) /ba∇dblΦ/ba∇dbl2(1+2σ 2−σd) LHS(U;H)/parenleftBig λ−1−2σ 2−σd+1/parenrightBig +/ba∇dbl∇Φ/ba∇dbl2 LHS(U;H). This proves ( 2.21) form= 1. Form>1, we have by Itˆ o formula (2.35)d˜H(u(t))m≤m˜H(u(t))m−1d˜H(u(t))+m(m−1) 2˜H(u(t))m−22r(t)dt, where r(t) =∞/summationdisplay j=1[Re/an}b∇acketle{t∆u(t)+|u(t)|2σu(t),Φej/an}b∇acket∇i}ht]2+4G2(1+2σ 2−σd)2M(u(t))4σ 2−σd∞/summationdisplay j=1[Re/an}b∇acketle{tu(t),Φej/an}b∇acket∇i}ht]2. Keeping in mind the previous estimates we get r(t)/lessorsimilar/ba∇dbl∇u(t)/ba∇dbl2 H/ba∇dblΦ/ba∇dbl2 LHS(U;V)+/ba∇dblu(t)/ba∇dbl2(2σ+1) L2σ+2(Rd)/ba∇dblΦ/ba∇dbl2 LHS(U;V) +4G2(1+2σ 2−σd)2M(u(t))1+4σ 2−σd/ba∇dblΦ/ba∇dbl2 LHS(U;H). Now we use ( 2.13), i.e./ba∇dbl∇u(t)/ba∇dbl2 H≤4˜H(u), and by means of ( 2.7) we get /ba∇dblu/ba∇dbl2(2σ+1) L2σ+2(Rd)≤ǫ 4/ba∇dbl∇u/ba∇dbl22σ+1 σ+1 H+Cǫ,σM(u)2σ+1 σ+1(1+2σ 2−σd) ≤ǫ˜H(u)2σ+1 σ+1+Cǫ,σM(u)2σ+1 σ+1(1+2σ 2−σd) 11for anyǫ>0. Thus ˜H(u(t))m−2r(t)/lessorsimilar˜H(u(t))m−1/ba∇dblΦ/ba∇dbl2 LHS(U;V)+˜H(u(t))m−1 σ+1/ba∇dblΦ/ba∇dbl2 LHS(U;V) +˜H(u(t))m−2M(u)2σ+1 σ+1(1+2σ 2−σd)/ba∇dblΦ/ba∇dbl2 LHS(U;V) +˜H(u(t))m−2M(u(t))1+4σ 2−σd/ba∇dblΦ/ba∇dbl2 LHS(U;H). In (2.35) we insert this estimate and the previous estimates for d˜H(u(t)), integrate in time, take the mathematical expectation to get rid of t he stochastic integrals and use Young inequality; hence we obtain d dtE[˜H(u(t))m]+2m(1−σ)λE[˜H(u(t))m]≤m(1−σ)λE[˜H(u(t))m] +Cλ1−mE[M(u(t))m(1+2σ 2−σd)] +Cλ1−m 2/parenleftBig /ba∇dblΦ/ba∇dblm LHS(U;V)E[M(u(t))m 22σ+1 σ+1(1+2σ 2−σd)]+/ba∇dblΦ/ba∇dblm LHS(U;H)E[M(u(t))m 2(1+4σ 2−σd)]/parenrightBig +C/parenleftbigg /ba∇dblΦ/ba∇dbl2 LHS(U;V)+/ba∇dblΦ/ba∇dbl2+4σ 2−σd LHS(U;H)/parenrightbiggm λ1−m+/ba∇dblΦ/ba∇dbl2m(σ+1) LHS(U;V)λ1−m(σ+1). Bearing in mind the estimates ( 2.16) for the mass, we get an inequality for E[˜H(u(t))m] thanks to Gronwall lemma. Then a repeated use of the Young inequality with long but elementary calculations provides that there e xist a constanta=a(d,σ)>0 and a smooth function φ2=φ2(d,σ,λ,m, Φ), strictly decreasing w.r.t. λ, such that E˜H(u(t))m≤e−maλt/parenleftBig ˜H(u0)m+C(λ−m+λ−m−1 2)M(u0)m(1+2σ 2−σd)/parenrightBig +Cφm 2λ−m for anyt≥0. Merging the results for the mass and the energy, we obtain the res ult for the V-norm. Indeed /ba∇dblu/ba∇dbl2 V=/ba∇dbl∇u/ba∇dbl2 H+/ba∇dblu/ba∇dbl2 Hand /ba∇dbl∇u/ba∇dbl2 H= 2H(u)+α σ+1/ba∇dblu/ba∇dbl2σ+2 L2σ+2(Rd). Forα=−1 we trivially get /ba∇dblu/ba∇dbl2 V≤2H(u)+M(u). Forα= 1, we have from ( 2.13) /ba∇dblu/ba∇dbl2 V≤4˜H(u)+M(u). Hereφ1andφ2are the functions appearing in Proposition 2.8. Corollary 2.9. Letd≥2andu0∈V. Under Assumptions 2.2and2.3, for everym≥1we have the following estimates: i)whenα=−1 (2.36)E[/ba∇dblu(t)/ba∇dbl2m V]/lessorsimilare−mλt[H(u0)m+M(u0)m]+[φ1+/ba∇dblΦ/ba∇dbl2 LHS(U;H)]mλ−m 12for anyt≥0; ii)whenα= 1, there is a positive constant a=a(d,σ)such that (2.37) E[/ba∇dblu(t)/ba∇dbl2m V]/lessorsimilare−maλt[˜H(u0)m+(λ−m+λ−m−1 2)M(u0)m(1+2σ 2−σd)+M(u0)m] +[φ2+/ba∇dblΦ/ba∇dbl2 LHS(U;H)]mλ−m for anyt≥0. The constant providing the above estimates /lessorsimilardepends on m,σanddbut not onλ. 3 Regularity results for the solution Forthesolutionofequation( 2.3)weknowthat u∈C([0,+∞);V)a.s. ifu0∈V. Now we look for the L∞(Rd)-space regularity of the paths. When d= 1, this follows directly from the Sobolev embedding H1(R)⊂L∞(R). But such an embedding does not hold for d>1. However for d= 2 ord= 3 one can obtain theL∞(Rd)-regularity by means of the deterministic and stochastic Strichar tz estimates. Letφ1andφ2be the functions appearing in Proposition 2.8. Proposition 3.1. Letd= 2ord= 3. In addition to the Assumptions 2.2and 2.3we suppose that σ<1+√ 17 4whend= 3. Given any finite T >0andu0∈Vthe solution of equation (2.3)is in L2σ(Ω;L2σ(0,T;L∞(Rd))). Moreover there exist positive constants b1=b1(σ), b2=b2(σ)andC=C(σ,d,T)such that (3.1)E/ba∇dblu/ba∇dbl2σ L2σ(0,T;L∞(Rd)) ≤C/parenleftBig /ba∇dblu0/ba∇dbl2σ V+λ−b1ψ(u0)σ(2σ+1)+φσ(2σ+1) 3λ−σ(2σ+1)+/ba∇dblΦ/ba∇dbl2σ LHS(U;V)/parenrightBig . where (3.2)ψ(u0) =/braceleftBigg H(u0)+M(u0), α =−1 ˜H(u0)+(λ−1+λ−b2)M(u0)1+2σ 2−σd+M(u0), α= 1 and (3.3)φ3(d,σ,λ,Φ) =/braceleftBigg φ1(σ,λ,Φ)+/ba∇dblΦ/ba∇dbl2 LHS(U;H), α =−1 φ2(d,σ,λ,σ(2σ+1),Φ)+/ba∇dblΦ/ba∇dbl2 LHS(U;H), α= 1 soλ/ma√sto→φ3(d,σ,λ,Φ)is a strictly decreasing function. Proof.Firstletusconsider d= 2. Werepeatedlyusetheembedding H1,q(R2)⊂ L∞(R2) valid for any q >2. So our target is to prove the estimate for the L2σ(Ω;L2σ(0,T;H1,q(R2)))-norm of ufor someq>2. We introduce the operator Λ := −iA0+λ. It generates the semigroup e−iΛt=e−λteiA0t,t≥0. 13Let us fixT >0. We write equation ( 2.3) in the mild form (see [ DBD03]) iu(t) = ie−Λtu0+/integraldisplayt 0e−Λ(t−s)Fα(u(s))ds+i/integraldisplayt 0e−Λ(t−s)ΦdW(s) =:I1(t)+I2(t)+I3(t) (3.4) and estimate E/ba∇dblIi/ba∇dbl2σ L2σ(0,T;H1,q(R2)), i= 1,2,3 for someq>2. For the estimate of I1we set (3.5) q=/braceleftBigg 2σ σ−1ifσ>1 6 3−σif 0<σ≤1 Notice that q >2. Now, before using the homogeneous Strichartz inequality (A.1) we neglect the term e−λt, sincee−λt≤1. First, assuming σ>1 we work with the admissible Strichartz pair (2 σ,2σ σ−1) and get /ba∇dblI1/ba∇dbl L2σ(0,T;H1,2σ σ−1(R2))=/vextenddouble/vextenddouble/vextenddoublee−λ·eiA0·A1/2 1u0/vextenddouble/vextenddouble/vextenddouble L2σ(0,T;L2σ σ−1(R2)) ≤/vextenddouble/vextenddouble/vextenddoubleeiA0·A1/2 1u0/vextenddouble/vextenddouble/vextenddouble L2σ(0,T;L2σ σ−1(R2)) /lessorsimilar/ba∇dblA1/2 1u0/ba∇dblL2(R2)=/ba∇dblu0/ba∇dblV For smaller values, i.e. 0 <σ≤1, we choose ˜ σ=3 σ>2>σso2˜σ ˜σ−1=6 3−σand /ba∇dblI1/ba∇dbl L2σ(0,T;H1,2˜σ ˜σ−1(R2))/lessorsimilar/ba∇dblI1/ba∇dbl L2˜σ(0,T;H1,2˜σ ˜σ−1(R2))/lessorsimilar/ba∇dblu0/ba∇dblV by the previous computations. For the estimate of I2, we use the Strichartz inequality ( A.2) and then the estimate from Lemma C.1on the nonlinearity. We bear in mind the notation γ′ for the conjugate exponent of γ∈(1,∞), i.e.1 γ+1 γ′= 1. First, consider σ>1; the pair (2σ,2σ σ−1) is admissible. Then /ba∇dblI2/ba∇dbl L2σ(0,T;H1,2σ σ−1(R2))=/ba∇dblA1/2 1I2/ba∇dbl L2σ(0,T;L2σ σ−1(R2)) /lessorsimilar/ba∇dblA1/2 1Fα(u)/ba∇dblL4 3(0,T;L4 3(R2))by (A.2) =/ba∇dblFα(u)/ba∇dblL4 3(0,T;H1,4 3(R2)) /lessorsimilar/ba∇dblu/ba∇dbl2σ+1 L4 3(2σ+1)(0,T;V)by (C.1) and (C.2) For 0<σ≤1 we proceed in a similar way; consideringthe admissible Strichartz pair (2+σ,2+4 σ) we have /ba∇dblI2/ba∇dblL2σ(0,T;H1,2+4 σ(R2))/lessorsimilar/ba∇dblI2/ba∇dblL2+σ(0,T;H1,2+4 σ(R2)) =/ba∇dblA1/2 1I2/ba∇dblL2+σ(0,T;L2+4 σ(R2)) /lessorsimilar/ba∇dblA1/2 1Fα(u)/ba∇dblLγ′(0,T;Lr′(R2))by (A.2) =/ba∇dblFα(u)/ba∇dblLγ′(0,T;H1,r′(R2)) 14where (r,γ) is an admissible Strichartz pair. According to ( C.1) we choose (3.6) (1 ,2)∋r′=/braceleftBigg 2 1+2σ,0<σ<1 2 4 3,1 2≤σ≤1 Hence (3.7) γ′=2r′ 3r′−2=/braceleftBigg 1 1−σ,0<σ<1 2 4 3,1 2≤σ≤1 In this way by means of the estimate ( C.2) of the polynomial nonlinearity /ba∇dblFα(u))/ba∇dblH1,r′(R2)/lessorsimilar/ba∇dblu/ba∇dbl1+2σ Vwe obtain /ba∇dblI2/ba∇dblL2σ(0,T;H1,2+4 σ(R2))/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 Lγ′(2σ+1)(0,T;V). Summing up, we have shown that for any σ >0 there exists q >2 andγ′ such that (3.8) E/ba∇dblI2/ba∇dbl2σ L2σ(0,T;H1,q(R2))/lessorsimilarE/parenleftBigg/integraldisplayT 0/ba∇dblu(t)/ba∇dblγ′(2σ+1) Vdt/parenrightBigg2σ γ′ . Bearing in mind Corollary 2.9, we get the second and third terms in the r.h.s. of (3.1). The details are given in the Appendix D.1. It remains to estimate the term I3. We choose qas in (3.5). Using the stochastic Strichartz estimate ( A.3), we get for σ>1 E/ba∇dblI3/ba∇dbl2σ L2σ(0,T;H1,2σ σ−1(R2))=E/ba∇dblA1/2 1I3/ba∇dbl2σ L2σ(0,T;L2σ σ−1(R2)) /lessorsimilar/ba∇dblA1/2 1Φ/ba∇dbl2σ LHS(U;H)=/ba∇dblΦ/ba∇dbl2σ LHS(H;V). For smaller values of σ, we proceed as before for I1. Now consider d= 3. The additional assumption on σappears because of the stronger conditions on the parameters given later on. Forq≥1 we haveHθ,q(R3)⊂L∞(R3) whenθq >3. So for each Iiin (3.4) we look for an estimate in the norm L2σ(0,T;Hθ,q(R3)) for some parameters θq>3. We estimate I1for any 0<σ<2. When 0 <σ≤1 we consider the admis- sible Strichartz pair (2 ,6). By means of the homogeneous Strichartz estimate (A.1) we proceed as before /ba∇dblI1/ba∇dblL2σ(0,T;H1,6(R3))/lessorsimilar/ba∇dblI1/ba∇dblL2(0,T;H1,6(R3)) =/vextenddouble/vextenddouble/vextenddoublee−λ·eiA0·A1/2 1u0/vextenddouble/vextenddouble/vextenddouble L2(0,T;L6(R3)) ≤/vextenddouble/vextenddouble/vextenddoubleeiA0·A1/2 1u0/vextenddouble/vextenddouble/vextenddouble L2(0,T;L6(R3)) /lessorsimilar/ba∇dblA1/2 1u0/ba∇dblL2(R3)=/ba∇dblu0/ba∇dblV. Whenσ>1 we work with the admissible Strichartz pair (2 σ,6σ 3σ−2) and get /ba∇dblI1/ba∇dbl L2σ(0,T;H1,6σ 3σ−2(R3))/lessorsimilar/ba∇dblA1/2 1u0/ba∇dblL2(R3)=/ba∇dblu0/ba∇dblV; 15since6σ 3σ−2>3 for 1<σ<2, we obtain the L∞(R3)-norm estimate. The estimate for I2is more involved and therefore we postpone it to the Appendix D.2. It remains to estimate the term I3. For any σ >0 we use the H¨ older inequality and the stochastic Strichartz estimate ( A.3) for the admissible pair (2+σ2 2,64+σ2 4+3σ2); therefore E/ba∇dblI3/ba∇dbl2σ L2σ(0,T;H1,64+σ2 4+3σ2(R3))/lessorsimilarTE/ba∇dblI3/ba∇dbl2σ L2+σ2 2(0,T;H1,64+σ2 4+3σ2(R3)) /lessorsimilar/ba∇dblΦ/ba∇dbl2σ LHS(U;V). Noticethattherestriction σ<1+√ 17 4onthepowerofthe nonlinearityaffects only the defocusing case. We conclude this section by remarking that there is no similar result fo r d≥4. Remark 3.2. For larger dimension, there is no result similar to those in t his section. Indeed, if one looks for u∈L2σ(0,T;H1,q(Rd))⊂L2σ(0,T;L∞(Rd))it is necessary that q>d in order to have H1,q(Rd)⊂L∞(Rd). Already the estimate for I1does not hold under this assumption. Indeed the homogeneous Stricha rtz estimate (A.1) provides I1∈C([0,T];H1(Rd))∩L2σ(0,T;H1,q(Rd)) if 1 σ=d/parenleftbigg1 2−1 q/parenrightbigg and 2≤q≤2d d−2. Since2d d−2≤4ford≥4, the latter condition q≤2d d−2and the condition q >d are incompatible for d≥4. Let us notice that also in the deterministic setting the resu lts on the attractors are known for d≤3, see [L95]. 4 The support of the invariant measures Nowweshowsomemorepropertiesonthe invariantmeasuresandth eirsupport. In dimension d= 2 andd= 3, thanks to the regularity results of Section 3we provide an estimate for the moments in the VandL∞(Rd)-norm. Proposition 4.1. Letd= 2ord= 3and the Assumptions 2.2and2.3hold. Letµbe an invariant measure for equation (2.3). Then for any finite m≥1we have (4.1)/integraldisplay /ba∇dblx/ba∇dbl2m Vdµ(x)≤φm 4λ−m for some smooth function φ4=φ4(d,σ,λ,m, Φ)which is strictly decreasing w.r.t.λ. 16Moreover, supposing in addition that σ<1+√ 17 4whend= 3, we have (4.2)/integraldisplay /ba∇dblx/ba∇dbl2σ L∞dµ(x)≤φ5(λ), whereφ5(λ)is a smooth decreasing function, depending also on the param eters d,σand on/ba∇dblΦ/ba∇dblLHS(U;V), Proof.As far as ( 4.1) is concerned, we define the bounded mapping Ψ konVas Ψk(x) =/braceleftBigg /ba∇dblx/ba∇dbl2m V,if/ba∇dblx/ba∇dblV≤k k2m,otherwise By the invariance of µand the boundedness of Ψ k, we have (4.3)/integraldisplay VΨkdµ=/integraldisplay VPsΨkdµ∀s>0. So PsΨk(x) =E[Ψk(u(s;x))]≤E/ba∇dblu(s;x)/ba∇dbl2m V. Moreover, from Corollary 2.9we get an estimate for E/ba∇dblu(s;x)/ba∇dbl2m V, and letting s→+∞the first terms in ( 2.36) and (2.37) vanish so we get lim s→+∞PsΨk(x)≤φm 4λ−m∀x∈V where φ4=/braceleftBigg φ1+/ba∇dblΦ/ba∇dbl2 LHS(U;H),forα=−1 φ2+/ba∇dblΦ/ba∇dbl2 LHS(U;H),forα= 1 The same holds for the integral, that is lim s→+∞/integraldisplay VPsΨk(x)dµ(x)≤φm 4λ−m by the Bounded Convergence Theorem. From ( 4.3) we get /integraldisplay VΨkdµ≤φm 4λ−m as well. Since Ψ kconverges pointwise and monotonically from below to /ba∇dbl·/ba∇dbl2m V, the Monotone Convergence Theorem yields ( 4.1). As far as ( 4.2) is concerned, we consider the estimate ( 3.1) forT= 1 and set˜Ψ(u) =/ba∇dblu/ba∇dbl2σ L∞(Rd); this defines a mapping ˜Ψ :V→R+∪ {+∞}. Its approximation ˜Ψk:V→R+, given by ˜Ψk(u) =/braceleftBigg /ba∇dblu/ba∇dbl2σ L∞(Rd),if/ba∇dblu/ba∇dblL∞(Rd)≤k k2σ, otherwise defines a bounded mapping ˜Ψk:V→R+. It obviously holds /integraldisplay V˜Ψkdµ=/integraldisplay1 0/parenleftbigg/integraldisplay V˜Ψkdµ/parenrightbigg ds. 17By the invariance of µand the boundedness of ˜Ψk, it also holds /integraldisplay V˜Ψkdµ=/integraldisplay VPs˜Ψkdµ∀s>0. Thus, by Fubini-Tonelli Theorem, since ˜Ψk(u) =/ba∇dblu/ba∇dbl2σ L∞(Rd)∧k2σ≤ /ba∇dblu/ba∇dbl2σ L∞(Rd), we get by Proposition 3.1 /integraldisplay V˜Ψkdµ=/integraldisplay1 0/integraldisplay VPs˜Ψkdµds=/integraldisplay V/integraldisplay1 0E/bracketleftBig ˜Ψk(u(s;x))/bracketrightBig dsdµ(x) ≤/integraldisplay VE/integraldisplay1 0/ba∇dblu(s;x)/ba∇dbl2σ L∞(Rd)dsdµ(x) ≤C/integraldisplay V/parenleftBig /ba∇dblx/ba∇dbl2σ V+λ−bψ(x)σ(2σ+1)+φσ(2σ+1) 3λ−σ(2σ+1)+/ba∇dblΦ/ba∇dbl2σ LHS(U;V)/parenrightBig dµ(x) from (3.1). Keeping in mind ( 3.2) and (4.1) we obtain the existence of a smooth decreasing functionφ5(λ), depending alsoon the parameters d,σand on/ba∇dblΦ/ba∇dblLHS(U;V), such that /integraldisplay V˜Ψkdµ≤φ5(λ) for anyk. Since˜Ψkconvergespointwise and monotonicallyfrom belowto ˜Ψ, the Mono- tone ConvergenceTheorem yields the same bound for/integraltext V/ba∇dblx/ba∇dbl2σ L∞(Rd)dµ(x). This proves (4.2). 5 Uniqueness of the invariant measure for suffi- ciently large damping We will prove that, if the damping coefficient λis sufficiently large, then the invariant measure is unique. Theorem 5.1. Letd= 2ord= 3. In addition to the Assumptions 2.2and2.3 we suppose that σ<1+√ 17 4whend= 3. If (5.1) λ>2φ5(λ) whereφ5is the function appearing in Proposition 4.1, then there exists a unique invariant measure for equation (2.3). Proof.The existence of an invariant measure comes from Theorem 2.6. Now we prove uniqueness by means of a reductio ad absurdum. Let us su ppose that there exists more than one invariant measure. In particular there exist two different ergodic invariant measures µ1andµ2. For both of them Proposition 4.1holds. Fix either i= 1 ori= 2 and consider any f∈L1(µi). Then by the Birkhoff Ergodic Theorem (see, e.g., [ CFS82]) forµi-a.e.xi∈Vwe have (5.2) lim t→+∞1 t/integraldisplayt 0f(u(s;xi))ds=/integraldisplay Vf dµiP−a.s. 18Hereu(t;x) is the solution at time t, with initial value u(0) =x∈V. Now fix two initial data x1andx2belonging, respectively, to the support of the measure µ1andµ2. We have /integraldisplay Vf dµ1−/integraldisplay Vf dµ2= lim t→+∞1 t/integraldisplayt 0[f(u(s;x1))−f(u(s;x2))]ds P-a.s.. Taking any arbitrary fin the set G0defined in ( B.2), we get /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Vf dµ1−/integraldisplay Vf dµ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Llim t→+∞1 t/integraldisplayt 0/ba∇dblu(s;x1)−u(s;x2)/ba∇dblHds. If we prove that (5.3) lim t→+∞/ba∇dblu(t;x1)−u(t;x2)/ba∇dblH= 0P−a.s., then we conclude that /integraldisplay Vf dµ1−/integraldisplay Vf dµ2= 0 soµ1=µ2thanks to Lemma B.1. So let us focus on the limit ( 5.3). With a short notation we write ui(t) =u(t;xi). Then consider the difference w=u1−u2fulfilling /braceleftBigg d dtw(t)−iA0w(t)+iFα(u1(t))−iFα(u2(t))+γw(t) = 0 w(0) =x1−x2 so 1 2d dt/ba∇dblw(t)/ba∇dbl2 H+γ/ba∇dblw(t)/ba∇dbl2 H≤/integraldisplay Rd/vextendsingle/vextendsingle/vextendsingle[|u1(t)|2σu1(t)−|u2(t)|2σu2(t)]w(t)/vextendsingle/vextendsingle/vextendsingledy. Using the elementary estimate ||u1|2σu1−|u2|2σu2| ≤Cσ[|u1|2σ+|u2|2σ]|u1−u2|, we bound the nonlinear term in the r.h.s. as /integraldisplay Rd|[|u1|2σu1−|u2|2σu2]w|dy≤[/ba∇dblu1/ba∇dbl2σ L∞(Rd)+/ba∇dblu2/ba∇dbl2σ L∞(Rd)]/ba∇dblw/ba∇dbl2 L2(Rd). Therefore d dt/ba∇dblw(t)/ba∇dbl2 H+2λ/ba∇dblw(t)/ba∇dbl2 H≤2/parenleftBig /ba∇dblu1(t)/ba∇dbl2σ L∞(Rd)+/ba∇dblu2(t)/ba∇dbl2σ L∞(Rd)/parenrightBig /ba∇dblw(t)/ba∇dbl2 H. Gronwall inequality gives /ba∇dblw(t)/ba∇dbl2 H≤ /ba∇dblw(0)/ba∇dbl2 He−2λt+2/integraltextt 0/parenleftBig /bardblu1(s)/bardbl2σ L∞(Rd)+/bardblu2(s)/bardbl2σ L∞(Rd)/parenrightBig ds that is (5.4) /ba∇dblw(t)/ba∇dbl2 H≤ /ba∇dblx1−x2/ba∇dbl2 He−2t/bracketleftBig λ−1 t/integraltextt 0/parenleftBig /bardblu1(s)/bardbl2σ L∞(Rd)+/bardblu2(s)/bardbl2σ L∞(Rd)/parenrightBig ds/bracketrightBig . This is a pathwise estimate. 19We know from Proposition 4.1thatf(x) =/ba∇dblx/ba∇dbl2σ L∞(Rd)∈L1(µi); therefore (5.2) becomes lim t→+∞1 t/integraldisplayt 0/ba∇dblu(s;xi)/ba∇dbl2σ L∞(Rd)ds=/integraldisplay V/ba∇dblx/ba∇dbl2σ L∞(Rd)dµi(x)≤φ5(λ) P-a.s., for either i= 1 ori= 2. Therefore, if λ>2φ5(λ), the exponential term in the r.h.s. of ( 5.4) vanishes as t→+∞. This proves (5.3) and concludes the proof. A Strichartz estimates In this section we recall the deterministic and stochastic Strichart z estimates on Rd. Definition A.1. We say that a pair (p,r)is admissible if 2 p+d r=d 2and(p,r)/ne}ationslash= (2,∞) and 2≤r≤2d d−2ford≥3 2≤r<∞ford= 2 2≤r≤ ∞ ford= 1 If (p,r) is an admissible pair, then 2 ≤p≤ ∞. Given 1≤γ≤ ∞, we denote by γ′its conjugate exponent, i.e.1 γ+1 γ′= 1. Lemma A.2. Let(p,r)be an admissible pair of exponents. Then the following properties hold i)for everyϕ∈L2(Rd)the function t/ma√sto→eitA0ϕbelongs toLp(R;Lr(Rd))∩ C(R;L2(Rd)). Furthermore, there exists a constant Csuch that (A.1) /ba∇dblei·A0ϕ/ba∇dblLp(R;Lr(Rd))≤C/ba∇dblϕ/ba∇dblL2(Rd),∀ϕ∈L2(Rd). ii)LetIbe an interval of Rand0∈J=I. If(γ,ρ)is an admissible pair andf∈Lγ′(I;Lρ′(Rd)), then the function t/ma√sto→Gf(t) =/integraltextt 0ei(t−s)A0f(s)ds belongs toLq(I;Lr(Rd))∩C(J;L2(Rd)). Furthermore, there exists a constant C, independent of I, such that (A.2) /ba∇dblGf/ba∇dblLp(I;Lr(Rd))≤C/ba∇dblf/ba∇dblLγ′(I;Lρ′(Rd)),∀f∈Lγ′(I;Lρ′(Rd)). Proof.See [C03, Proposition 2.3.3]. Lemma A.3 (stochastic Strichartz estimate) .Let(p,r)be an admissible pair. Then for any a∈(1,∞)andT <∞there exists a constant Csuch that (A.3)/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplay· 0ei(·−s)A0Ψ(s)dW(s)/vextenddouble/vextenddouble/vextenddouble/vextenddouble La(Ω,Lp(0,T;Lr(Rd)))≤C/ba∇dblΨ/ba∇dblL2(0,T;LHS(U,L2(Rd))) for anyΨ∈L2(0,T;LHS(U,L2(Rd))). Proof.See [H18, Proposition 2]. 20B Determining sets The set (B.1) G1=/braceleftBigg f∈Cb(V) : sup u/negationslash=vf(u)−f(v) /ba∇dblu−v/ba∇dblV<∞/bracerightBigg is a determining set for measures on V(see, e.g., [ B13] Theorem 1.2). This means that given two probability measures µ1andµ2onVwe have /integraldisplay Vf dµ1=/integraldisplay Vf dµ2∀f∈ G1=⇒µ1=µ2. Following Remark 2.2 in [ GHMR17 ] we can consider as a determining set for measures on Vthe set (B.2) G0=/braceleftBigg f∈Cb(V) : sup u/negationslash=vf(u)−f(v) /ba∇dblu−v/ba∇dblH<∞/bracerightBigg involving the weaker H-norm instead of the V-norm. Indeed we have Lemma B.1. Letµ1andµ2be two invariant measures. If /integraldisplay Vf dµ1=/integraldisplay Vf dµ2∀f∈ G0, thenµ1=µ2. Proof.We show the proof since we work in Rdwhereas [ GHMR17 ] deals with a bounded domain. SetPNxto be the element whose Fourier transform is 1 |ξ|≤NF(x); hence /ba∇dblPNx/ba∇dblV≤√ 1+N2/ba∇dblx/ba∇dblH. Now we show that any function f∈ G1can be approximated by a function fN∈ G0by settingfN(x) =f(PNx). Indeed |fN(x)−fN(y)| ≤L/ba∇dblPNx−PNy/ba∇dblV≤L/radicalbig 1+N2/ba∇dblx−y/ba∇dblH By assumption we know that /integraldisplay VfNdµ1=/integraldisplay VfNdµ2 Taking the limit as N→+∞, by the bounded convergence theorem we get the same identity for f∈ G1. Henceµ1=µ2. C Estimate of the nonlinearity We consider F(u) =|u|2σu. Lemma C.1. Letd= 2. For anyσ>0, ifp∈(1,2)is defined as (C.1) p=/braceleftBigg 2 2σ+1,0<σ<1 2 4 3, σ ≥1 2 21then (C.2) /ba∇dblF(u)/ba∇dblH1,p(R2)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 H1(R2)∀u∈H1(R2). Letd= 3. For anyσ∈(0,3 2]we have (C.3) /ba∇dblF(u)/ba∇dbl H1,6 2σ+3(R3)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 H1(R3)∀u∈H1(R3) and for any σ∈[1,3 2]we have (C.4) /ba∇dblF(u)/ba∇dblH2−σ,6 5(R3)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 H1(R3)∀u∈H1(R3). Proof.We start with the case d= 2. To estimate the H1,p-norm ofFit is enough to deal with /ba∇dblF/ba∇dblLp(Rd)and/ba∇dbl∂F/ba∇dblLp(Rd). We compute (C.5)∂F(u) =σ|u|2σ−2(¯u∂u+u∂¯u)u+|u|2σ∂u,for an arbitrary u∈V, and thus |∂F(u)|/lessorsimilarσ|u|2σ|∂u|. We have /ba∇dblF(u)/ba∇dblLp(Rd)=/ba∇dblu/ba∇dbl2σ+1 L(2σ+1)p(Rd)(C.6) and the H¨ older inequality, for 1 ≤p<2, gives /ba∇dbl∂F(u)/ba∇dblLp(Rd)≤ /ba∇dbl|u|2σ/ba∇dbl L2p 2−p(Rd)/ba∇dbl∂u/ba∇dblL2(Rd) (C.7) ≤ /ba∇dblu/ba∇dbl2σ L4σp 2−p(Rd)/ba∇dblu/ba∇dblV We recall the Sobolev embedding H1(R2)⊂Lr(R2) for any 2 ≤r<∞. Therefore, if 2≤(2σ+1)p 2≤4σp 2−p then both the r.h.s. of ( C.6) and (C.7) can be estimated by a quantity involving theH1(R2)-norm. The two latter inequalities are the same as p≥2 2σ+1 sooneeasilyseesthat the choice( C.1) allowsto fulfil the tworequiredestimates, i.e./ba∇dblF(u)/ba∇dblLp(Rd)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 Vand/ba∇dbl∂F(u)/ba∇dblLp(Rd)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 V. This proves ( C.2). Ford= 3, first we show that for any σ∈(0,3 2] (C.8) /ba∇dblF(u)/ba∇dblH1,p(R3)/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 H1(R3)∀u∈H1(R3) withp=6 2σ+3∈[1,2). 22To this end we notice that the r.h.s. of ( C.6) and (C.7) are estimated by a quantity involving the H1(R3)-norm ifH1(R3)⊂L(2σ+1)p(R3) andH1(R3)⊂ L4σp 2−p(R3). Recalling the Sobolev embedding H1(R3)⊂Lr(R3) for any 2 ≤r≤6 we get the conditions 2≤(2σ+1)p≤6 (equivalent to2 2σ+1≤p≤6 2σ+1) (C.9) 2≤4σp 2−p≤6 (equivalent to2 2σ+1≤p≤6 2σ+3) (C.10) Notice that ( C.10) is stronger than ( C.9); moreover ( C.10) has a solution p∈ [1,2) only ifσ∈(0,3 2]. Choosing (C.11) p=6 2σ+3∈[1,2) we fulfil all the requirements and so we have proved ( C.3). Now for 1 ≤σ≤3 2there is the continuous embedding H1,6 2σ+3(R3)⊆ H2−σ,6 5(R3). Hence from ( C.3) we get ( C.4). D Computations in the proof of Proposition 3.1 D.1 From (3.8)to(3.1) From (3.8) we proceed as follows. We distinguish different values of the param- eterσ. •σ∈(0,1 4): wehaveγ′=1 1−σ, so2σ γ′= 2σ(1−σ)<3 8andγ′(2σ+1) =2σ+1 1−σ<2. With the H¨ older inequality twice E/parenleftBigg/integraldisplayT 0/ba∇dblu(t)/ba∇dblγ′(2σ+1) Vdt/parenrightBigg2σ γ′ ≤/parenleftBigg E/integraldisplayT 0/ba∇dblu(t)/ba∇dblγ′(2σ+1) Vdt/parenrightBigg2σ γ′ /lessorsimilarT/parenleftBigg E/integraldisplayT 0/ba∇dblu(t)/ba∇dbl2 Vdt/parenrightBigg2σ γ′γ′2σ+1 2 We conclude by means of the estimate of Corollary 2.9form= 1; for instance in casei) /parenleftBigg E/integraldisplayT 0/ba∇dblu(t)/ba∇dbl2 Vdt/parenrightBiggσ(2σ+1) /lessorsimilard,σ/parenleftBigg/integraldisplayT 0/bracketleftBig e−λt[H(u0)+M(u0)]+[φ1+/ba∇dblΦ/ba∇dbl2 LHS(U;H)]λ−1/bracketrightBig dt/parenrightBiggσ(2σ+1) /lessorsimilard,σ,T/parenleftBig [H(u0)+M(u0)]λ−1+[φ1+/ba∇dblΦ/ba∇dbl2 LHS(U;H)]λ−1/parenrightBigσ(2σ+1) /lessorsimilard,σ,T[H(u0)+M(u0)]σ(2σ+1)λ−σ(2σ+1)+[φ1+/ba∇dblΦ/ba∇dbl2 LHS(U;H)]σ(2σ+1)λ−σ(2σ+1) 23•σ∈[1 4,1 2): wehaveγ′=1 1−σ, so2σ γ′= 2σ(1−σ)≤1 2andγ′(2σ+1) =2σ+1 1−σ≥2. With the H¨ older inequality E/parenleftBigg/integraldisplayT 0/ba∇dblu(t)/ba∇dblγ′(2σ+1) Vdt/parenrightBigg2σ γ′ ≤/parenleftBigg E/integraldisplayT 0/ba∇dblu(t)/ba∇dblγ′(2σ+1) Vdt/parenrightBigg2σ γ′ and then we conclude by means of the estimate of Corollary 2.9for 2m= γ′(2σ+1); for instance in case ii) /parenleftBigg E/integraldisplayT 0/ba∇dblu(t)/ba∇dblγ′(2σ+1) Vdt/parenrightBigg2σ γ′ /lessorsimilar/parenleftBig/integraldisplayT 0e−aγ′ 2(2σ+1)λtdt[˜H(u0)γ′ 2(2σ+1)+(λ−γ′ 2(2σ+1)+λ−γ′ 2(2σ+1)−1 2)M(u0)γ′ 2(2σ+1)(1+2σ 2−σd) +M(u0)γ′ 2(2σ+1)]+T[φ2+/ba∇dblΦ/ba∇dbl2 LHS(U;H)]γ′ 2(2σ+1)λ−γ′ 2(2σ+1)/parenrightBig2σ γ′ /lessorsimilara,σ,Tλ−2σ(1−σ)/bracketleftbig˜H(u0)+(λ−1+λ−4σ−1 4σ(1−σ)(2σ+1))M(u0)1+2σ 2−σd+M(u0)/bracketrightbigσ(2σ+1) +[φ2+/ba∇dblΦ/ba∇dbl2 LHS(U;H)]σ(2σ+1)λ−σ(2σ+1) •σ∈[1 2,2 3): we have γ′=4 3, so2σ γ′=3 2σ <1 andγ′(2σ+1) =4 3(2σ+1)≥8 3. So we proceed as in the previous case. •σ≥2 3: we haveγ′=4 3, so2σ γ′≥1 andγ′(2σ+ 1)≥28 9. With the H¨ older inequality E/parenleftBigg/integraldisplayT 0/ba∇dblu(t)/ba∇dblγ′(2σ+1) Vdt/parenrightBigg2σ γ′ /lessorsimilarTE/integraldisplayT 0/ba∇dblu(t)/ba∇dblγ′(2σ+1)2σ γ′ Vdt and then we conclude by means ofthe estimate ofCorollary 2.9form= 2σ(2σ+ 1). D.2 Estimate of I2whend= 3 We distinguish two ranges of values for σ. •For 0<σ≤1 we haveL2(0,T;L6(R3))⊆L2σ(0,T;L6(R3)). So we con- sider the admissible Strichartz pair (2 ,6) and get for any admissible Strichartz pair (γ,r) /ba∇dblI2/ba∇dblL2σ(0,T;H1,6(R3))/lessorsimilar/ba∇dblI2/ba∇dblL2(0,T;H1,6(R3)) =/ba∇dblA1/2 1I2/ba∇dblL2(0,T;L6(R3)) /lessorsimilar/ba∇dblA1/2 1Fα(u)/ba∇dblLγ′(0,T;Lr′(R3))by (A.2) /lessorsimilar/ba∇dblFα(u)/ba∇dblLγ′(0,T;H1,r′(R3)) The parameters are such that γ′=4r′ 7r′−6. From the Definition A.1we have the condition 2 ≤r≤6, equivalent to6 5≤r′≤2. Choosing r′=6 3+2σ, 24we haver′∈[6 5,2) when 0<σ≤1 andγ′=2 2−σ∈(1,2]; thus we can use ( C.3) to estimate the nonlinearity Fα(u). Summing up, we have /ba∇dblI2/ba∇dblL2σ(0,T;L∞(R3))/lessorsimilar/ba∇dblI2/ba∇dblL2σ(0,T;H1,6(R3))/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 L22σ+1 2−σ(0,T;V). Hence E/ba∇dblI2/ba∇dbl2σ L2σ(0,T;L∞(R3))/lessorsimilarE/parenleftBigg/integraldisplayT 0/ba∇dblu(t)/ba∇dbl2 2−σ(2σ+1) Vdt/parenrightBiggσ(2−σ) /lessorsimilar/parenleftBigg E/integraldisplayT 0/ba∇dblu(t)/ba∇dbl22σ+1 2−σ Vdt/parenrightBiggσ(2−σ) by H¨ older inequality since σ(2−σ)≤1. From here, bearing in mind Corollary 2.9we conclude as in the previous sub- section and we obtain the second and third terms in the r.h.s. of ( 3.1). •Forσ>1 we use the admissible Strichartz pair (2 σ,6σ 3σ−2) so /ba∇dblI2/ba∇dbl L2σ(0,T;Hθ,6σ 3σ−2(R3))=/ba∇dblAθ/2 1I2/ba∇dbl L2σ(0,T;L6σ 3σ−2(R3)) /lessorsimilar/ba∇dblAθ/2 1Fα(u)/ba∇dblL2(0,T;L6 5(R3)) /lessorsimilar/ba∇dblFα(u)/ba∇dblL2(0,T;Hθ,6 5(R3)) where we used ( A.2) withγ′= 2 andρ′=6 5, corresponding to the admissible Strichartzpair( γ,ρ) withγ= 2andρ= 6. Notice that6 5isthe minimal allowed value forρ′whend= 3. Now, assuming 1 <σ≤3 2we use the estimate ( C.4) withθ= 2−σ. Summing up, we obtain /ba∇dblI2/ba∇dbl L2σ(0,T;H2−σ,6σ 3σ−2(R3))/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 L2(2σ+1)(0,T;H1(R3)). When (2−σ)6σ 3σ−2>3 we haveH2−σ,6σ 3σ−2(R3)⊂L∞(R3). This gives the condition σ<1+√ 17 4. Hence /ba∇dblI2/ba∇dblL2σ(0,T;L∞(R3))/lessorsimilar/ba∇dblu/ba∇dbl2σ+1 L2(2σ+1)(0,T;V). Sinceσ>1, we conclude with the H¨ older inequality that E/ba∇dblI2/ba∇dbl2σ L2σ(0,T;L∞(R3))/lessorsimilarE/parenleftBigg/integraldisplayT 0/ba∇dblu(t)/ba∇dbl2(2σ+1) Vdt/parenrightBiggσ /lessorsimilarTE/integraldisplayT 0/ba∇dblu(t)/ba∇dbl2σ(2σ+1) Vdt Finally we obtain the second and third term of ( 3.1) by means of Corollary 2.9 as before. 25Acknowledgements B.FerrarioandM.Zanellagratefullyacknowledgesfinancialsuppor tfromGNAMPA- INdAM. References [BRZ14] V. Barbu, M. R¨ ockner and D. Zhang. 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Invariant measu res for a stochastic nonlinear and damped 2D Schr¨ odinger equation. arXiv:2106.07043 [BHW19] Z. Brze´ zniak, F. Hornung and L. Weis. Martingale solutions for the stochastic nonlinear Schr¨ odinger equation in the energy spac e. Probab. Theory Related Fields , 174(3-4):1273–1338, 2019. [C03] T. Cazenave. Semilinear Schr¨ odinger Equations , volume 10. Amer- ican Mathematical Soc., 2003. [CFS82] I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai. Ergodic Theory , Grundlehren der mathematischen Wissenschaften 245. Springer- Verlag, 1982. [CHS19] J. Cui, J. Hong and L. Sun. On global existence and blow-up f or damped stochastic nonlinear Schr¨ odinger equation. Discrete Con- tin. Dyn. Syst. , Ser. B 24(12): 6837–6854, 2019. [DPZ96] G. Da Prato and J. Zabczyk. Ergodicity for infinite-dimensional systems. London Mathematical Society Lecture Note Series, 229. Cambridge University Press, Cambridge, 1996. 26[DBD99] A. De Bouard and A. Debussche. A stochastic nonlinear Schr¨ odinger equation with multiplicative noise. Commun. Math. Phys., 205(1):161–181, 1999. [DBD03] A. De Bouard and A. Debussche, The Stochastic Nonlinear Schr¨ odinger Equation in H1.Stochastic Analysis and Applications 21:1, 97-126, 2003. [DO05] A. Debussche and C. Odasso. Ergodicity for a weakly damped stochastic non-linear Schr¨ odinger equation. J. Evol. Equ. 5(3):317– 356, 2005. [EKZ17] I. Ekren, I. Kukavicaand M. Ziane. Existence of invariant m easures for the stochastic damped Schr¨ odingerequation. Stoch. Partial Dif- fer. Equ. Anal. Comput. 5(3): 343–367, 2017. [GHMR21] N.E. Glatt-Holtz, V.R. Martinez and G.H. Richards. On the lon g- time statistical behavior of smooth solutions of the weakly damped, stochastically-driven KdV equation. arXiv:2103.12942v1 [GHMR17] N.E. Glatt-Holtz, J. Mattingly and G.H. Richards. On Unique E r- godicity in Nonlinear Stochastic Partial Differential Equations. J. Stat. Phys. 1–32, 2016. [HW19] J. Hong and X. Wang. Invariant measures for stochastic nonlinear Schr¨ odinger equations. Numerical Approximations and Symplectic Structures, Springer, 2019. [H18] F. Hornung. The nonlinear stochastic Schr¨ odinger equation via stochastic Strichartz estimates. J. Evol. Equ. 18, 1085–1114, 2018. [K06] J.U. Kim. Invariant Measures for a Stochastic Nonlinear Schr¨ odinger Equation Indiana University Mathematics Journal , 55(2): 687–717, 2006 [L95] P. Lauren¸ cot. Long-time behaviour for weakly damped drive n non- linear Schr¨ odinger equations in RN,N≤3.NoDEA 2, 357–369, 1995. 27

2022-05-26

We study the nonlinear Schr\"odinger equation with linear damping, i.e. a zero order dissipation, and additive noise. Working in $R^d$ with d = 2 or d = 3, we prove the uniqueness of the invariant measure when the damping coefficient is sufficiently large.

Ergodic results for the stochastic nonlinear Schrödinger equation with large damping

2205.13364v1

arXiv:1402.6899v1 [cond-mat.mtrl-sci] 27 Feb 2014On the longitudinal spin current induced by a temperature gr adient in a ferromagnetic insulator S. R. Etesami,1,2L. Chotorlishvili,2A. Sukhov,2and J. Berakdar2 1Max-Planck-Institut f¨ ur Mikrostrukturphysik, 06120 Hal le, Germany 2Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle- Wittenberg, 06120 Halle, Germany (Dated: March 3, 2022) Based on the solution of the stochastic Landau-Lifshitz-Gi lbert equation discretized for a ferro- magnetic chain subject to a uniform temperature gradient, w e present a detailed numerical study of the spin dynamics with a focus particularly on finite-size effects. We calculate and analyze the net longitudinal spin current for various temperature grad ients, chain lengths, and external static magnetic fields. In addition, we model an interface formed by a nonuniformly magnetized finite-size ferromagnetic insulator and a normal metal and inspect the e ffects of enhanced Gilbert damping on the formation of the space-dependent spin current within th e chain. A particular aim of this study is the inspection of the spin Seebeck effect beyond the linear response regime. We find that within our model the microscopic mechanism of the spin Seebeck curr ent is the magnon accumulation effect quantified in terms of the exchange spin torque. According to our results, this effect drives the spin Seebeck current even in the absence of a deviation between th e magnon and phonon temperature profiles. Our theoretical findings are in line with the recent ly observed experimental results by M. Agrawal et al., Phys. Rev. Lett. 111, 107204 (2013). PACS numbers: 85.75.-d, 73.50.Lw, 72.25.Pn, 71.36.+c I. INTRODUCTION Thermal magneto- and electric effects have a long his- tory and are the basis for a wide range of contemporary devices. Research activities revived substantially upon the experimental demonstration of the correlation be- tween an applied temperature gradient and the observed spindynamics, includingaspincurrentalongthetemper- aturegradientinanopen-circuitmagneticsample,theso- called spin Seebeck effect (SSE)1. Meanwhile an impres- sive body work has accumulated on thermally induced spin- and spin-dependent currents1–11(for a dedicated discussion we referto the topical review12). The SSE was observed not only in metallic ferromagnets (FMs) like Co2MnSi or semiconducting FMs, e.g. GaMnAs, (Ref.4), but also in magnetic insulators LaY 2Fe5O12(Ref.5) and (Mn, Ze)Fe 2O4(Ref.7). The Seebeck effect is usually quantified by the Seebeck coefficient Swhich is defined, in a linear response manner, as the ratio of the gener- ated electric voltage ∆ Vto the temperature difference ∆T:S=−∆V ∆T. The magnitude of the Seebeck coeffi- cientSdepends on the scattering rate and the density of electron states at the Fermi level, and thus it is ma- terial dependent variable. In the case of SSE, the spin voltage is formally determined by µ↑−µ↓, whereµ↓(↑) are the electrochemical potentials for spin-up and spin- down electrons, respectively. The density of states and the scattering rate for spin-up and spin-down electrons are commonly different, which results in various Seebeck constants for the two spin channels. In a metallic magnet subjected to a temperature gradient, one may think of the electrons in different spin channels to generate differ- ent driving forces, leading to a spin voltage that induces a nonzero spin current. When a magnetic insulator is in contact with a normal metal (NM) and the system issubjected to a thermal gradient, the total spin current flowing through the interface is a sum of two oppositely directed currents. The current emitted from the FM into the NM, is commonly identified as a spin pumping cur- rentIspand originatesfromthe thermally activatedmag- netization dynamics in the FM, while the other current Iflis associated with the thermal fluctuations in the NM and is known as spin torque13. The competition between the spin pump and the spin torque currents defines the directionofthetotalspincurrentwhichisproportionalto the thermal gradient applied to the system. The theory ofthe magnon-drivenSSE5presupposesthat the magnon temperature follows the phonon temperature profile and inalinearresponseapproximationprovidesagoodagree- ment with experiments. In a recent study14the theory of the magnon-driven SSE was extended beyond the linear response approxi- mation. In particular, it was shown that the nonlinearity leads to a saturation of the total spin current and nonlin- ear effects become dominant when the following inequal- ity holds H0/Tm F< kB/(MsV), where H0is the constant magnetic field applied to the system, Tm Fis the magnon temperature, Msis the saturation magnetization and V is the volume of the sample. The macrospin formulation ofthestochasticLandau-Lifshitz-Gilbert(LLG)equation and the Fokker-Planck approach utilized in Ref.14is in- appropriate for non-uniformly magnetized samples with characteristic lengths exceeding several 10 nm. Beyond the macrospin formulation the SSE effect for nonuni- formly magnetized samples can be described by intro- ducing a local magnetization vector15/vector m(/vector r,t). In this case, however, the correspondingFokker-Planckequation turns into an integro-differential equation and can only be solved after a linearization16. Recently17, the lon- gitudinal SSE was studied in a NM-FM-NM sandwich2 structure in the case of a nonuniform magnetization pro- file. The linear regime, however, can not totaly embrace nontrivial and affluent physics of the SSE. In the present study we inspect the SSE for a nonuni- formlymagnetized finite-size FM-NM interfacesubjected to an arbitrary temperature gradient. Our purpose is to go beyond linear response regime which is relevant for the nonlinear magnetization dynamics. It is shown that in analogy with the macrospin case14the spin current in the nonlinear regime depends not only on the tempera- ture gradient, but on the absolute values of the magnon temperature as well. In finite-size non-uniformly magne- tized samples, however, the site-dependent temperature profile may lead to new physical important phenomena. For instance, we show that the key issue for the spin cur- rent flowing through a nonuniformly magnetized mag- netic insulator is the local exchange spin torque and the local site-dependent magnon temperature profile, result- ing in a generic spatial distribution of the steady state spin current in a finite chain subject to a uniform tem- peraturegradient. The maximalspincurrentispredicted to be located at the middle of the chain. II. THEORETICAL FRAMEWORK For the description of the transversal magnetiza- tion dynamics we consider propagation of the normal- ized magnetization direction /vector m(/vector r,t) as governed by the Laundau-Lifshitz-Gilbert (LLG) equation18,36 ∂/vector m ∂t=−γ/bracketleftBig /vector m×/vectorHeff/bracketrightBig +α/bracketleftbigg /vector m×∂/vector m ∂t/bracketrightbigg −γ/bracketleftBig /vector m×/vectorh(/vector r,t)/bracketrightBig ,(1) where the deterministic effective field /vectorHeff=−1 MSδF δ/vector mde- rives from the free energy density Fand is augmented by a Gaussian white-noise random field h(/vector r,t) with a space- dependent local intensity and autocorrelation function. αis the Gilbert damping, γ= 1.76·1011[1/(Ts)] is the gyromagnetic ratio and MSis the saturation magnetiza- tion.Freads F=1 V/integraldisplay/bracketleftbiggA 2|/vector∇m|2+Ea(/vector m)−µ0MS/vectorH0·/vector m/bracketrightbigg dV,(2) where/vectorH0is the external constant magnetic field, Ea(/vector m) is the anisotropy energy density and Ais the exchange stiffness. Vis the system volume. We employ a dis- cretized version of the integro-differential equation (1) by defining Ncells with a characteristic length a= /radicalbig 2A/µ0M2sof the exchange interaction between the magnetic moments. a3= Ω0is the volume of the re- spective cell. Assuming negligible variations of /vector m(/vector r,t) over a small a, one introduces a magnetization vector /vectorMnaveraged over the nth cell/vectorMn=MS V/integraltext Ω0/vector m(/vector r,t)dVand the total energy density becomes ε=−/vectorH0·/summationdisplay n/vectorMn+K1 M2 S/summationdisplay n/parenleftbig M2 S−(Mz n)2/parenrightbig −2A a2M2 S/summationdisplay n/vectorMn·/vectorMn+1.(3) /vectorH0is the external magnetic field and K1is the uniaxial anisotropy density (with the easy axis: /vector ez). The effective magnetic field actingon the n-th magnetic moment reads /vectorHeff n=−∂ε ∂/vectorMn=/vectorH0+2K1 M2 SMz n/vector ez +2A a2M2 S/parenleftBig /vectorMn+1+/vectorMn−1/parenrightBig .(4) Thermal activation is introduced by adding to the total effectivefieldastochasticfluctuatingmagneticfield /vectorhn(t) so that /vectorHeff n(t) =/vectorH0+/vectorHanis n+/vectorHexch n+/vectorhn(t).(5) Here/vectorHanis nis the magnetic anisotropy field, /vectorHexch nis the exchange field. The random field /vectorhn(t) has a thermal origin and simulates the interaction of the magnetization with a thermal heat bath (cf. the review Ref. [19] and references therein). The site dependence of /vectorhn(t) reflects theexistenceofthelocalnonuniformtemperatureprofile. On the scale ofthe volumeΩ 0the heat bath is considered uniform at a constant temperature. The random field is characterized via the standard statistical properties of the correlation function /angb∇acketlefthik(t)/angb∇acket∇ight= 0, /angb∇acketlefthik(t)hjl(t+∆t)/angb∇acket∇ight=2kBTiαi γMSa3δijδklδ(∆t).(6) iandjdefine the corresponding sites of the FM-chain andk,lcorrespond to the cartesian components of the random magnetic field, Tiandαiare the site-dependent local temperature and the dimensionless Gilbert damp- ing constant, respectively, kB= 1.38·10−23[J/K] is the Boltzmann constant. In what follows we employ for the numerical calcu- lations the material parameters related to YIG, e.g. as tabulated in Ref.6(Table I). Explicitlythe exchangestiff- ness isA≈10 [pJ/m], the saturation magnetization has a value of 4 πMS≈106[A/m]. The anisotropy strength K1can be derived from the estimate for the frequency ω0=γ2K1/MS≈10·109[1/s]6. The size of the FM cell is estimated from a=/radicalbig 2A/µ0M2syielding about 20 [nm]. Fordampingparameterwetakethevalue α= 0.01, which exceeds the actual YIG value5,6. This is done to optimize the numerical procedure in order to obtain rea- sonable calculation times. We note that although the quasi-equilibrium is assured when tracking the magne- tization trajectories on the time scale longer than the relaxation time, the increased αquantitatively alters the3 FIG. 1: a) Schematics of the FM chain considered in the calculations. b) Suggested alignment for measurements. strength in the correlation function (eq. (6)) and there- fore indirectly has an impact on the values of the spin current. We focus on a system representing a junction of a FM insulatorand a NM which is schematically shownin FIG. 1. This illustration mimics the experimental setup for measuring the longitudinal SSE20, even though the anal- ysis performed here does not include all the aspects of the experimental setting. The direction of the magnetic moments in the equilibrium is parallel to the FM-NM in- terface. Experimentally it was suggested to pick up the longitudinal spin current by means of the inverse spin Hall effect20. If it is so possible then, the electric field generated via the inverse spin Hall effect (ISHE) reads− →E=D− →Is×− →σ. Here− →Edenotes electric field related to the inverse spin Hall effect,− →Isdefines the spatial di- rection of the spin current, and− →σis spin polarization of the electrons in the NM, and Dis the constant. We note, however, that our study is focused on the spin dynamics only and makes no statements on ISHE. III. DEFINITION OF THE SPIN CURRENT For convenience we rewrite the Gilbert equation with the total energy density (3) in the form suggested in Ref.17 ∂/vectorSn ∂t+γ/bracketleftBig /vectorSn×/parenleftBig /vectorHeff n(t)−/vectorHex/parenrightBig/bracketrightBig +αγ MS/bracketleftBigg /vectorSn×∂/vectorSn ∂t/bracketrightBigg +∇·/vectorJ/vector s n= 0, (7)where/vectorSn=−/vectorMn/γand the expression for the spin cur- rent density tensor reads ∇·/vectorJ/vector s n=γ/bracketleftBig /vectorSn×/vectorHex n/bracketrightBig . (8) Here /vectorQn=−γ[/vectorSn×/vectorHex n] (9) is the local exchange spin torque. For the particular geometry (FIG. 1) the only nonzero components of the spin current tensor are Isxn,Isy n,Iszn. Taking into account eqs. (4) and (7), we consider a dis- crete version of the gradient operator and for the com- ponents of the spin current tensor Is n=a2Js nwe deduce: Iα n=Iα 0−2Aa M2 Sn/summationdisplay m=1Mβ m(Mγ m−1+Mγ m+1)εαβγ,(10) whereεαβγis the Levi-Civita antisymmetric tensor, Greek indexes define the current components and the Latin ones denote sites of the FM-chain. In what fol- lows we will utilize eq. (10) for quantifying the spin cur- rent in the spin chain. We consider different temperature gradients applied to the system taking into account the dependence of the magnon temperature on the phonon temperature profile5. Since the temperature in the chain is not uniform, we expect a rich dynamics of different magnetic moments /vectorMn. In this case only nonuniform site-dependent spin current Incan fulfil the equation (7). In order to prove this we will consider different configu- rations of magnetic fields for systems of different lengths. Modeling the interface effects between the FM insulator and the NM proceeds by invoking the concept of the en- hanced Gilbert damping proposed in a recent study21. The increased damping constant in the LLG equation of the last magnetic moment describes losses of the spin current due to the interface effect. In order to evaluate the spin current flowing from the NM to the FM insula- tor we assume that the dynamics of the last spin in the insulator chain is influenced by the spin torque flowing from the NM to the magnetic insulator. The magnetic anisotropy is considered to have an easy axis5. IV. NUMERICAL RESULTS ON ISOLATED FERROMAGNETIC INSULATOR CHAIN For the study of thermally activated magnetization dynamics we generate from 1000 to 10000 random tra- jectories for each magnetic moment of the FM-chain. All obtained observables are averaged over the statis- tical ensemble of stochastic trajectories. The number of realizations depends on the thermal gradient applied to the system. For long spin chains (up to 500 mag- netic moments) the calculations are computationally in- tensive even for the optimized advanced numerical Heun- method22, which converges in quadratic mean to the so- lution of the LLG equation when interpreted in the sense4 of Stratonovich23. For the unit cell of the size 20 [nm], the FM-chain of 500 spins is equivalent to the magnetic insulator sample of the width around 10 [ µm]. We make sure in our calculations that the magnetization dynam- ics is calculated on the large time scale exceeding the system’s relaxation time which can be approximated via τrel≈MS/(γ2K1α)≈10 [ns]24. A. Role of the local temperature and local spin exchange torque Prior to studying a realistic finite-size system we con- sider a toy model of three coupled magnetic moments. Our aim is to better understand the role of local tem- perature and local exchange spin torque Qn(eq. (9)) in the formation of the spin current In. Considering eqs. (8, 9), we can utilize a recursive relation for the site- dependent spin current Inand the local exchange spin torqueIn=In−1+a3 γQnfor different temperatures of the site in the middle of the chain above T2> Tavand belowT2< Tav. The mean temperature in the system is Tav=/parenleftbig T1+T2+T3/parenrightbig /3. The calculations are performed for different values of the site temperatures. We find that the exchange spin torques Qnrelated to magnetic mo- mentsMnwith a temperature above the mean temper- atureTn> Tavhave a positive contribution to the spin current in contrast to the exchange spin torques Qmof the on-average-”cold”magnetic momentswith Tm< Tav. This finding hints on the existence of a maximum spin current in a finite chain of magnetic moments and/or strong temperature gradient. This means that the site- dependent spin current Inincreases if Qn>0 until the local site temperature drops below the mean tempera- tureTn< Tav, in which case the exchange spin torque becomes negative Qn<0 and the spin current decreases. In order to prove that the negative contribution in the spin current of the on-average-cold magnetic moments is not an artefact of the three magnetic moments only, we studied long spin chains which mimic non-uniformly magnetized magnetic insulators. In the thermodynamic limit for a large number of magnetic moments N≫1 we expect to observe a formation of the equilibrium pat- terns in the spin current profile correspondingto the zero exchange spin torque Qn= 0 between nearest adjacent moments. B. Longitudinal spin current In FIG. 2 a dependence of distinct components of the spin current on the site is plotted. As inferred from the figure the current is not uniformly distributed along the chain. Evidently, the spin current has a maximum in the middle of the chain. The site-dependent spin cur- rent is an aftermath of the nonuniform magnon temper-/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/SolidSquare/SolidSquare/SolidSquare /SolidSquare/SolidSquare/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/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/SolidSquareInSx /SquareInSy /SolidCircleInSz 0 10 20 30 40 5001234 Sitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1 FIG. 2: Different cartesian components of the statistically averaged longitudinal spin current as a function of the site number. Numerical parameters are ∆ T= 50 [K], α= 0.01 andH0= 0 [T]. The temperature gradient is defined ∆ T= T1−T50, whereT1= 50 [K]. The only nonzero component of the spin current is ISzn. Other two components ISxn, ISy nare zero because of the uniaxial magnetic anisotropy field which preserves XOYsymmetry of the magnetization dynamics. ature profile applied to the system. This effect was not observed in the single macro spin approximation and is onlyrelevantforthe non-uniformlymagnetizedfinite-size magnetic insulator sample. In addition one observesthat the amplitude of the spin current increases with increas- ingthe thermalgradient. Thisispredictablynatural; less sohowever, is the presenceof amaximum ofthe spin cur- rent observed in the middle of the chain. We interpret this observation in terms of a collective cumulative av- eraged influence of the surrounding magnetic moments on particular magnetic moment. For a linear tempera- ture gradient, as in FIG. 2, we have ∆ T=T1−TN aNwhich means that half of the spins with i < N/2 possess tem- peratures abovethe mean temperature of the chain T1/2, while the other half have temperatures below the mean temperature. Further, the main contributors in the total spin current are the hot magnetic moments with temper- atures above the mean temperature Tn> Tavand with a positive exchange spin torque Qn>0. While magnetic moments with a temperature below the mean tempera- tureTn< Tav,Qn<0absorbthe spincurrentandhavea negativecontribution in the total spin current. This non- equivalence of magnetic moments results in a maximum of the total spin current in the center of the chain. In what follows the magnetic moments with temperatures higher than the mean temperature in the chain are re- ferred to as hot magnetic moments, while the magnetic moments with temperatures lower than Tavwe refer to as cold magnetic moments (i.e., our reference tempera- ture isTav). The idea we are following is that the hot magnetic moments form the total spin current which is partly utilized for the activation of the cold magnetic moments. FIG. 3 illustrates the motivation of this state- ment. The maximum of the spin current (solid circles) is5 /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/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 /SolidUpTriangle/ScriptA3 ΓQnz 0 10 20 30 40 5001234 Sitenumber, n1011/LBracket1/HBars/Minus1/RBracket1 FIG. 3: Z-componentofthestatistically averaged spincurr ent ISzn(blue solid circles) and the distribution of the exchange spin torquea3 γQz n(red solid triangles), both site-dependent. Direct correlation between the behavior of the spin current and the exchange spin torque can be observed: the change of the sign of the exchange spin torque exactly matches the maximum of the spin current. observed in the vicinity of the sites where the exchange spin torque term Qnchanges its sign from positive to negative (solid triangles), highlighting the role of the hot and cold magnetic moments in finite-size systems. To further affirm we consider two different temperature pro- files - linear and exponential - with slightly shifted values of the mean temperature (FIG. 4). The dependence of the maximum spin current on the mean temperature is a quite robust effect and a slight shifts of the mean tem- perature to the left lead to a certain shifting of the spin current’s maximum. The effect of the nonuniform spin current passing through the finite-size magnetic insula- tormightbetestedexperimentallyusingtheSSEsetupin which the spin current’s direction is parallel to the tem- perature gradient. One may employ the inverse spin Hall effect using FM insulator covered by a stripe of param- agnetic metal, e.g. Ptat different sites (cf Ref.20), albeit the chain must be small ( <∼1µm). Furthermore, from FIG. 2 we infer that the only nonzero component of the spin current is Iz n. Due to the uniaxial magnetic anisotropy all orientations of the magnetic moments in the XOYplane are equivalent and Ix n,Iy ncomponents of the spin current vanish. C. Role of boundary conditions To elaborate on the origin of the observed maximum of the spin current we inspect the role of boundary con- ditions. In fact, in spite of employing different bound- ary conditions for the chain we observe the same effect (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 /SolidSquare/SolidSquare /SolidSquare /SolidSquare /SolidSquare /SolidSquare /SolidSquare /SolidSquare /SolidSquare /SolidSquare /SolidSquare /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 /Circle /Circle /Circle /Circle /Circle /Circle /Circle /Circle /Circle /Circle/SolidSquareTn/EquΑlLinear /CircleTn/EquΑlExponential 0 10 20 30 40 500.00.51.01.52.02.5 Sitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1 1 25 502031.234.450 nTn/LBracket1K/RBracket1Temperature profile FIG. 4: Z-component of the statistically averaged spin cur- rent for the linear ∆ T=T1−T50and exponential ∆ T(n) = 50[K]e−(n−1)/50temperature gradients. The slight shift of the mean temperature to the left leads to a certain shifting of th e maximum spin current to the left. the cold and hot magnetic moments is inherent to the spin dynamics within the chain, which is independent from the particular choice of the boundary conditions. Furthermore, we model the situation with the extended region at the ends of the FM chain (FIG. 6), in which the end temperatures are constant (i.e., one might imag- ine the heat reservoirs to have finite spatial extensions). Modeling the ends of the FM-chain with zero temper- ature gradient by means of the LLG equations is cer- tainly an approximation, which can be improved by em- ploying the Landau-Lifshitz-Bloch equations reported in Ref.12. It captures, however, the main effects at rela- tively low temperatures: the flow of the spin current for the decaying spin density away from the T= const-∆ T- interfaceand anon-zerointegralspin currentfor the sites 0< n <50 and 150 < n <200. As we see even in the fragments of the chain with a zero temperature gradient thespincurrentisnotzero. Thereasonisthattheforma- tion of the spin current profile is a collective many body effect of the interacting magnetic moments. Therefore, the fragment of the chain with nonzero temperature gra- dient (sites 50 < n <150) has a significant influence on the formation of the spin current profiles in the left and rightregionsofthe chainwhere the temperaturegradient vanishes. D. Temperature dependence of the longitudinal spin current InFIG.7thedependenceofthe z-componentoftheav- eraged longitudinal spin current on the temperature gra- dient is shown. The dependence In(∆T) (inset ofFIG. 7) is linear and the amplitude of the spin current increases6 /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/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 /UpTriangle /UpTriangle /UpTriangle /UpTriangle /UpTriangle /UpTriangle /UpTriangle /UpTriangle /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 /DownTriangle /DownTriangle /DownTriangle /DownTriangle /DownTriangle /DownTriangle /DownTriangle /DownTriangle /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 /Square /Square /Square /Square /Square /Square /Square /Square /Square/SolidCircleM0/EquΑlMN/Plus1/EquΑl/LParen10,0,0/RParen1 /UpTriangleM0/EquΑl/LParen10,0,0/RParen1,MN/Plus1/EquΑl/LParen10,0,Ms/RParen1 /DownTriangleM0/EquΑl/LParen10,0,Ms/RParen1,MN/Plus1/EquΑl/LParen10,0,0/RParen1 /SquareM0/EquΑlMN/Plus1/EquΑl/LParen10,0,Ms/RParen1 0 10 20 30 40 5001234 Sitenumber, nSpinnurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1 FIG. 5: Effect of different boundary conditions on the aver- aged spin current. Numerical parameters are ∆ T= 50 [K], α= 0.01 andH0= 0 [T]. The temperature gradient is defined ∆T=T1−T50, whereT1= 50 [K]. Inspite of different bound- ary conditions we observe the same maximal spin current for the site number corresponding to the mean temperature of the system. Thus, the effect of the cold and hot magnetic moments is independent of the particular choice of boundary conditions. /Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet /Bullet 0 50 100 150 200/Minus1012345 Sitenumber, nSpinnurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1 1501001502002060100 nTn/LBracket1K/RBracket1Temperature profile FIG. 6: Effect of boundary conditions in the case of different temperature profiles at the boundaries: linear temperature gradient (thick curve), constant temperature for 0 < n <50 and 150 < n <200 (thin curve). Even in the fragments of the chain with zero temperature gradient the spin current is not zero, which results from the formation of the spin cur- rent profile as a collective many body effect of the interactin g magnetic moments. Therefore, the fragment of the chain with nonzero temperature gradient (sites 50 < n <150) has a sig- nificant influence on the formation of the spin current profile s in the left and right zero temperature gradient parts of the chain. with the temperature gradient. This result is consistent with the experimental facts (Refs.4,5) and our previous analytical estimations obtained via the single macrospin model14./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 /SolidSquare /SolidSquare /SolidSquare /SolidSquare /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 /UpTriangle /UpTriangle /UpTriangle /UpTriangle /UpTriangle /UpTriangle /UpTriangle /UpTriangle /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 /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle /SolidUpTriangle 0 10 20 30 40 5001234 Sitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1 /Circle /Circle/SolidCircle /SolidCircle/Square /Square/SolidSquare /SolidSquare/UpTriangle /UpTriangle/SolidUpTriangle /SolidUpTriangle 0 5003 /DifferenceDeltaT/LBracket1K/RBracket1I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1 /DifferenceDeltaT=50 [K] /DifferenceDeltaT=40 [K] /DifferenceDeltaT=0 [K]/DifferenceDeltaT=10 [K]/DifferenceDeltaT=20 [K]/DifferenceDeltaT=30 [K] FIG. 7: Dependence of the averaged spin current on the strength of the temperature gradient. Numerical parameter s areα= 0.01 andH0= 0 [T]. The temperature gradient is defined as ∆ T=T1−T50, whereT1= 50 [K]. The inset shows the averaged spin current for the 26-th site. The maximum current increases with elevating the temperature gradient . E. Finite-size effects Finite-size effects are considered relevant for the ex- perimental observations (e.g. Ref.4). In the thermody- namic limit N≫1 we expect the formation of equilib- rium patterns in the spin currentprofile correspondingto the zero exchange spin torque Qn= 0 between nearest adjacent moments. To address this issue, the spin cur- rent for chains of different lengths is shown in FIG. 8. As we see in the case of N= 500 magnetic moments large pattern of the uniform spin current corresponding to the sites 50< n <450 is observed. In order to understand such a behavior of the spin current for a large system size, we plotted the dependence on the site number of the exchange spin torque Qn(FIG. 9). As we see, the exchange spin torque corresponding to the spin current plateau is characterizedby largefluctuations aroundzero value, while nonzero positive (negative) values of the ex- change spin torque Qnobserved at the left (right) edges correspond to the nonmonotonic left and right wings of the spin torque profile. One may try to interpret the ob- servedresults in terms ofthe so called magnon relaxation length (MRL) λm≈2/radicalBig (DkBT/¯h2)τmmτmp(Refs.5,6), whereDis the spin-wave stiffness constant and τmm,mp are the magnon-magnon, and the magnon-phonon relax- ation times, respectively. The MRL is a characteristic length which results from the solution of the heat-rate equation for the coupled magnon-phonon system5. The physical meaning of λmis an exponential drop of the space distribution of the local magnon temperature for the given external temperature gradient ∆ T. In other words, although the externally applied temperature bias is kept constant, the thermal distribution for magnons is not necessarily linear. In general, one may suggest a sinh(x)-like spatial dependence5and a temperature de-7 pendence λm(T). Estimates of the MRL for the material parameters related to YIG (suppl. mater. of Ref.5) and TN= 0.2 [K] yield the following λm≈10 [µm]26. As seen from FIG. 8 the length starting from which the sat- uration of the spin current comes into play as long as the FM-chain exceeds the length 20 [nm] ×100≈2 [µm]. However,werecallthat MRLisawitnessofthe deviation between the magnon and phonon temperature profiles. Therefore, for interpretingthe nonmonotonicpartsofthe spin current profile (FIG. 8) in terms of the MRL one has to prove the pronounced deviation between magnon and phonontemperaturesattheboundaries. Forfurtherclar- ification we calculate the magnon temperature profile. This can be done self-consistently via the Langevin func- tion< Mz n>=L/parenleftbig < Mz n> Hn/kBTm n/parenrightbig . HereHz nis the zcomponent of the local magnetic field which depends on the external magnetic field and the mean values of the adjacent magnetic moments < Mz n−1>, < Mz n+1>( see eq. (4)). As inferred from the FIG. 10 the magnon temperature profile follows the phonon temperature pro- file. Prominent deviation between the phonon and the magnon temperatures is observed only at the beginning of the chain and gradually decreases and becomes small on the MRL scale. Close to the end of the chain the temperature difference becomes almost zero. This means that left nonmonotonic parts of the spin current profile FIG. 8 can be interpreted in terms of none-equilibrium processes. Comparing this result with the exchange spin torque profile (FIG. 9) we see that in this part of the spin chain the exchange spin torque is positive. This is the reason why the spin current Inis increasing with the site number n. The saturated plateau of the spin current shown in FIG. 8 corresponds to the zero ex- change spin torque Qn= 0 (cf. FIG. 9) and the decay of the spin Seebeck current Inat the right edge corre- sponds to the negative spin exchange torque Qn<0. Thus, for the formation of the convex spin current profile the key issue is not the difference between magnon and phonontemperatures,whichasweseeisprettysmall,but the magnon temperature profile itself. The existence of the hot(cold) magnetic moments with the local magnon temperature up (below) the mean magnon temperature generates the spin current. This difference in the local magnon temperature of the different magnetic moments drives the spin current in the chain. On the other hand any measurement of the spin current done in the vicin- ity of the right edge of the current profile will demon- strate a non-vanishing spin current in the absence of the deviation between the magnon and phonon temperature profiles. This may serve as an explanation of the re- cent experiment25, where a non-vanishing spin current was observed in the absence of the deviation between the magnon and the phonon temperature profiles. We note that zero values of the spin current shown in FIG. 8 is the artefact of isolated magnetic insulator chain. Real measurement of the spin currents usually involve FM- insulator/NM-interfaces. As will be shown below the in- terface effect described by an enhanced Gilbert dampingN=50N=500 N=200 N=150 N=100 1 100 200 300 400 500012 Sitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1 FIG. 8: The dependence of the averaged spin current on the length of the FM-chain. Numerical parameters are α= 0.01 andH0= 0 [K]. The temperature gradient is linear and the maximum temperature is on the left-hand-side of the chain (T1= 100 [K]). In all cases the per-site temperature gradient is ∆T/N= 0.2 [K]. 0100200300400500/Minus0.10/Minus0.050.000.050.100.150.20 Sitenumber, n/ScriptA3 ΓQnz/Multiply1011/LBracket1/HBars/Minus1/RBracket1 FIG. 9: The dependence of the exchange spin torque on the site number. Numerical parameters are α= 0.01 and H0= 0 [K]. The temperature gradient is linear and the maximum temperature is on the left-hand-side of the chain (T1= 100 [K]). The per-site temperature gradient is ∆ T/N= 0.2 [K]. The exchange spin torque profile consists of three parts, the positive part corresponds to the high temperatur e domain and low temperature domain corresponds to the neg- ative exchange spin torque. In the middle of the chain where the spin current is constant, the exchange spin torque fluctu - ates in the vicinity of the zero value. and the spin torque lead to a nonzero spin current at the interfaces which is actually measured in the experiment. F. Role of the external magnetic filed ( H0/negationslash= 0) It follows from our calculations that the dependence of the longitudinal spin current on the magnetic field is not8 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Tp Tm Tp 0100 200 300 400 500020406080100 Sitenumber, nTemperature/LBracket1K/RBracket1 FIG. 10: The magnon temperature profile (line) formed in the system. Numerical parameters are α= 0.01 andH0= 0 [K]. Blue line corresponds to the applied linear phonon tempera- ture profile. The maximum temperature on the left-hand-side of the chain is( T1= 100 [K]). The per-site temperature gradi- ent is ∆T/N= 0.2 [K]. The maximal deviation between the phonon and magnon temperatures is observed only at the left edge of chain. The difference between temperatures graduall y decreases and becomes almost zero for the sites with n >400. trivial. Once the external static magnetic field is applied perpendicularly to the FM-chain and along the easy axis at the same time, we can suppress the spin current at ele- vated magnetic fields (FIG. 11). The threshold magnetic field is - asexpected - the strength ofthe anisotropyfield, i.e. 2K1/MS∼0.056 [T]. By applying magnetic fields much higher than 0 .056 [T], the magnetic moments are fully aligned along the field direction and hence the X-, Y-components of the magnetization required to form the Z-component of the longitudinal averaged spin current vanish. In the case of the magnetic field being applied perpen- dicularly to the easy axis, the behavior becomes more rich (FIG. 12). In analogy with the situation observed in FIG. 11 there are no sizeable changes for the In(∆T)- dependence at low static fields. This is the regime where the anisotropy field is dominant. In contrast to the Hz 0 applied field, the spin current does not linearly depend on the strength of the field (inset of FIG. 11), which is explained by the presence of different competing contri- butions in the total energy density and not a simple cor- rection of the Z-component of the anisotropy field illus- trated in the previous figure. Surprisingly, the magnetic field oriented along the FM-chain can also suppress the appearance of the spin current’s profile. Also in this case the strong magnetic field destroys the formation of the magnetization gradient resulting from the applied tem- perature bias./SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle /Circle/Circle/Circle/Circle/Circle/Circle /SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare /Square/Square/Square/Square/Square/Square/SolidCircleH0z/EquΑl0/LBracket1T/RBracket1 /CircleH0z/EquΑl10/Minus2/LBracket1T/RBracket1 /SolidSquareH0z/EquΑl10/Minus1/LBracket1T/RBracket1 /SquareH0z/EquΑl1/LBracket1T/RBracket1 0 10 20 30 40 500123 /DifferenceDeltaT/LBracket1K/RBracket1Spincurrent I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1 FIG. 11: Effect of the external magnetic field applied paralle l to the easy axis on the averaged spin current. Numerical parameters are ∆ T= 50 [K], α= 0.01 andN= 50. The temperature gradient is linear and the maximum temperature is on the left-hand-side of the chain. /SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle /Circle/Circle/Circle/Circle/Circle/Circle /SolidSquare/SolidSquare/SolidSquare/SolidSquare /SolidSquare/SolidSquare /Square/Square/Square/Square/Square/Square/SolidCircleH0x/EquΑl0/LBracket1T/RBracket1 /CircleH0x/EquΑl10/Minus2/LBracket1T/RBracket1 /SolidSquareH0x/EquΑl10/Minus1/LBracket1T/RBracket1 /SquareH0x/EquΑl1/LBracket1T/RBracket1 0 10 20 30 40 5001234 /DifferenceDeltaT/LBracket1K/RBracket1Spincurrent I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1 /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle /SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/DifferenceDeltaT/EquΑl50/LBracket1K/RBracket1 0.020.040.060.080.1001234 H0x/LBracket1T/RBracket1I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1 FIG. 12: Effect of the external magnetic field applied perpen- dicularly to the easy axis on the averaged spin current. Nu- merical parameters are ∆ T= 50 [K], α= 0.01 andN= 50. The temperature gradient is linear and the maximum tem- perature is on the left-hand-side of the chain. V. INTERFACE EFFECTS The experimental setup to detect the spin current might involve a NM adjacent to the spin-current gener- ating substance, e.g. a FM insulator. This NM converts the injected spin current from the FM to an electric cur- rent via ISHE1,5,27. So it is of interest to see the effect of the adjacent NM on the generated spin current in the considered chain. Obviously, the main effects appear in the FM-NM interface. The interface effect can be di- vided into two parts which is described in the following subsections.9 A. Spin pumping and enhanced Gilbert damping Inmagnetic insulators , chargedynamicsislessrelevant (in our model, anyway), and in some cases the dissipa- tive losses associated with the magnetization dynamics are exceptionally low (e.g. in YIG28α= 6.7×10−5). When a magnetic insulator is brought in contact with anormal metal , magnetization dynamics results in spin pumping, which in turn causes angular momentum being pumped to the NM. Because of this nonlocal interaction, the magnetization losses become enhanced21. If we consider the normal metal as a perfect spin sinkwhich remains in equilibrium even though spins are pumped into it (which means there is a rapid spin relax- ation and no back flow of spin currents to the magnetic insulator), the magnetization dynamics is described by the LLG equation with an additional torque originating from the FM-insulator/NM interfacial spin pumping21 ∂/vectorM ∂t=−γ/bracketleftBig /vectorM×/vectorHeff/bracketrightBig +α MS/bracketleftBigg /vectorM×∂/vectorM ∂t/bracketrightBigg +/vector τsp,(11) where /vector τsp=γ¯h 4πM2 Sgeffδ(x−L)/bracketleftBigg /vectorM×∂/vectorM ∂t/bracketrightBigg ,(12) whereLis the position of the interface, eis the elec- tron charge and geffis the real part of the effective spin-mixing conductance. In the YIG-Pt bilayer the maximum measured effective spin-mixing conductance is geff= 4.8×1020[m−2] Ref.21. In fact if the spin pumping torque should be completely described, one should add another torque containing the imaginary part of geff29. However, we omit this imaginary part here because it has been found to be too small at FM-NM interfaces30. The aforementioned spin pumping torque concerns the cases that we characterized with /vectorM. In our discrete model which includes a chain of Nferromagnetic cells, we describe the above phenomena as follows ∂/vectorMn ∂t=−γ/bracketleftBig /vectorMn×/vectorHeff n/bracketrightBig +α MS/bracketleftBigg /vectorMn×∂/vectorMn ∂t/bracketrightBigg +/vector τsp n, (13) where /vector τsp n=γ¯h2 2ae2M2 Sg⊥δnN/bracketleftbigg /vectorMn×∂Mn ∂t/bracketrightbigg ,(14) which means the spin pumping leads to an enhanced Gilbert damping in the last site ∆α=γ¯h 4πaMsgeff. (15) As mentioned, the above enhanced Gilbert damping could solely describe the interfacial effects as long as we treat the adjacent normal metal as a perfect spinsink without any back flow of the spin current from the NM17,21. The latter is driven by the accumulated spins in the normal metal. If we model the normal metal as a perfect spin sink for the spin current, spin accumu- lation does not build up. This approximation is valid when the spin-flip relaxation time is very small and so it prevents any spin-accumulation build-up. So the spins injected by pumping decayand/orleavethe interfacesuf- ficiently fast and there won’t be any backscattering into the ferromagnet13,31. We note by passing that in a re- centstudy concerningthis phenomena, it hasbeen shown that spin pumping (and so enhanced Gilbert damping) depends on the transverse mode number and in-plane wave vector21. B. Spin transfer torque It was independently proposed by Slonczewski32and Berger33that the damping torque in the LLG equation could have a negative sign as well, corresponding to a negative sign of α. This means that the magnetization vector could move into a final position antiparallel to the effective field. In order to achieve this, energy has to be supplied to the FM system to make the angle between the magnetization and the effective field larger. This en- ergy is thought to be provided by the injection of a spin current/vectorIincidentto the FM13,29,34 /vector τs=−γ M2 SV/bracketleftBig /vectorM×/bracketleftBig /vectorM×/vectorIinjected/bracketrightBig/bracketrightBig ,(16) which describes the dynamics of a monodomain ferro- magnet of volume Vthat is subject to the spin current /vectorIincidentand modifies the right-hand side of the LLG equation as a source term. In general, a torque-term additional to the Slonczewskis torque (eq. (16)) is also allowed29,35 /vector τsβ=−γ MSVβ/bracketleftBig /vectorM×/vectorIincident/bracketrightBig , (17) whereβgives the relative strength with respect to the Slonczewski’s torque (eq. (16)). For the case of a FM-chain, again we assume that the abovespin-transfertorques act solelyon the last FM cell. C. Numerical results for interface effects InordertosimulatetheenhancedGilbertdampingand the spin–transfer torque we assume that they act only on thechainend(motivatedbytheiraforementionedorigin). So the dynamics of our FM-chain is described by the following LLG18,36equations ∂/vectorMn ∂t=−γ 1+α2/bracketleftBig /vectorMn×/vectorHeff n/bracketrightBig −γα (1+α2)MS/bracketleftBig /vectorMn×/bracketleftBig /vectorMn×/vectorHeff n/bracketrightBig/bracketrightBig , n= 1,...,(N−1),(18)10 and ∂/vectorMN ∂t=−γ 1+α2 N/bracketleftBig /vectorMN×/vectorHeff N/bracketrightBig −γαN (1+α2 N)MS/bracketleftBig /vectorMN×/bracketleftBig /vectorMN×/vectorHeff N/bracketrightBig/bracketrightBig −γ M2 Sa3/bracketleftBig /vectorMN×/bracketleftBig /vectorMN×/vectorIinjected/bracketrightBig/bracketrightBig −γ MSa3β/bracketleftBig /vectorMN×/vectorIincident/bracketrightBig ,(19) whereαN=α+γ¯hgeff/(4πaMs). Eq.(18) and (19) describe the magnetizationdynamics in the presence of the interface effects and include both spinpump and spintorqueeffects. Results inthe absence of the spin torque are presented at the FIG. 13. The en- hanced Gilbert damping captures losses of the spin cur- rent associated with the interface effect. A nonzero spin current corresponding to the last n= 500 spin quantifies the amount of the spin current pumped into the normal metal from the magnetic insulator. However, the con- vex profile of the spin current is observed as well in the presence ofthe interface effects. The influence ofthe spin torqueonthespincurrentprofileisshowninFIG.14. We see from these results, the large spin torque reduces the totalspincurrentfollowingthroughtheFM-insulato/NM interfaces. The spin torque current is directed opposite to the spin pump current and therefore compensates it. 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/SmallCircle 0 100 200 300 400 500012 Sitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1 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/SmallCircle 450 475 500012 nWith enhanced damping, /DifferenceDeltaΑ=0.5 Without enhanced damping, /DifferenceDeltaΑ=0.5 FIG. 13: Statistically averaged spin current in the chain of N= 500-sites. Numerical parameters are ∆ T= 100 [K], α= 0.01 andH0= 0 [T]. The temperature gradient is lin- ear and the maximum temperature is on the left-hand-side of the chain ( T1). The blue curve shows the averaged spin current when no enhanced Gilbert damping and no spin– transfer torque is present. The red curve shows the aver- aged spin current when the enhanced Gilbert damping with geff= 1.14×1022[m−2] is present. The inset shows the aver- aged spin current of the last fifty sites 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/SmallCircle 0 100 200 300 400 500012 Sitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1 /SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/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 450 475 500012 nIn both cases /DifferenceDeltaΑ=0.5 incidentIincident/EquΑl 1.03/Multiply1015/LParen1/Minus1, 0, 0/RParen1/LBracket2/HBars/Minus1/RBracket2 Iincident/EquΑl 5.15/Multiply1015/LParen1/Minus1, 0, 0/RParen1/LBracket2/HBars/Minus1/RBracket2 FIG. 14: Statistically averaged spin current in the chain of N= 500 when there are both the enhanced Gilbert damp- ing and the spin–transfer torque. Numerical parameters are ∆T= 100 [K], α= 0.01,H0= 0 [T], geff= 1.14×1022[m−2] andβ= 0.01. The temperature gradient is linear and the maximum temperature is on the left-hand-side of the chain (T1). The blue curve has /vectorIincident= 1×1015(−1,0,0) [¯hs−1] and the red curve is with /vectorIincident= 5×1015(−1,0,0) [¯hs−1]. VI. MECHANISMS OF THE FORMATION OF SPIN EXCHANGE TORQUE AND SPIN SEEBECK CURRENT In the previous sections we demonstrated the direct connection between the spin Seebeck current profile and the exchange spin torque. Here we consider the mecha- nisms of the formation of the exchange spin torque. For this purpose we investigate changes in the magnetization profile associatedwith the changeof the magnontemper- ature<∆Mz n>=< Mz n>−< Mz 0n>, where< Mz n> is the mean component of the magnetization moment for the case of the applied linear thermal gradient, while < Mz 0n>correspondstothemeanmagnetizationcompo- nent in the absence of thermal gradient ∆ T= 0. Quan- tity<∆Mz n>defines the magnon accumulation as the differencebetweentherelativeequilibriummagnetization profile and excited one Ref.37and is depicted in FIG. 15. We observe a direct connection between the magnon ac- cumulation effect and the exchange spin torque. A pos- itive magnon accumulation, meaning an excess of the magnonscomparedtotheequilibriumstateisobservedin the high temperature part of the chain. While in the low temperature part the magnon accumulation is negative indicatingalackofmagnonscomparedtotheequilibrium state. The exchange spin torque is positive in the case of the positive magnon accumulation and is negative in the case of the negative magnon accumulation (the exchange spin torque vanishes in the equilibrium state). From the physical point of view, the result is comprehensible: the spin Seebeck current is generated by the magnon accu- mulation, transmitted through the equilibrium part of the chain and partially absorbed in the part of the chain with a negative magnon accumulation.11 /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 50 100 150 200/Minus4/Minus202 Sitenumber, nMagnon accumulation10/Minus5/Multiply/LParen1m/Minusm0/RParen1 15010015020020 nTn/LBracket1K/RBracket1 050100150200/Minus0.10/Minus0.050.000.050.100.15 n/ScriptA3 ΓQnz/Multiply1011/LBracket1/HBars/Minus1/RBracket1 FIG. 15: Site dependence of the exchange spin torque and the magnon accumulation effect. We observe a direct connec- tion between magnon accumulation effect and the exchange spin torque. A positive magnon accumulation, i.e. excess of the magnons, is observed in the high temperature part of the chain. While in the low temperature part magnon accumu- lation is negative (lack of magnons compare to the equilib- rium state). The exchange spin torque is positive for posi- tive magnon accumulation, and negative for negative magnon accumulation. The spin Seebeck current is generated by ex- cess magnons, transmitted through the equilibrium part of the chain and partially absorbed in the region with magnon drain. VII. CONCLUSIONS Based on the solution of the stochastic Landau- Lifshitz-Gilbert equation discretized for a ferromagnetic chain in the presence of a temperature gradient formed along the chain, we studied the longitudinal spin See- beck effect with a focus on the space-dependent effects. In particular, we calculated a longitudinal averaged spin current as a function of different temperature gradients, temperature gradient strengths, distinct chain lengths and differently oriented external static magnetic fields. Our particular interest was to explain the mechanisms of the formation of the spin Seebeck current beyond the linear response regime. The merit was in pointing out a microscopicmechanismfortheemergenceofthespinSee-beck current in a finite-size system. We have shown that, within our model, the microscopic mechanism of the spin Seebeck current is the magnon accumulation effect quan- tified in terms of the exchange spin torque. We proved that the magnon accumulation effect drives the spin See- beck current even in the absence of significant deviation between magnon and phonon temperature profiles. Our theoretical findings are in line with recently observed ex- perimental results25where non-vanishing spin Seebeck current was observed in the absence of a temperature difference between phonon and magnon baths. Concerningthe influence ofthe external constant mag- netic fields on the spin Seebeck current we found that their role is nontrivial: An external static magnetic field applied perpendicularly to the FM-chain and along the easy axis may suppress the spin current at elevated mag- netic fields (FIG. 11). The threshold magnetic field has a strengthofthe anisotropyfield, i.e. 2 K1/MS∼0.056[T]. In the case of the magnetic field applied perpendicu- larly to the easy axis, we observe a more complex be- havior (FIG. 12). In analogy with the situation seen in FIG. 11 there are no sizeable changes for the In(∆T)- dependence at low static fields. This is the regime where the anisotropy field is dominant. In contrast to the Hz 0 applied field, it does not linearly depend on the strength of the field (inset of FIG. 11), which is explained by the presence of different competing contributions in the total energyand not a simple correctionof the Z-componentof the anisotropy field. Notably, the magnetic field oriented along the FM-chain can also suppress the emergence of the spin current’s profile. Also in this case a strong mag- netic field destroys the formation of the magnetization gradient resulting from the applied temperature bias. In addition, we modeled an interface formed by a nonuniformly magnetized finite size ferromagnetic insu- lator and a normal metal (e.g., YIG-Platinum junction) to inspect the effects of the enhanced Gilbert damping on the formation of space-dependent spin current within the chain. VIII. ACKNOWLEDGEMENTS The financial support by the Deutsche Forschungsge- meinschaft (DFG) is gratefully acknowledged. 1K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. 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2014-02-27

Based on the solution of the stochastic Landau-Lifshitz-Gilbert equation discretized for a ferromagnetic chain subject to a uniform temperature gradient, we present a detailed numerical study of the spin dynamics with a focus particularly on finite-size effects. We calculate and analyze the net longitudinal spin current for various temperature gradients, chain lengths, and external static magnetic fields. In addition, we model an interface formed by a nonuniformly magnetized finite-size ferromagnetic insulator and a normal metal and inspect the effects of enhanced Gilbert damping on the formation of the space-dependent spin current within the chain. A particular aim of this study is the inspection of the spin Seebeck effect beyond the linear response regime. We find that within our model the microscopic mechanism of the spin Seebeck current is the magnon accumulation effect quantified in terms of the exchange spin torque. According to our results, this effect drives the spin Seebeck current even in the absence of a deviation between the magnon and phonon temperature profiles. Our theoretical findings are in line with the recently observed experimental results by M. Agrawal et al., Phys. Rev. Lett. 111, 107204 (2013).

On the longitudinal spin current induced by a temperature gradient in a ferromagnetic insulator

1402.6899v1

Current Induced Damping of Nanosized Quantum Moments in the Presence of Spin-Orbit Interaction Farzad Mahfouziand Nicholas Kioussisy Department of Physics and Astronomy, California State University, Northridge, CA, USA (Dated: November 10, 2021) Motivated by the need to understand current-induced magnetization dynamics at the nanoscale, we have developed a formalism, within the framework of Keldysh Green function approach, to study the current-induced dynamics of a ferromagnetic (FM) nanoisland overlayer on a spin-orbit-coupling (SOC) Rashba plane. In contrast to the commonly employed classical micromagnetic LLG simula- tions the magnetic moments of the FM are treated quantum mechanically . We obtain the density matrix of the whole system consisting of conduction electrons entangled with the local magnetic moments and calculate the eective damping rate of the FM. We investigate two opposite limiting regimes of FM dynamics: (1) The precessional regime where the magnetic anisotropy energy (MAE) and precessional frequency are smaller than the exchange interactions, and (2) The local spin- ip regime where the MAE and precessional frequency are comparable to the exchange interactions. In the former case, we show that due to the nite size of the FM domain, the \Gilbert damping"does not diverge in the ballistic electron transport regime, in sharp contrast to Kambersky's breathing Fermi surface theory for damping in metallic FMs. In the latter case, we show that above a critical bias the excited conduction electrons can switch the local spin moments resulting in demagnetization and reversal of the magnetization. Furthermore, our calculations show that the bias-induced anti- damping eciency in the local spin- ip regime is much higher than that in the rotational excitation regime. PACS numbers: 72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg I. INTRODUCTION Understanding the current-induced magnetization switching (CIMS) at the nanoscale is mandatory for the scalability of non-volatile magnetic random access mem- ory (MRAM) of the next-generation miniaturized spin- tronic devices. However, the local magnetic moments of a nanoisland require quantum mechanical treatment rather than the classical treatment of magnetization commonly employed in micromagnetic simulations, which is the cen- tral theme of this work. The rst approach of CIMS employs the spin transfer torque (STT)1,2in magnetic tunnel junctions (MTJ) con- sisting of two ferromagnetic (FM) layers (i.e., a switch- able free layer and a xed layer) separated by an insulat- ing layer, which involves spin-angular-momentum trans- fer from conduction electrons to local magnetization3,4. Although STT has proven very successful and brings the precious benet of improved scalability, it requires high current densities ( 1010A/cm2) that are uncomfort- ably high for the MTJ's involved and hence high power consumption. The second approach involves an in-plane current in a ferromagnet-heavy-metal bilayer where the magnetization switching is through the so-called spin- orbit torque (SOT) for both out-of-plane and in-plane magnetized layers.5{8The most attractive feature of the SO-STT method is that the current does not ow through the tunnel barrier, thus oering potentially faster and more ecient magnetization switching compared to the MTJs counterparts. As in the case of STT, the SO-STT has two compo- nents: a eld-like and an antidamping component. Whilethe eld-like component reorients the equilibrium direc- tion of the FM, the antidamping component provides the energy necessary for the FM dynamics by either enhanc- ing or decreasing the damping rate of the FM depending on the direction of the current relative to the magneti- zation orientation as well as the structural asymmetry of the material. For suciently large bias the SOT can overcome the intrinsic damping of the FM leading to ex- citation of the magnetization precession.8The underlying mechanism of the SOT for both out-of-plane and in-plane magnetized layers remains elusive and is still under de- bate. It results from either the bulk Spin Hall Eect (SHE)9{12, or the interfacial Rashba-type spin-orbit cou- pling,13{16or both17{19. Motivated by the necessity of scaling down the size of magnetic bits and increasing the switching speed, the objective of this work is to develop a fully quantum me- chanical formalism, based on the Keldysh Green function (GF) approach, to study the current-induced local mo- ment dynamics of a bilayer consisting of a FM overlayer on a SOC Rashba plane, shown in Fig. 1. Unlike the commonly used approaches to investigate the magnetization dynamics of quantum FMs, such as the master equation20, the scattering21or quasi-classical22 methods, our formalism allows the study of magnetiza- tion dynamics in the presence of nonequilibrium ow of electrons. We consider two dierent regimes of FM dynamics: In the rst case, which we refer to as the single domain dynamics, the MAE and the precession frequency are smaller than the exchange interactions, and the FM can be described by a single quantum magnetic moment, of a typically large spin, S, whose dynamics are governedarXiv:1702.08408v2 [cond-mat.mes-hall] 27 Apr 20172 FIG. 1: (Color online) Schematic view of the FM/Rashba plane bilayer where the FM overlayer has length Lxand is innite (nite) along the y-direction for the case of a single domain (nano-island) discussed in Sec. III (IV). The magne- tization,~ m, of the FM precesses around the direction denoted by the unit vector, ~ nM, with frequency !and cone angle, . The Rashba layer is attached to two normal (N) leads which are semi-innite along the x-direction, across which an exter- nal bias voltage, V, is applied. mainly by the quantized rotational modes of the magne- tization. We show that the magnetic degrees of freedom entering the density matrix of the conduction electron- local moment entagled system simply shift the chemical potential of the Fermi-Dirac distribution function by the rotational excitations energies of the FM from its ground state. We also demonstrate that the eective damping rate is simply the netcurrent along the the auxiliarym- direction, where m= -S, -S+1, :::, +S, are the eigenval- ues of the total Szof the FM. Our results for the change of the damping rate due to the presence of a bias volt- age are consistent with the anti-damping SOT of clas- sical magnetic moments,16,23, where due to the Rashba spin momentum locking, the anti-damping SOT, to low- est order in magnetic exchange coupling, is of the form, ~ m(~ m^y), where ^yis an in-plane unit vector normal to the transport direction. In the adiabatic and ballistic transport regimes due to the nite S value of the nanosize ferromagnet our formal- ism yields a nite \Gilbert damping", in sharp contrast to Kambersky's breathing Fermi surface theory for damp- ing in metallic FMs.24On the other hand, Costa and Muniz25and Edwards26demonstrated that the prob- lem of divergent Gilbert damping is removed by takinginto account the collective excitations. Furthermore, Ed- wards points out26the necessity of including the eect of long-range Coulomb interaction in calculating damping for large SOC. In the second case, which corresponds to an indepen- dent local moment dynamics, the FM has a large MAE and hence the rotational excitation energy is compara- ble to the local spin- ip excitation (exchange energy). We investigate the eect of bias on the damping rate of the local spin moments. We show that above a criti- cal bias voltage the owing conduction electrons can ex- cite (switch) the local spin moments resulting in demag- netization and reversal of the magnetization. Further- more, we nd that, in sharp contrast to the single do- main precessional dynamic, the current-induced damping is nonzero for in-plane and out-of-plane directions of the equilibrium magnetization. The bias-induced antidamp- ing eciency in the local moment switching regime is much higher than that in the single domain precessional dynamics. The paper is organized as follows. In Sec. II we present the Keldysh formalism for the density matrix of the en- tagled quantum moment-conduction electron system and the eective dampin/antdamping torque. In Sec. III we present results for the current-induced damping rate in the single domain regime. In Sec. IV we present results for the current-induced damping rate in the independent local regime. We conclude in Sec. V. II. THEORETICAL FORMALISM Fig. 1 shows a schematic view of the ferromagnetic heterostructure under investigation consisting of a 2D ferromagnet-Rashba plane bilayer attached to two semi- innite normal (N) leads whose chemical potentials are shifted by the external bias, Vbias. The magnetization of the FM precesses around the axis specied by the unit vector, ~ nM, with frequency !and cone angle . The FM has length, LFM x, along the transport direction. The total Hamiltonian describing the coupled conduc- tion electron-localized spin moment system in the het- erostructure in Fig. 1 can be written as, Htot=X rr0;0Trfsdgh 1s^H0 rr0+rr001sr+rr0Jsd~ 0
~sd(r) +0rr0HM  fs0 dgr00 fsdgri : (1) Here,~sd(r) is the local spin moment at atomic position r, the trace is over the dierent congurations of the lo- cal spin moments, fsdg, fsdgr=jfsdgi e ris the quasi-particle wave-function associated with the conduc-tion electron ( e) entangled to the FM states ( jfsdgi), Jsdis thesdexchange interaction, 1sis the identity ma- trix in spin conguration space, and ^ x;y;z are the Pauli matrices. We use the convention that, except for r, bold3 symbols represent operators in the magnetic congura- tion space and symbols with hat represent operators in the single particle Hilbert space of the conduction elec- trons. The magnetic Hamiltonian HMis given by HM=gBX r~Bext(r)
~sd(r) (2) X hr;r0iJdd rr0 s2 d~sd(r0)
~sd(r)X rJsd sd~sc(r)
~sd(r); where, the rst term is the Zeeman energy due to the external magnetic eld, the second term is the magnetic coupling between the local moments and the third term is the energy associated with the intrinsic magnetic eld acting on the local moment, ~sd(r), induced by the local spin of the conduction electrons, ~sc(r). The Rashba model of a two-dimensional electron gas with spin orbit coupling interacting with a system of localized magnetic moments has been extensively em- ployed14,27,28to describe the eect of enhanced spin-orbit coupling solely at the interface on the current-induced torques in ultrathin ferromagnetic (FM)/heavy metal (HM) bilayers. The eects of (i) the ferromagnet induc- ing a moment in the HM and (ii) the HM with strong spin-orbit coupling inducing a large spin-orbit eect in the ferromagnet (Rashba spin-orbit coupling) lead to a thin layer where the magnetism and the spin-orbit cou- pling coexist.27 The single-electron tight-binding Hamiltonian29for the conduction electrons of the 2D Rashba plane, H0 rr0 which is nite along the transport direction xand innite along theydirection is of the form, ^H0 xx0(kya) = [tcos(kya)0tsosin(kya)x 0]xx0(3) +t(x;x0+1+x+1;x0)0+itso(x;x0+1x+1;x0)y 0: Here,x;x0denote atomic coordinates along the trans- port direction, ais the in-plane lattice constant, and tso is the Rashba SOI strength. The values of the local eec- tive exchange interaction, Jsd= 1eV, and of the nearest- neighbor hopping matrix element, t=1 eV, represent a realistic choice for simulating the exchange interaction of 3dferromagnetic transition metals and their alloys (Fe, Co).30{32The Fermi energy, EF=3.1 eV, is about 1 eV below the upper band edge at 4 eV consistent with the ab initio calculations of the (111) Pt surface33. Further- more, we have used tso=0.5 eV which yields a Rashba parameter, R=tsoa1.4 eV A (a=2.77 A is the in- plane lattice constant of the (111) Pt surface) consis- tent with the experimental value of about 1-1.5 eV A34 and the ab initio value of 1 eV A28. However, because other experimental measurements for Pt/Co/Pt stacks report35a Rashba parameter which is an order of mag- nitude smaller, in Fig.3 we show the damping rate for dierent values of the Rashba SOI.. For the results in Sec. IV, we assume a real space tight binding for propa- gation along y-axis.The single particle propagator of the coupled electron- spin system is determined from the equation of motion of the retarded Green function,  Ei^^HHMJsd 2^~ 
^~sd ^Gr(E) =^1;(4) where,is the broadening of the conduction electron states due to inelastic scattering from defects and/or phonons, and for simplicity we ignore writing the identity matrices ^1 and 1in the expression. The density matrix of the entire system consisting of the noninteracting elec- trons (fermionic quasi-particles) and the local magnetic spins is determined (see Appendix A for details of the derivation for a single FM domain) from the expression, ^=ZdE ^Gr(E)f(E^HM)^Ga(E): (5) It is important to emphasize that Eq. (5) is the central result of this formalism which demonstrates that the ef- fect of the local magnetic degrees of freedom is to shift the chemical potential of the Fermi-Dirac distribution func- tion by the eigenvalues, "m, ofHMjmi="mjmi, i.e., the excitation energies of the FM from its ground state. Here,jmiare the eigenstates of the Heisenberg model describing the FM. The density matrix can then be used to calculate the local spin density operator of the con- duction electrons, [ ~sc(r)]mm0=P ss0mm0 ss0;rr~ ss0=2, which along with Eqs. (2), (4), and (5) form a closed set of equations that can be solved self consistently. Since, the objective of this work is the damping/anti-damping (transitional) behavior of the FM in the presence of bias voltage, we only present results for the rst iteration. Eq. (5) shows that the underlying mechanism of the damping phenomenon is the ow of conduction electrons from states of higher chemical potential to those of lower one where the FM state relaxes to its ground state by transferring energy to the conduction electrons. There- fore, the FM dynamical properties in this formalism is completely governed by its coupling to the conduction electrons, where conservation of energy and angular mo- mentum dictates the excitations as well as the uctua- tions of the FM sate through the Fermi distribution func- tion of the electrons coupled to the reservoirs. This is dierent from the conventional Boltzmann distribution function which is commonly used to investigate the ther- mal and quantum uctuations of the magnetization. Due to the fact that the number of magnetic congura- tions (i.e. size of the FM Hilbert space) grows exponen- tially with the dimension of the system it becomes pro- hibitively expensive to consider all possible eigenstates of theHMoperator. Thus, in the following sections we consider two opposite limiting cases of magnetic cong- urations. In the rst case we assume a single magnetic moment for the whole FM which is valid for small FMs with strong exchange coupling between local moments and small MAE. In this case the dynamics is mainly gov- erned by the FM rotational modes and local spin ips can4 FIG. 2: (Color online) Schematic representation of the quasi- particles of the FM and conduction electron entangled states. The horizontal planes denote the eigenstates, jS;miof the totalSzof the FM with eigenvalues m=S;S+ 1;:::; +S along the auxiliarym-direction. Excitation of magnetic state induces a shift of the chemical potential of the Fermi-Dirac distribution function leading to ow of quisiparticles along the m-direction which corresponds to the damping rate of the FM. The FM damping involves two processes: (1) An intra-plane process involving spin reversal of the conduction electron via the SOC; and (2) An inter-plane process involving quasiparti- cle ow of majority (minority) spin along the ascending (de- scending)m-direction due to conservation of total angular momentum, where the interlayer hopping is accompanied by a spin ip of conduction electrons. be ignored. In the second case we ignore the correlation between dierent local moments and employ a mean eld approximation such that at each step we focus on an indi- vidual atom by considering the local moment under con- sideration as a quantum mechanical object while the rest of the moments are treated classically. We should men- tion that a more accurate modeling of the system should contain both single domain rotation of the FM as well as the local spin ipping but also the eect of nonlocal correlations between the local moments and conduction electrons, which are ignored in this work. III. SINGLE DOMAIN ROTATIONAL SWITCHING In the regime where the energy required for the excita- tion of a single local spin moment ( meV ) is much larger than the MAE (eV) the low-energy excited states cor- respond to rotation of the total angular momentum of the FM acting as a single domain and the eects of local spin ips described as the second term in Eq 2, can be ignored. In this regime all of the local moments behave collectively and the local moment operators can be replaced by the average spin operator, ~sd(r) =P r0~sd(r0)=Nd=sd~S=S, whereNdis the number of local moments and ~Sis the total angular momentum with amplitude S. The mag- netic energy operator is given by HM=~B
S, where, ~B=gB~Bext+Jsd~ sc. Here, for simplicity we assume ~ scto be scalar and independent of the FM state. The eigenstates of HMoperator are then simply the eigen-states,jS;mi, of the total angular momentum Sz, with eigenvalues m!=S!;:::; +S!, where!=Bzis the Larmor frequency. Thus, the wave function of the cou- pled electron-spin conguration system, shown schemat- ically in Fig. 2 is of the form, ms0r(t) =jS;mi s0r(t). One can see that the magnetic degrees of freedom corre- sponding to the dierent eigenstates of the Szoperator, enters as an additional auxiliary dimension for the elec- tronic system where the variation of the magnetic energy, hS;mjHMjS;mi=m!, shifts the chemical potentials of the electrons along this dimension. The gradient of the chemical potential along the auxiliary direction, is the Larmor frequency ( eVGHz ) which appears as an eective \electric eld"in that direction. Substituting Eq (5) in Eq (A1)(b) and averaging over one precession period we nd that the average rate of angular momentum loss/gain, which we refer to as the eective \ damping rate "per magnetic moment, can be written as Tm=1 2=(T mT+ m); (6) where, T m=Jsd 2SNdTrel[^S m^m;m1]: (7) is the current along the auxiliarym-direction in Fig. 2 from them$m+ 1 (sign) state of the total Szof the FM. Here,Trel, is the trace over the conduction electron degrees of freedom, and S m=p S(S+ 1)m(m1) are the ladder operators. It is important to note that within this formalism the damping rate is simply the net current across the mth-layer along the auxiliary direction associated with the transition rate of the FM from state mto its nearest-neighbor states ( m1). Fig. 3 shows the damping rate as a function of the pre- cession cone angle, = cos1(m S), for dierent values of bias and for an in-plane eective magnetic eld (a) along and (b) normal to the transport direction, and (c) an out- of-plane magnetic eld. For cases (a) and (c) the damp- ing rate is negative and relatively independent of bias for low bias values. A negative damping rate implies that the FM relaxes towards the magnetic eld by losing its angu- lar momentum, similar to the Gilbert damping rate term in the classical LLG equation, where its average value over the azimuthal precession angle, '=!t, is of the form,T=sdRd' 2~ m(~ m~B)
~ nM, which is nonzero (zero) when the unit vector ~ nMis along (perpendicular to) the eective magnetic eld. The dependence of the damping rate on the bias voltage when the eective mag- netic eld~Bis inplane and normal to the transport direc- tion can be understood by the spin- ip re ection mech- anism accompanied by Rashba spin-momentum locking described in Ref.16. One can see that a large enough bias can result in a sign reversal of the damping rate and hence a magnetization reversal of the FM. It's worth mention- ing that due to the zero-point quantum uctuations of5 the magnetization, at = 0;(i.e.m=S) we have T 6=0 which is inversely proportional to the size of the magnetic moment, S. In Fig. 4(a) we present the eective damping rate ver- sus bias for dierent values of the Rashba SOC. The re- sults show a linear response regime with respect to the bias voltage where both the zero-bias damping rate and the slope,dT=dV increases with the Rashba SOC. This is consistent with Kambersky's mechanism of Gilbert damping due to the SOC of itinerant electrons,24and the SOT mechanism16. Fig. 4(b) shows that in the absence of bias voltage the damping rate is proportional to t2 soand the eect of the spin current pumped into the left and right reservoirs is negligible. This result of the t2 sodepen- dence of the zero-bias damping rate is in agreement with recent calculations of Costa and Muniz25and Edwards26 which took into account the collective excitations. In the presence of an external bias, Tvaries linearly with the SOC, suggesting that to the lowest order it can be tted to T= sin2()tso(c1tso~!+c2eVbias); (8) wherec1andc2are tting parameters. The bias-induced eciency of the anti-damping SOT, ~!(T(Vbias)T(0))=eVbiasT(0), describes how e- cient is the energy conversion between the magnetization dynamics and the conduction electrons. Accordingly, for a given bias-induced eciency, , one needs to apply an external bias equal to ~!=e to overcome the zero-bias damping of the FM. Fig. 5 displays the anti-damping ef- ciency versus the position of the Fermi energy of the FM from the bottom (-4 t=-4 eV) to the top (4 t=4eV) of the conduction electron band for the two-dimensional square lattice. The result is independent of the bias voltage and the Larmor frequency in the linear response regime ( i.e. Vbias;!t). We nd that the eciency peaks when the Fermi level is in the vicinity of the bottom or top of the energy band where the transport is driven by electron- or hole-like carriers and the Gilbert damping is minimum. The sign reversal of the antidamping SOT is due to the electron- or hole-like driven transport similar to the Hall eect.36 Classical Regime of the Zero Bias Damping rate | In the following we show that in the case of classical magnetic moments ( S!1 ) and the adiabatic regime (!!0), the formalism developed in this paper leads to the conventional expressions for the damping rate. In this limit the system becomes locally periodic and one can carry out a Fourier transformation from mSzspace to azimuthal angle of the magnetization orientation, ', space. Conservation of the angular momentum suggests that the majority- (minority-) spin electrons can propa- gate only along the ascending (descending) m-direction, where the hopping between two nearest-neighbor m- layers is accompanied by a spin- ip. As shown in Fig. 2 the existence of spin- ip hopping requires the presence of intralayer SOC-induced noncollinear spin terms which rotate the spin direction of the conduction electrons as 04590135180−10−505B=[|B|,0,0]Damping Rate ( µeV) 4590135180B=[0,|B|,0] Cone Angle, θ (deg)4590135180B=[0,0,|B|] Vbias=−5 mV Vbias=0 Vbias=5 mV(a) (b) (c) Student Version of MATLABFIG. 3: (Color online) Eective damping rate for a single FM domain as a function of the precession cone angle, , for various bias values under an eective magnetic eld which is in-plane (a) along and (b) normal to the transport direction and (c) out-of-plane. The length of the FM along the xdi- rection isLx= 25awhile it is assumed to be innite in the y-direction, ~!= 10eV, the broadening parameter = 0, kBT= 10meV and the domain magnetic moment S= 200. The results are robust with larger values of Sin either the ballistic,~!, or dirty,~!, regimes. −2−1012−3−2−101 Bias Voltage (mV)Damping Rate ( µeV) −0.5 0 0.5−15−10−505 Spin Orbit Coupling, tso (eV) tso=0 tso=0.1 eV tso=0.2 eVVbias=−5 mV Vbias=0 Vbias=5 mV(a) (b) Student Version of MATLAB FIG. 4: (Color online) Damping torque versus (a) bias voltage and (b) spin-orbit coupling strength, for m= 0 corresponding to the precession cone angle of 90o. The precession axis of the FM is along the y-direction and the rest of the parameters are the same as in Fig. 3. The zero-bias damping rate versus SOC shows at2 sodependence while the damping rate under non- zero bias exhibits nearly linear SOC dependence. they propagate in each m-layer. This is necessary for the persistent ow of electrons along the 'auxiliary di- rection and therefore damping of the magnetization dy- namics. Using the Drude expression of the longitudinal conductivity along the '-direction for the damping rate, we nd that, within the relaxation time approximation, =!!1 , where the relaxation time of the excited con- duction electrons is much shorter than the time scale of6 −4−3−2−101234−3 −2 −1 0 1 2 3 Fermi Energy (eV)Antidamping Efficiency (%) −4−2024−40−200Damping Rate ( µeV) Vbias=5 mV Vbias=0 Vbias=−5 mV Student Version of MATLAB FIG. 5: (Color online) Bias-induced precessional anti- damping eciency,  = ~!(T(Vbias)T(0))=eVbiasT(0), ver- sus the Fermi energy of the 2D Rashba plane in Fig.1, where the energy band ranges from -4 eV to +4 eV. The magnetiza- tion precesses around the in-plane direction ( yaxis) normal to the transport direction and the rest of the parameters of the system are the same as in Fig. 3. Note, for magnetization precession around the xandzaxis,T(Vbias) =T(0) for all precession cone angles and hence =0. Inset shows the damp- ing rate versus the Fermi energy for dierent bias values used to calculate precessional anti-damping eciency. the FM dynamics, Tis given by T=! X nZdkxdkyd' (2)3(v' n~k)2f0("n~k(')):(9) Here,v' n~k=@"n~k(')=@' is the group velocity along the '-direction in Fig. 2, and "n;~k="0(j~kj)j~h(~k)jfor the 2D-Rashba plane, where "0(j~kj) is the spin independent dispersion of the conduction electrons and ~h=atso^ez ~k+1 2Jsd~ m, is the spin texture of the electrons due to the SOC and the sdexchange interaction. For small precession cone angle, , the Gilbert damping constant can be determined from =T=sd!sin2(), where the zero-temperature Tis evaluated by Eq. (9). We nd that 1 t2 so (k+ Fa)2D+(EF) + (k Fa)2D(EF) (1+cos2( )); (10) whereD+()(E) is the density of states of the majority (minority) band, is the angle between the precession axis and the normal to the Rashba plane, and the Fermi wave-vectors ( k F) are obtained from, "0(k F) =EF Jsd=2. Eq. (10) shows that the Gilbert damping increases as the precession axis changes from in-plane ( ==2) to out of plane ( = 0),37which can also be seen in Fig. 3. It is important to emphasize that in contrast to Eq. (9) the results shown in Fig. 4 correspond to the ballistic regime with = 0 in the central region and the relaxation of the excited electrons occurs solely inside the metallic reservoirs. To clarify how the damping rate changes from 10−810−610−410−2100−80−60−40−20020 Broadening (eV)Damping Rate ( µeV) S=200, Vbias=0 S=200, Vbias=3 mV S=300, Vbias=0 S=300, Vbias=3 mV Student Version of MATLABFIG. 6: (Color online) Precessional damping rate versus broadening of the states in the presence (solid lines) and ab- sence (dashed lines) of bias voltage for two values of the do- main sizeS= 200 and S= 300. In both ballistic, =!0, and diusive, =!1, regimes the precessional damping rate is independent of the domain size, while in the intermediate case, the amplitude of the minimum of damping rate shows a linear dependence versus S. Note that the value of the broad- ening at which the damping rate is minimum varies inversely proportional to the domain size, S. the ballistic to the diusive regime we present in Fig. 6 the damping rate versus the broadening, , of states in the presence (solid line) and absence (dashed line) of bias voltage. We nd that in both ballistic ( =!0) and diusive ( =!1) regimes the damping rate is independent of the size of the FM domain, S. On the other hand, in the intermediate regime the FM dynamics become strongly dependent on the eective domain size where the minimum of the damping rate varies linearly withS. This can be understood by the fact that the eective chemical potential dierence between the rst, m=Sand last,m=Slayers in Fig.3 is proportional toSand for a coherent electron transport the conduc- tance is independent of the length of the system along the transport direction. Therefore, in this case the FM motion is driven by a coherent dynamics. IV. DEMAGNETIZATION MECHANISM OF SWITCHING In Sec. III we considered the case of a single FM do- main where its low-energy excitations, involving the pre- cession of the total angular momentum, can be described by the eigenstates jmiofSzand local spin ip processes were neglected. However, for ultrathin FM lms or FM nanoclusters, where the MAE per atom ( meV ) is com- parable to the exchange energy between the local mo- ments (Curie temperature), the low-energy excitations involve both magnetization rotation and local moments spin- ips due to conduction electron scattering which can in turn change also S. In this case the switching is ac-7 FIG. 7: (Color online) Spatial dependence of the local damp- ing rate for the spin-1 =2 local moments of a FM island under dierent bias voltages ( 0:4V) and magnetization directions. For the parameters we chose the size of the FM island to be 2525a2, the eective magnetic eld, jBj= 20meV , the broadening, = 0, andkBT= 10 meV. companied by the excitation of local collective modes that eectively lowers the amplitude of the magnetic ordering parameter. For simplicity we employ the mean eld ap- proximation for the 2D FM nanocluster where the spin under consideration at position ris treated quantum me- chanically interacting with all remaining spins through an eective magnetic eld, ~B. The spatial matrix ele- ments of the local spin operator are [^~sd;r]r1r2=~ sd(r1)r1r2(1r1r)1s+1 2r1r2r1r~ ;(11) where,~s are the Pauli matrices. The magnetic energy can be expressed as, HM(r) =~B(r)
~=2, where, the eective local magnetic eld is given by, ~B(r) =gB~Bext+ 4X r0Jdd rr0~ sd(r0) + 2Jsd~ sc(r):(12) The equation of motion for the single particle propa- gator of the electronic wavefunction entangled with the local spin moment under consideration can then be ob- tained from,  E^HM(r)^HJsd 2^~ 
^~sd;r ^Gr r(E) =^1:(13) The density matrix is determined from Eq. (5) which can in turn be used to calculate the spin density of the conduction electrons, ~ sc(r) =Tr(^~ ^rr)=2, and the di- rection and amplitude of the local magnetic moments, ~ sd(r) =Tr(~^rr)=2. Fig. 7 shows the spatial dependence of the spin-1 2local moment switching rate for a FM/Rashba bilayer (Fig. −101−30−20−100102030 Damping Torque (meV)B=[|B|,0,0] −101 Bias Voltage (V)B=[0,|B|,0] −101B=[0,0,|B|] |B|=1 meV |B|=20 meV(c) (a) (b) Student Version of MATLABFIG. 8: (Color online) Bias dependence of the average (over all sites) damping rate of the FM island for in-plane eective magnetic eld (or equilibrium magnetization) (a) along and (b) normal to the transport direction and (c) out-of-plane magnetic eld for two values of jBj. 1) for two bias values ( Vbias=0:4V) and for an in- plane eective magnetic eld (a) along and (b) normal to the transport direction, and (c) an out-of-plane mag- netic eld. The size of the FM island is 25 a25a, where ais the lattice constant. Negative local moment switch- ing rate (blue) denotes that, once excited, the local mo- ment relaxes to its ground state pointing along the di- rection of the eective magnetic eld; however positive local damping rate (red) denotes that the local moments remain in the excited state during the bias pulse dura- tion. Therefore, the damping rate of the local moments under bias voltage can be either enhanced or reduced and even change sign depending on the sign of the bias voltage and the direction of the magnetization. We nd that the bias-induced change of the damping rate is high- est when the FM magnetization is in-plane and normal to the transport directions similar to the single domain case. Furthermore, the voltage-induced damping rate is peaked close to either the left or right edge of the FM (where the reservoirs are attached) depending on the sign of the bias. Note that there is also a nite voltage-induced damping rate when the magnetization is in-plane and and along the transport direction ( x) or out-of-the-plane ( z). Fig. 8 shows the bias dependence of the average (over all sites) damping rate for in- (a and b) and out-of-plane (c) directions of the eective magnetic eld (direction of the equilibrium magnetization) and for two values of jBj. This quantity describes the damping rate of the amplitude of the magnetic order parameter. For an in- plane magnetization and normal to the transport direc- tion (Fig. 8) the bias behavior of the damping rate is lin- ear and nite in contrast to the single domain [Fig. 3(a)] where the damping rate was found to have a negligible response under bias. On the other hand, the bias behav- ior of the current induced damping rate shows similar behavior to the single domain case when the equilibrium8 −4 −2 0 2 4−30−20−100102030 Fermi Energy (eV)Antidamping Efficiency (%) B=[20,0,0] meV B=[0,20,0] meV B=[0,0,20] meV Student Version of MATLAB FIG. 9: (Color online) Bias-induced local anti-damping eciency due to local spin- ip,  = jBj(T(Vbias) T(0))=eVbiasT(0), versus Fermi energy for dierent equilib- rium magnetization orientations. For the calculation we chose Vbias= 0:2 V and the rest of the Hamiltonian parameters are the same as in Fig. 7. magnetization direction is in-plane and normal to the transport direction (Fig. 8(b)). For an out-of-plane ef- fective magnetic eld [Fig. 8(c)] the damping torque has an even dependence on the voltage bias. In order to quantify the eciency of the voltage in- duced excitations of the local moments, we calculate the relative change of the average of the damping rate in the presence of a bias voltage and present the result versus the Fermi energy for dierent orientations of the magne- tization in Fig 9. We nd that the eciency is maximum for an in-plane equilibrium magnetization normal to the transport direction and it exhibits an electron-hole asym- metry. The bias-induced antidamping eciency due to spin- ip can reach a peak around 20% which is much higher than the peak eciency of about 2% in the sin- gle domain precession mechanism in Fig. 5 for the same system parameters. Future work will be aimed in determining the switch- ing phase diagram16by calculating the local antidamping and eld-like torques self consistently for dierent FM congurations. V. CONCLUDING REMARKS In conclusion, we have developed a formalism to in- vestigate the current-induced damping rate of nanoscale FM/SOC 2D Rashba plane bilayer in the quantum regime within the framework of the Kyldysh Green func- tion method. We considered two dierent regimes of FM dynamics, namely, the single domain FM and indepen- dent local moments regimes. In the rst regime we as- sume the rotation of the FM as the only degree of free- dom, while the second regime takes into account only the local spin- ip mechanism and ignores the rotation ofthe FM. When the magnetization (precession axis) is in- plane and normal to the transport direction, similar to the conventional SOT for classical FMs, we show that the bias voltage can change the damping rate of the FM and for large enough voltage it can lead to a sign reversal. In the case of independent spin-1 =2 local moments we show that the bias-induced damping rate of the local quantum moments can lead to demagnetization of the FM and has strong spatial dependence. Finally, in both regimes we have calculated the bias-induced damping eciency as a function of the position of the Fermi energy of the 2D Rashba plane. Appendix A: Derivation of Electronic Density Matrix Using the Heisenberg equation of motion for the an- gular momentum operator, ~S(t), and the commutation relations for the angular momentum, we obtain the fol- lowing Landau-Lifshitz equations of motion, i@ @tS(t) =hzS(t)h(t)Sz(t) (A1a) i@ @tSz(t) =1 2 h+(t)S(t)h(t)S+(t)
(A1b) ~hmm0(t) =1 ~X rJsd~smm0 c(r) +gBmm0~B(t);(A1c) where,S=SxSy(=xy), is the angu- lar momentum (spin) ladder operators, ~smm0 c(r) = 1 2P 0~ 0mm0 0;rris the local spin density of the con- duction electrons which is an operator in magnetic con- guration space. Here, is the density matrix of the system, and the subscripts, r;m; refer to the atomic cite index, magnetic state and spin of the conduction electrons, respectively. In the following we assume a pre- cessing solution for Eq (A1)(a) with a xed cone angle and Larmor frequency !=hz. Extending the Hilbert space of the electrons to include the angular momentum degree of freedom we dene ms0i(t) =jS;mi s0i(t). The equation of motion for the Green function (GF) is then given  Ei^H(k) +n!n 2SJsd(k)z ^Gr nm(E;k) (A2) p S(S+ 1)n(n+ 1) 2SJsd(k)^Gr n+1m(E;k) p S(S+ 1)n(n1) 2SJsd(k)+^Gr n1m(E;k) =^1nm where,n= (S;S+1;:::;S ) and the gauge transforma- tion ni(t)!ein!t ni(t) has been employed to remove the time dependence. The density matrix of the system is of the form ^nm=ei(nm)!tSX p=SZdE 2^Gr np2fp^^Ga pm (A3)9 where,fp^(E) =f(Ep!^) is the equilibrium Fermi distribution function of the electrons. Due to the fact thatp!are the eigenvalues of HM=gB~B
S, one can generalize this expression by transforming into a ba- sis where the magnetic energy is not diagonal which in turn leads to Eq (5) for the density matrix of the con- duction electron-local moment entagled system. Appendix B: Recursive Relation for GFs Since in this work we are interested in diagonal blocks of the GFs and in general for FMs at low temperaturewe haveS1, we need a recursive algorithm to be able to solve the system numerically. The surface Keldysh GFs corresponding to ascending ^ gu;r=<, and descending ^gd;r=<, recursion scheme read, ^gu;r n(E;k) =1 E!nin^H(k)^rn(E;k)n 2SJsd(k)z(S n)2 4S2Jsd(k)+^gu;r n1(E;k)Jsd(k)(B1) ^u;< n(E;k) =X  2in+^r n;(E;k)^a n;(E;k) fn+(S n)2 4S2Jsd+^gu;r n1^u;< n1^gu;a n1Jsd (B2) ^gd;r n(E;k) =1 E!nin^rn(E;k)^H(k)n 2SJsd(k)z(S+ n)2 4S2Jsd(k)^gu;r n+1(E;k)+Jsd(k)(B3) ^d;< n(E;k) =X  2in+^r n;(E;k)^a n;(E;k) fn+(S+ n)2 4S2Jsd^gd;r n+1^d;< n+1^gd;a n+1+Jsd (B4) where, ^r n(E;k) =P ^r (E!n;k) corresponds to the self energy of the leads, =L;R refers to the left and right leads in the two terminal device in Fig. 3 and S m=p S(S+ 1)m(m1). Using the surface GFs we can calculate the GFs as follows, ^Gr n;m(E;k) =1 E!nin^H(k)^rnn 2SJsd(k)z^r;u n^r;d n; n =m (B5) =S+ n 2S^gu;r n(E;k)Jsd(k)^Gr n+1;m(E;k); n6=m (B6) =S n 2S^gd;r n(E;k)Jsd(k)+^Gr n1;m(E;k); n6=m (B7) where the ascending and descending self energies are given by, ^r;u n=(S n)2 4S2Jsd(k)+^gu;r n1(E;k)Jsd(k) (B8) ^r;d n=(S+ n)2 4S2Jsd(k)^gd;r n+1(E;k)+Jsd(k) (B9) The average rate of angular momentum loss/gain can be obtained from the real part of the loss of angular momentum in one period of precession, T0 n=1 2(T0 nT0+ n) =1 2= X kTr[S n 2S+Jsd(k)^nn+1(k)S+ n 2SJsd(k)^nn1(k)]! (B10)10 which can be interpreted as the current owing across the layer n. 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2017-02-27

Motivated by the need to understand current-induced magnetization dynamics at the nanoscale, we have developed a formalism, within the framework of Keldysh Green function approach, to study the current-induced dynamics of a ferromagnetic (FM) nanoisland overlayer on a spin-orbit-coupling (SOC) Rashba plane. In contrast to the commonly employed classical micromagnetic LLG simulations the magnetic moments of the FM are treated {\it quantum mechanically}. We obtain the density matrix of the whole system consisting of conduction electrons entangled with the local magnetic moments and calculate the effective damping rate of the FM. We investigate two opposite limiting regimes of FM dynamics: (1) The precessional regime where the magnetic anisotropy energy (MAE) and precessional frequency are smaller than the exchange interactions, and (2) The local spin-flip regime where the MAE and precessional frequency are comparable to the exchange interactions. In the former case, we show that due to the finite size of the FM domain, the \textquotedblleft Gilbert damping\textquotedblright does not diverge in the ballistic electron transport regime, in sharp contrast to Kambersky's breathing Fermi surface theory for damping in metallic FMs. In the latter case, we show that above a critical bias the excited conduction electrons can switch the local spin moments resulting in demagnetization and reversal of the magnetization. Furthermore, our calculations show that the bias-induced antidamping efficiency in the local spin-flip regime is much higher than that in the rotational excitation regime.

Current Induced Damping of Nanosized Quantum Moments in the Presence of Spin-Orbit Interaction

1702.08408v2

Qualitative properties of solutions to a nonlinear transmission problem for an elastic Bresse beam Tamara Fastovska1,2,∗, Dirk Langemann3and Iryna Ryzhkova1 January 23, 2024 1Department of Mathematics and Computer Science, V.N. Karazin Kharkiv National University, Kharkiv, Ukraine 2Institut f¨ ur Mathematik, Humboldt-Universit¨ at zu Berlin, Berlin, Germany 3Institut f¨ ur Partielle Differentialgleichungen,Technische Universit¨ at Braun- schweig, Braunschweig, Germany Abstract 1 We consider a nonlinear transmission problem for a Bresse beam, which consists of two parts, damped and undamped. The mechanical damp- ing in the damped part is present in the shear angle equation only, and the damped part may be of arbitrary positive length. We prove well-posedness of the corresponding PDE system in energy space and establish existence of a regular global attractor under certain conditions on nonlinearities and coefficients of the damped part only. Moreover, we study singular limits of the problem when l→0 orl→0 simultaneously with ki→+∞and perform numerical modelling for these processes. 2 Keywords: Bresse beam, transmission problem, global attractor, singular limit 1arXiv:2309.15171v2 [math.AP] 22 Jan 20243 Introduction In this paper we consider a contact problem for the Bresse beam. Originally the mathematical model for homogeneous Bresse beams was derived in [ 4]. We use the variant of the model described in [ 20, Ch. 3]. Let the whole beam occupies a part of a circle of length Land has the curvature l=R−1. We consider the beam as a one-dimensional object and measure the coordinate xalong the beam. Thus, we say that the coordinate xchanges within the interval (0 , L). The parts of the beam occupying the intervals (0 , L0) and (L0, L) consist of different materials. The part lying in the interval (0 , L0) is partially subjected to a structural damping (see Figure 1). The Bresse system Figure 1: Composite Bresse beam. describes evolution of three quantities: transversal displacement, longitudinal displacement and shear angle variation. We denote by φ,ψ, and ωthe transversal displacement, the shear angle variation, and the longitudinal displacement of the left part of the beam lying in (0 , L0). Analogously, we denote by u,v, and wthe transversal displacement, the shear angle variation, and the longitudinal displacement of the right part of the beam occupying the interval ( L0, L). We assume the presence of the mechanical dissipation in the equation for the shear angle variation for the left part of the beam. We also assume that both ends of the beam are clamped. Nonlinear oscillations 2of the composite beam can be described by the following system of equations ρ1φtt−k1(φx+ψ+lω)x−lσ1(ωx−lφ) +f1(φ, ψ, ω ) =p1(x, t), (1) β1ψtt−λ1ψxx+k1(φx+ψ+lω) +γ(ψt) +h1(φ, ψ, ω ) =r1(x, t), x∈(0, L0), t >0, (2) ρ1ωtt−σ1(ωx−lφ)x+lk1(φx+ψ+lω) +g1(φ, ψ, ω ) =q1(x, t), (3) and ρ2utt−k2(ux+v+lw)x−lσ2(wx−lu) +f2(u, v, w ) =p2(x, t), (4) β2vtt−λ2vxx+k2(ux+v+lw) +h2(u, v, w ) =r2(x, t), x ∈(L0, L), t >0, (5) ρ2wtt−σ2(wx−lu)x+lk2(ux+v+lw) +g2(u, v, w ) =q2(x, t), (6) where ρj, βj, kj, σj, λjare positive parameters, fj, gj, hj:R3→Rare nonlinear feedbacks, pj, qj, rj: (0, L)×R3→Rare known external loads, γ:R→Ris a nonlinear damping. The system is subjected to the Dirichlet boundary conditions φ(0, t) =u(L, t) = 0 , ψ(0, t) =v(L, t) = 0 , ω(0, t) =w(L, t) = 0 ,(7) the transmission conditions φ(L0, t) =u(L0, t), ψ(L0, t) =v(L0, t), ω(L0, t) =w(L0, t),(8) k1(φx+ψ+lω)(L0, t) =k2(ux+v+lw)(L0, t), (9) λ1ψx(L0, t) =λ2vx(L0, t), (10) σ1(ωx−lφ)(L0, t) =σ2(wx−lu)(L0, t), (11) and supplemented with the initial conditions φ(x,0) = φ0(x), ψ(x,0) = ψ0(x), ω(x,0) = ω0(x), (12) φt(x,0) = φ1(x), ψ t(x,0) = ψ1(x), ω t(x,0) = ω1(x), (13) 3u(x,0) = u0(x), v(x,0) = v0(x), w (x,0) = w0(x), (14) ut(x,0) = u1(x), v t(x,0) = v1(x), w t(x,0) = w1(x). (15) One can observe patterns in the problem which appear to have physical meaning: Qi(ξ, ζ, η ) =ki(ξx+ζ+lη) are shear forces , Ni(ξ, ζ, η ) =σi(ηx−lξ) are axial forces , Mi(ξ, ζ, η ) =λiζxare bending moments for damped ( i= 1) and undamped ( i= 2) parts respectively. Later we will use them to rewrite the problem in a compact and physically natural form. The paper is devoted to the well-posedness and long-time behaviour of the system (1)-(15). Our main goal is to establish conditions under which the assumed amount of dissipation is sufficient to guarantee the existence of a global attractor. The paper is organized as follows. In Section 2 we represent functional spaces and pose the problem in an abstract form. In Section 3 we prove that the problem is well-posed and possesses strong solutions provided nonlinearities and initial data are smooth enough. Section 4 is devoted to the main result on the existence of a compact attractor. The nature of dissipation prevents us from proving dissipativity explicitly, thus we show that the corresponding dynamical system is of gradient structure and asymptotically smooth. We establish the unique continuation property by means of the observability inequality obtained in [ 27] to prove the gradient property. The compensated compactness approach is used to prove the asymptotic smoothness. In Section 5 we show that solutions to (1)-(15)tend to solutions to a transmission problem for the Timoshenko beam when l→0 and to solutions to a transmission problem for the Euler-Bernoulli beam when l→0 and ki→ ∞ as well as perform numerical modelling of these singular limits. 44 Preliminaries and Abstract formulation 4.1 Spaces and notations Let us denote Φ1= (φ, ψ, ω ),Φ2= (u, v, w ),Φ = (Φ1,Φ2). Thus, Φ is a six-dimensional vector of functions. Analogously, Fj= (fj, gj, hj) :R3→R3, F = (F1, F2), Pj= (pj, qj, rj) : [(0 , L)×R+]3→R3, P = (P1, P2), Rj=diag{ρj, βj, ρj}, R =diag{ρ1, β1, ρ1, ρ2, β2, ρ2}, Γ(Φ t) = (0 , γ(ψt),0,0,0,0), where j= 1,2. The static linear part of the equation system can be formally rewritten as AΦ = −∂xQ1(Φ1)−lN1(Φ1) −∂xM1(Φ1) +Q1(Φ1) −∂xN1(Φ1) +lQ1(Φ1) −∂xQ2(Φ2)−lN2(Φ2) −∂xM2(Φ2) +Q2(Φ2) −∂xN2(Φ2) +lQ2(Φ2) . (16) Then transmission conditions (8)-(11) can be written as follows Φ1(L0, t) = Φ2(L0, t), Q1(Φ1(L0, t)) =Q2(Φ2(L0, t)), M1(Φ1(L0, t)) =M2(Φ2(L0, t)), N1(Φ1(L0, t)) =N2(Φ2(L0, t)). Throughout the paper we use the notation ||·||for the L2-norm of a function and (·,·) for the L2-inner product. In these notations we skip the domain, on which functions are defined. We adopt the notation || · || L2(Ω)only when 5domain is not evident. We also use the same notations || · || and (·,·) for [L2(Ω)]3. To write our problem in an abstract form we introduce the following spaces. For the velocities of the displacements we use the space Hv={Φ = (Φ1,Φ2) : Φ1∈[L2(0, L0)]3,Φ2∈[L2(L0, L)]3} with the norm ||Φ||2 Hv=||Φ||2 v=2X j=1||p RjΦj||2, which is equivalent to the standard L2-norm. For the beam displacements we use the space Hd={Φ∈Hv: Φ1∈[H1(0, L0)]3,Φ2∈[H1(L0, L)]3, Φ1(0, t) = Φ2(L, t) = 0 ,Φ1(L0, t) = Φ2(L0, t) with the norm ||Φ||2 Hd=||Φ||2 d=2X j=1 ||Qj(Φj)||2+||Nj(Φj)||2+||Mj(Φj)||2
. This norm is equivalent to the standard H1-norm. Moreover, the equivalence constants can be chosen independent of lforlsmall enough (see [ 24], Remark 2.1). If we set Ψ(x) =( Φ1(x), x ∈(0, L0), Φ2(x), x ∈[L0, L) we see that there is isomorphism between Hdand [ H1 0(0, L)]3. 4.2 Abstract formulation The operator A:D(A)⊂Hv→Hvis defined by formula (16), where 6D(A) ={Φ∈Hd: Φ1∈H2(0, L0),Φ2∈H2(L0, L), Q1(Φ1(L0, t)) =Q2(Φ2(L0, t)), N1(Φ1(L0, t)) =N2(Φ2(L0, t)), M1(Φ1(L0, t)) =M2(Φ2(L0, t))} Arguing analogously to Lemmas 1.1-1.3 from [ 23] one can prove the following lemma. Lemma 4.1. The operator Ais positive and self-adjoint. Moreover, (A1/2Φ, A1/2B) =1 k1(Q1(Φ1), Q1(B1)) +1 σ1(N1(Φ1), N1(B1)) +1 λ1(M1(Φ1), M1(B1))+ 1 k2(Q2(Φ2), Q2(B2)) +1 σ2(N2(Φ2), N2(B2)) +1 λ2(M2(Φ2), M2(B2)) (17) andD(A1/2) =Hd⊂Hv. Thus, we can rewrite equations (1)-(6) in the form RΦtt+AΦ + Γ(Φ t) +F(Φ) = P(x, t), (18) boundary conditions (7) in the form Φ1(0, t) = Φ2(L, t) = 0 , (19) and transmission conditions (8)-(11) can be written as Φ1(L0, t) = Φ2(L0, t), (20) Q1(Φ1(L0, t)) =Q2(Φ2(L0, t)), (21) M1(Φ1(L0, t)) =M2(Φ2(L0, t)), (22) N1(Φ1(L0, t)) =N2(Φ2(L0, t)). (23) Initial conditions have the form Φ(x,0) = Φ 0(x), Φt(x,0) = Φ 1(x). (24) We use H=Hd×Hvas a phase space. 75 Well-posedness In this section we study strong, generalized and variational (weak) solutions to (18)-(24). Definition 5.1. Φ∈C(0, T;Hd)TC1(0, T;Hv)such that Φ(x,0) = Φ 0(x), Φt(x,0) = Φ 1(x)is said to be a strong solution to (18)-(24) if •Φ(t)lies in D(A)for almost all t; •Φ(t)is an absolutely continuous function with values in HdandΦt∈ L1(a, b;Hd)for0< a < b < T ; •Φt(t)is an absolutely continuous function with values in HvandΦtt∈ L1(a, b;Hv)for0< a < b < T ; •equation (18) is satisfied for almost all t. Definition 5.2. Φ∈C(0, T;Hd)TC1(0, T;Hv)such that Φ(x,0) = Φ 0(x), Φt(x,0) = Φ 1(x)is said to be a generalized solution to (18)-(24) if there exists a sequence of strong solutions Φ(n)to(18)-(24) with the initial data (Φ(n) 0,Φ(n) 1)and right hand side P(n)(x, t)such that lim n→∞max t∈[0,T] ||Φ(n)(·, t)−Φ(·, t)||d+||Φ(n) t(·, t)−Φt(·, t)||v = 0. We also need a definition of a variational solution. We use six-dimensional vector-functions B= (B1, B2),Bj= (βj, γj, δj) from the space FT={B∈L2(0, T;Hd), Bt∈L2(0, T;Hv), B(T) = 0} as test functions. Definition 5.3. Φis said to be a variational (weak) solution to (18)-(24) if •Φ∈L∞(0, T;Hd),Φt∈L∞(0, T;Hv); •Φsatisfies the following variational equality for all B∈FT 8−TZ 0(RΦt, Bt)(t)dt−(RΦ1, B(0)) +ZT 0(A1/2Φ, A1/2B)(t)dt+ ZT 0(Γ(Φ t), B)(t)dt+ZT 0(F(Φ), B)(t)dt−ZT 0(P, B)(t)dt= 0; (25) •Φ(x,0) = Φ 0(x). Now we state a well-posedness result for problem (18)-(24). Theorem 5.4 (Well-posedness) .Let fi, gi, hi:R3→Rare locally Lipschitz i.e. |fi(a)−fi(b)| ≤L(K)|a−b|,provided |a|,|b| ≤K; (N1) there exist Fi:R3→Rsuch that (fi, hi, gi) =∇Fi; there exists δ >0such that Fj(a)≥ −δfor all a∈R3; (N2) P∈L2(0, T;Hv); (R1) and the nonlinear dissipation satisfies γ∈C(R)and non-decreasing , γ(0) = 0 . (D1) Then for every initial data Φ0∈Hd,Φ1∈Hvand time interval [0, T]there exists a unique generalized solution to (18)-(24) with the following properties: •every generalized solution is variational; •energy inequality E(T) +ZT 0(γ(ψt), ψt)dt≤ E(0) +ZT 0(P(t),Φt(t))dt (26) 9holds, where E(t) =1 2h ||R1/2Φt(t)||2+||A1/2Φ(t)||2i +LZ 0F(Φ(x, t))dx and F(Φ(x, t)) =( F1(φ(x, t), ψ(x, t), ω(x, t)), x ∈(0, L0), F2(u(x, t), v(x, t), w(x, t)), x ∈(L0, L). •If, additionally, Φ0∈D(A),Φ1∈Hdand ∂tP(x, t)∈L2(0, T;Hv) (R2) then the generalized solution is also strong and satisfies the energy equality. Proof. The proof essentially uses monotone operators theory. It is rather standard by now (see, e.g., [ 9]), so in some parts we give only references to corresponding arguments. However, we give some details which demonstrate the peculiarity of 1D problems. Step 1. Abstract formulation. We need to reformulate problem (18)-(24)as a first order problem. Let us denote U= (Φ,Φt), U 0= (Φ 0,Φ1)∈H=Hd×Hv, TU= I0 0R−1! 0−I A0! U+ 0 Γ(Φ t)! . Consequently, D(T) =D(A)×Hd⊂H. In what follows we use the notations B(U) = I0 0R−1! 0 F(Φ)! ,P(x, t) = 0 P(x, t)! . 10Thus, we can rewrite problem (18)-(24) in the form Ut+TU+B(U) =P, U (0) = U0∈H. (27) Step 2. Existence and uniqueness of a local solution. Here we use Theorem 7.2 from [9]. For the reader’s convenience we formulate it below. Theorem 5.5 ([9]).Consider the initial value problem Ut+TU+B(U) =f, U (0) = U0∈H. (28) Suppose that T:D(T)⊂H→His a maximal monotone mapping, 0∈ T0 andB:H→His locally Lipschitz, i.e. there exits L(K)>0such that ||B(U)−B(V)||H≤L(K)||U−V||H,||U||H,||V||H≤K. IfU0∈D(T),f∈W1 1(0, t;H)for all t >0, then there exists tmax≤ ∞ such that(28) has a unique strong solution Uon(0, tmax). IfU0∈D(T),f∈L1(0, t;H)for all t >0, then there exists tmax≤ ∞ such that(28) has a unique generaized solution Uon(0, tmax). In both cases lim t→tmax||U(t)||H=∞provided tmax<∞. First, we need to check that Tis a maximal monotone operator. Mono- tonicity is a direct consequence of Lemma 4.1 and (D1). To prove Tis maximal as an operator from HtoH, we use Theorem 1.2 from [ 3, Ch. 2]. Thus, we need to prove that Range (I+T) =H. Let z= (Φ z,Ψz)∈Hd×Hv. We need to find y= (Φ y,Ψy)∈D(A)×Hd=D(T) such that −Ψy+ Φ y= Φ z, AΦy+ Ψ y+ Γ(Ψ y) = Ψ z, 11or, equivalently, find Ψ y∈Hdsuch that M(Ψy) =1 2AΨy+1 2AΨy+ Ψ y+ Γ(Ψ y) = Ψ z−AΦz= Θ z for an arbitrary Θ z∈H′ d=D(A1/2)′. Naturally, due to Lemma 4.1 Ais a duality map between HdandH′ d, thus the operator Mis onto if and only if1 2AΨy+ Ψ y+ Γ(Ψ y) is maximal monotone as an operator from Hdto H′ d. According to Corollary 1.1 from [ 3, Ch. 2], this operator is maximal monotone if1 2Ais maximal monotone (it follows from Lemma 4.1) and I+ Γ(·) is monotone, bounded and hemicontinuous from HdtoH′ d. The last statement is evident for the identity map, now let’s prove it for Γ. Monotonicity is evident. Due to the continuity of the embedding H1(0, L0)⊂ C(0, L0) in 1D every bounded set XinH1(0, L0) is bounded in C(0, L0) and thus due to (D1) Γ(X) is bounded in C(0, L0) and, consequently, in L2(0, L0). To prove hemicontinuity we take an arbitrary Φ = ( φ, ψ, ω, u, v, w )∈Hd, an arbitrary Θ = ( θ1, θ2, θ3, θ4, θ5, θ6)∈Hdand consider (Γ(Ψ y+tΦ),Θ) =ZL0 0γ(ψy(x) +tψ(x))θ2(x)dx, where Ψ y= (φy, ψy, ωy, uy, vy, wy). Since ψy+tψ→ψy,ast→0 inH1(0, L0) and in C(0, L0), we obtain that γ(ψy(x) +tϕ(x))→γ(ψy(x)) as t→0 for every x∈[0, L0], and has an integrable bound from above due to (D1). This implies γ(ψy(x) +tϕ(x))→γ(ψy(x)) in L1(0, L0) ast→0 . Since θ2∈H1(0, L0)⊂L∞(0, L0), then (Γ(Ψ y+tΦ),Θ)→(Γ(Ψ y),Θ), t→0. Hemicontinuity is proved. Further, we need to prove that Bis locally Lipschitz on H, that is, Fis locally Lipschitz from HdtoHv. The embedding H1/2+ε(0, L)⊂C(0, L) and (N1) imply |Fj(eΦj(x))−Fj(bΦj(x))| ≤C(max(||eΦ||d,||bΦ||d))||eΦj−bΦj||1 (29) 12for all x∈[0, L0], ifj= 1 and for all x∈[L0, L], ifj= 2. This, in turn, gives us the estimate ||F(eΦ)−F(bΦ)||v≤C(max(||eΦ||d,||bΦ||d))||eΦ−bΦ||d. Thus, all the assumptions of Theorem 5.5 are satisfied and existence of a local strong/generalized solution is proved. Step 3. Energy inequality and global solutions. It can be verified by direct calculations, that strong solutions satisfy energy equality. Using the same arguments, as in proof of Proposition 1.3 [11], and (D1) we can pass to the limit and prove (26) for generalized solutions. Let us assume that a local generalised solution exists on a maximal interval (0, tmax),tmax<∞. Then (26) implies E(tmax)≤ E(0). Since due to (N2) c1||U(t)||H≤ E(t)≤c2||U(t)||H, we have ||U(tmax)||H≤C||U0||H. Thus, we arrive to a contradiction which implies tmax=∞. Step 4. Generalized solution is variational (weak). We formulate the following obvious estimate as a lemma for future use. Lemma 5.6. Let(N1) holds and eΦ,bΦare two weak solutions to (18)- (24) with the initial conditions (eΦ0,eΦ1)and(bΦ0,bΦ1)respectively. Then the following estimate is valid for all x∈[0, L], t > 0andϵ∈[0,1/2): |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. Proof. The energy inequality and the embedding H1/2+ε(0, L)⊂C(0, L) imply that for every weak solution Φ max t∈[0,T],x∈[0,L]|Φ(x, t)| ≤C(||Φ0||d,||Φ1||v). Thus, using (N1) and (29), we prove the lemma. Evidently, (25)is valid for strong solutions. We can find a sequence of 13strong solutions Φ(n), which converges to a generalized solution Φ strongly inC(0, T;Hd), and Φ(n) tconverges to Φ tstrongly in C(0, T;Hv). Using Lemma 5.6, we can easily pass to the limit in nonlinear feedback term in (25). Since the test function B∈L∞(0, T;Hd)⊂L∞((0, T)×(0, L)), we can use the same arguments as in the proof of Proposition 1.6 [ 11] to pass to the limit in the nonlinear dissipation term. Namely, we can extract from Φ(n) ta subsequence that converges to Φ talmost everywhere and prove that it converges to Φ tstrongly in L1((0, T)×(0, L)). Remark 1. In space dimension greater then one we do not have the embed- ding H1(Ω)⊂C(Ω), therefore, we need to assume polynomial growth of the derivative of the nonlinearity to obtain estimates similar to Lemma 5.6. 6 Existence of attractors. In this section we study long-time behaviour of solutions to problem (18)-(24) in the framework of dynamical systems theory. From Theorem 5.4 we have Corollary 1. Let, additionally to conditions of Theorem 5.4, P(x, t) =P(x). Then (18)-(24) generates a dynamical system (H, S t)by the formula St(Φ0,Φ1) = (Φ( t),Φt(t)), where Φ(t)is the weak solution to (18)-(24) with initial data (Φ0,Φ1). To establish the existence of the attractor for this dynamical system we use Theorem 6.8 below, thus we need to prove the gradientness and the asymptotic smoothness as well as the boundedness of the set of stationary points. 6.1 Gradient structure In this subsection we prove that the dynamical system generated by (18)-(24) possesses a specific structure, namely, is gradient under some additional conditions on the nonlinearities. 14Definition 6.1 ([8,10,12]).LetY⊆Xbe a positively invariant set of (X, S t). •a continuous functional L(y)defined on Yis said to be a Lyapunov function of the dynamical system (X, S t)on the set Y, if a function t7→L(Sty)is non-increasing for any y∈Y. •the Lyapunov function L(y)is said to be strict onY, if the equality L(Sty) =L(y) for all t >0implies Sty=yfor all t >0; •A dynamical system (X, S t)is said to be gradient , if it possesses a strict Lyapunov function on the whole phase space X. The following result holds true. Theorem 6.2. Let, additionally to the assumptions of Corollary 1, the following conditions hold f1=g1= 0, h 1(φ, ψ, ω ) =h1(ψ), (N3) f2, g2, h2∈C1(R3), (N4) γ(s)s >0for all s̸= 0. (D2) Then the dynamical system (H, S t)is gradient. Proof. We use as a Lyapunov function L(Φ(t)) =L(t) =1 2 ||R1/2Φt(t)||2+||A1/2Φ(t)||2 +LZ 0F(Φ(x, t))dx+(P,Φ(t)). (30) Energy inequality (26) implies that L(t) is non-increasing. The equality L(t) =L(0) together with (D2) imply that ψt(t)≡0 on [0 , T]. We need to prove that Φ( t)≡const , which is equivalent to Φ( t+h)−Φ(t) = 0 for every h >0. In what follows we use the notation Φ( t+h)−Φ(t) =Φ(t) = (φ,ψ,ω,u,v,w)(t) . Step 1. Let us prove that Φ1≡0. In this step we use the distribution theory (see, e.g., [ 5]) because some functions involved in computations are of too low 15smoothness. Let us set the test function B= (B1,0) = ( β1, γ1, δ1,0,0,0). Then Φ(t) satisfies −TZ 0(R1Φ1 t, Bt)(t)dt−(R1(Φ1 t(h)−Φ1 1), B1(0))+ TZ 01 k1(Q1(Φ1), Q1(B1))(t)dt+1 σ1(N1(Φ1), N1(B1))(t) + TZ 0(h1(ψ(t+h))−h1(ψ(t)), γ1(t))dt= 0. The last term equals to zero due to (N3) and ψ(t)≡const . Setting in turn B= (0, γ1,0,0,0,0),B= (0,0, δ1,0,0,0),B= (β1,0,0,0,0,0) we obtain φx+lω= 0 almost everywhere on (0 , L0)×(0, T), (31) ρ1ωtt−lσ1(ωx−lφ)x= 0 almost everywhere on (0 , L0)×(0, T), (32) ρ1φtt−σ1(ωx−lφ) = 0 in the sense of distributions on (0 , L0)×(0, T). (33) These equalities imply φttx= 0,ωtt= 0 in the sense of distributions . (34) Similar to regular functions, if partial derivative of a distribution equals to zero, then the distribution ”does not depends” on the corresponding variable (see [5, Ch. 7], Example 2.) That is, ωt=c1(x)×1(t) in the sense of distributions 16However, Theorem 5.4 implies that ωtis a regular distribution, thus, we can treat the equality above as the equality almost everywhere. Furthermore, ω(x, t) =ω(x,0) +Zt 0c1(x)dτ=ω(x,0) +tc1(x). Since||ω(·, t)|| ≤Cfor all t∈R+,c1(x) must be zero. Thus, ω(x, t) =c2(x), (35) which together with (31) implies φx=−lc2(x), φ(x, t) =φ(0, t)−lxZ 0c2(y)dy=c3(x), φtt= 0. The last equality together with (31),(33)and boundary conditions (19)give us that φ,ωare solutions to the following Cauchy problem (with respect to x): ωx=lφ, φx=−lω, ω(0, t) =φ(0, t) = 0 . Consequently, ω≡φ≡0. Step 2. Let us prove, that u≡v≡w≡0. Due to (N4), we can use the Taylor expansion of the difference F2(Φ2(t+h))−F2(Φ2(t)) and thus ( u,v,w) 17satisfies on (0 , T)×(L0, L) ρ2utt−k2uxx+gu(∂xΦ2,Φ2) +∇f2(ζ1,h(x, t))·Φ2= 0, (36) β2vtt−λ2vxx+gv(∂xΦ2,Φ2) +∇h2(ζ2,h(x, t))·Φ2= 0, (37) ρ2wtt−σ2wxx+gw(∂xΦ2,Φ2) +∇g2(ζ3,h(x, t))·Φ2= 0 (38) u(L0, t) =v(L0, t) =w(L0, t) = 0 , (39) u(L, t) =v(L, t) =w(L, t) = 0 , (40) k2(ux+v+lw)(L0, t) = 0 , (41) vx(L0, t) = 0 , σ 2(wx−lu)(L0, t) = 0 , (42) Φ2(x,0) = Φ2(x, h)−Φ2 0,Φ2 t(x,0) = Φ2 t(x, h)−Φ2 1, (43) where gu, gv, gware linear combinations of ux, vx, wx, u, v, w with the constant coefficients, ζj,h(x, t) are 3D vector functions which components lie between u(x, t+h) and u(x, t),v(x, t+h) and v(x, t),w(x, t+h) and w(x, t) respectively. Thus, we have a system of linear equations on ( L0, L) with overdetermined boundary conditions. L2-regularity of ux, vx, wxon the boundary for solutions to a linear wave equation was established in [ 21], thus, boundary conditions (41)-(42) make sense. It is easy to generalize the observability inequality [ 27, Th. 8.1] for the case of the system of the wave equations. Theorem 6.3 ([27] ).For the solution to problem (36)-(43) the following estimate holds: ZT 0[|ux|2+|vx|2+|wx|2](L0, t)dt≥C(E(0) + E(T)), where E(t) =1 2 ||ut(t)||2+||vt(t)||2+||wt(t)||2+||ux(t)||2+||vx(t)||2+||wx(t)||2
. Therefore, if conditions (41),(42)hold true, then u=v=w= 0. The theorem is proved. 186.2 Asymptotic smoothness. Definition 6.4 ([8,10,12]).A dynamical system (X, S t)is said to be asymptotically smooth if for any closed bounded set B⊂Xthat is positively invariant ( StB⊆B) one can find a compact set K=K(B)which uniformly attracts B, i. e. sup{distX(Sty,K) :y∈B} →0ast→ ∞ . In order to prove the asymptotical smoothness of the system considered we rely on the compactness criterion due to [ 17], which is recalled below in an abstract version formulated in [12]. Theorem 6.5. [12] Let (St, H)be a dynamical system on a complete metric space Hendowed with a metric d. Assume that for any bounded positively invariant set BinHand for any ε >0there exists T=T(ε, B)such that d(STy1, STy2)≤ε+ Ψ ε,B,T(y1, y2), yi∈B, (44) where Ψε,B,T(y1, y2)is a function defined on B×Bsuch that lim inf m→∞lim inf n→∞Ψε,B,T(yn, ym) = 0 for every sequence yn∈B. Then (St, H)is an asymptotically smooth dynamical system. To formulate the result on the asymptotic smoothness of the system considered we need the following lemma. Lemma 6.6. Let assumptions (D1) hold. Let moreover, there exists a positive constant Msuch that γ(s1)−γ(s2) s1−s2≤M, s 1, s2∈R, s1̸=s2. (D3) Then for any ε >0there exists Cε>0such that L0Z 0(γ(ξ1)−γ(ξ2))ζdx≤ε∥ζ∥2+CεL0Z 0(γ(ξ1)−γ(ξ2))(ξ1−ξ2)dx (45) 19for any ξ1, ξ2, ζ∈L2(0, L0). The proof is similar to that given in [12, Th.5.5]). Theorem 6.7. Let assumptions of Theorem 5.4, (D3), and m≤γ(s1)−γ(s2) s1−s2, s 1, s2∈R, s1̸=s2. (D4) withm > 0hold. Let, moreover, k1=σ1 (46) ρ1 k1=β1 λ1. (47) Then the dynamical system (H, S t)generated by problem (1)–(11) is asymp- totically smooth. Proof. In this proof we perform all the calculations for strong solutions and then pass to the limit in the final estimate to justify it for weak solutions. Let us consider strong solutions ˆU(t) = ( ˆΦ(t),ˆΦt(t)) and ˜U(t) = ( ˜Φ(t),˜Φt(t)) to problem (1)–(11)with initial conditions ˆU0= (ˆΦ0,ˆΦ1) and ˜U0= (˜Φ0,˜Φ1) lying in a ball, i.e. there exists R >0 such that ∥˜U0∥H+∥ˆU0∥H≤R. (48) Denote U(t) =˜U(t)−ˆU(t) and U0=˜U0−ˆU0. Obviously, U(t) is a weak 20solution to the problem ρ1φtt−k1(φx+ψ+lω)x−lσ1(ωx−lφ) +f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω) = 0 (49) β1ψtt−λ1ψxx+k1(φx+ψ+lω) +γ(˜ψt)−γ(ˆψt) +h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω) = 0 (50) ρ1ωtt−σ1(ωx−lφ)x+lk1(φx+ψ+lω) +g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω) = 0 (51) ρ2utt−k2(ux+v+lw)x−lσ2(wx−lu) +f2(˜u,˜v,˜w)−f2(ˆu,ˆv,ˆw) = 0 (52) β2vtt−λ2vxx+k2(ux+v+lw) +h2(˜u,˜v,˜w)−h2(ˆu,ˆv,ˆw) = 0 , (53) ρ2wtt−σ2(wx−lu)x+lk2(ux+v+lw) +g2(˜u,˜v,˜w)−g2(ˆu,ˆv,ˆw) = 0 (54) with boundary conditions (7),(8)–(11)and the initial conditions U(0) = ˜U0−ˆU0. It is easy to see by the energy argument that E(U(T)) +TZ tL0Z 0(γ(˜ψs)−γ(ˆψs))ψsdxds =E(U(t)) +TZ tH(ˆU(s),˜U(s))ds, (55) where H(ˆU(t),˜U(t)) =L0Z 0(f1( ˆφ,ˆψ,ˆω)−f1( ˜φ,˜ψ,˜ω))φtdx+L0Z 0(h1( ˆφ,ˆψ,ˆω)−h1( ˜φ,˜ψ,˜ω))ψtdx +L0Z 0(g1( ˆφ,ˆψ,ˆω)−g1( ˜φ,˜ψ,˜ω))ωtdx+LZ L0(f2(ˆu,ˆv,ˆw)−f2(˜u,˜v,˜w))utdx +LZ L0(h2(ˆu,ˆv,ˆw)−h2(˜u,˜v,˜w))vtdx+LZ L0(g2(ˆu,ˆv,ˆw)−g2(˜u,˜v,˜w))wtdx, (56) 21and E(t) =E1(t) +E2(t), (57) here E1(t) =ρ1L0Z 0ω2 tdxdt +ρ1L0Z 0φ2 tdxdt +β1L0Z 0ψ2 tdx+σ1L0Z 0(ωx−lφ)2dx+ +k1L0Z 0(φx+ψ+lω)2dx+λ1L0Z 0ψ2 xdx(58) and E2(t) =ρ2L0Z 0w2 tdxdt +ρ2L0Z 0u2 tdxdt +β2L0Z 0v2 tdx+σ2L0Z 0(wx−lu)2dx+ +k2L0Z 0(ux+v+lw)2dx+λ2L0Z 0v2 xdx. (59) Integrating in (55) over the interval (0 , T) we come to TE(U(T))+TZ 0TZ tL0Z 0(γ(˜ψs)−γ(ˆψs))ψsdxdsdt =TZ 0E(U(t))dt+TZ 0TZ tH(ˆU(s),˜U(s))dsdt. (60) Now we estimate the first term in the right-hand side of (60). In what follows we present formal estimates which can be performed on strong solutions. Step 1. We multiply equation (51)byωandx·ωxand sum up the results. After integration by parts with respect to twe obtain 22ρ1TZ 0L0Z 0ωtxωtxdxdt +ρ1TZ 0L0Z 0ω2 tdxdt +σ1TZ 0L0Z 0(ωx−lφ)xxωxdxdt +σ1TZ 0L0Z 0(ωx−lφ)xωdxdt −k1lTZ 0L0Z 0(φx+ψ+lω)xωxdxdt −k1lTZ 0L0Z 0(φx+ψ+lω)ωdxdt −TZ 0L0Z 0(g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω))(xωx+ω)dxdt =ρ1L0Z 0ωt(x, T)xωx(x, T)dx+ρ1L0Z 0ωt(x, T)ω(x, T)dx −ρ1L0Z 0ωt(x,0)xωx(x,0)dx−ρ1L0Z 0ωt(x,0)ω(x,0)dx. (61) Integrating by parts with respect to xwe get ρ1TZ 0L0Z 0ωtxωtxdxdt =−ρ1 2TZ 0L0Z 0ω2 tdxdt +ρ1L0 2TZ 0ω2 t(L0, t)dt (62) and 23σ1TZ 0L0Z 0(ωx−lφ)xxωxdxdt−k1lTZ 0L0Z 0(φx+ψ+lω)xωxdxdt =σ1TZ 0L0Z 0(ωx−lφ)xx(ωx−lφ)dxdt +σ1lTZ 0L0Z 0(ωx−lφ)xxφdxdt −k1lTZ 0L0Z 0(φx+ψ+lω)xωxdxdt =−σ1 2TZ 0L0Z 0(ωx−lφ)2dxdt +σ1L0 2TZ 0(ωx−lφ)2(L0, t)dt−σ1lTZ 0L0Z 0(ωx−lφ)φdxdt −2σ1lTZ 0L0Z 0(ωx−lφ)x(φx+ψ+lω)dxdt +σ1lTZ 0L0Z 0(ωx−lφ)x(ψ+lω)dxdt −σ1lL0TZ 0(ωx−lφ)(L0, t)φ(L0, t)dt−k1l2TZ 0L0Z 0(φx+ψ+lω)xφdxdt. (63) Analogously, σ1TZ 0L0Z 0(ωx−lφ)xωdxdt =−σ1TZ 0L0Z 0(ωx−lφ)2dxdt +σ1TZ 0(ωx−lφ)(L0, t)ω(L0, t)dt−lσ1TZ 0L0Z 0(ωx−lφ)φdxdt. (64) It follows from Lemma 5.6, energy relation (26), and property (N2) that TZ 0L0Z 0|g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω)|2dxdt≤C(R, T) max t∈[0,T]∥Φ(·, t)∥2 H1−ϵ,0< ϵ < 1/2. (65) 24Therefore, for every ε >0 TZ 0L0Z 0(g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω))(xωx+ω)dxdt≤εTZ 0∥ωx−lφ∥2dt+C(ε, R, T )lot, (66) where we use the notation lot= max t∈[0,T](∥φ(·, t)∥2 H1−ϵ+∥ψ(·, t)∥2 H1−ϵ+∥ω(·, t)∥2 H1−ϵ +∥u(·, t)∥2 H1−ϵ+∥v(·, t)∥2 H1−ϵ+∥w(·, t)∥2 H1−ϵ),0< ϵ < 1/2.(67) Similar estimates hold for nonlinearities g2,fi,hi,i= 1,2. We note that for any η∈H1(0, L0) (or analogously η∈H1(L0, L)) η(L0)≤sup (0,L0)|η| ≤C∥η∥H1−ϵ,0< ϵ < 1/2. (68) Since due to (46) 2σ1lTZ 0L0Z 0(ωx−lφ)x(φx+ψ+lω)dxdt ≤σ1 16TZ 0L0Z 0(ωx−lφ)2dxdt + 16k1l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt, the following estimate can be obtained from (61)– (66) ρ1 2TZ 0L0Z 0ω2 tdxdt +ρ1L0 2TZ 0ω2 t(L0, t)dt+13σ1L0 8TZ 0(ωx−lφ)2(L0, t)dt ≤13σ1 8TZ 0L0Z 0(ωx−lφ)2dxdt+17k1l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt+C(R, T)lot +C(E(0) + E(T)),(69) 25where C >0. Step 2. Multiplying equation (51)byωand ( x−L0)·ωxand arguing as above we come to the estimate ρ1 2TZ 0L0Z 0ω2 tdxdt+13σ1L0 8TZ 0(ωx−lφ)2(0, t)dt≤13σ1 8TZ 0L0Z 0(ωx−lφ)2dxdt + 17k1l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt +C(R, T)lot+C(E(0) + E(T)).(70) Summing up estimates (69)and(70)and multiplying the result by1 2we get ρ1 2TZ 0L0Z 0ω2 tdxdt +ρ1L0 4TZ 0ω2 t(L0, t)dt+3σ1L0 16TZ 0(ωx−lφ)2(L0, t)dt +3σ1L0 16TZ 0(ωx−lφ)2(0, t)dt≤13σ1 8TZ 0L0Z 0(ωx−lφ)2dxdt + 17k1l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt +C(R, T)lot+C(E(0) + E(T)).(71) Step 3. Next we multiply equation (49)by−1 l(ωx−lφ), equation (51)by 1 lφx, summing up the results and integrating by parts with respect to twe arrive at 26ρ1 lTZ 0L0Z 0φt(ωtx−lφt)dxdt +k1 lTZ 0L0Z 0(φx+ψ+lω)x(ωx−lφ)dxdt +σ1TZ 0L0Z 0(ωx−lφ)2dxdt−1 lTZ 0L0Z 0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))(ωx−lφ)dxdt +ρ1 lTZ 0L0Z 0ωtφtxdxdt +σ1 lTZ 0L0Z 0(ωx−lφ)xφxdxdt −k1TZ 0L0Z 0(φx+ψ+lω)φxdxdt−TZ 0L0Z 0(g1( ˜φ,˜ψ,˜ω)−g1( ˆφ,ˆψ,ˆω))φxdxdt =ρ1 lL0Z 0φt(x, T)(ωx−lφ)(x, T)dx−ρ1 lL0Z 0φt(x,0)(ωx−lφ)(x,0)dx +ρ1 lL0Z 0ωt(x, T)φx(x, T)dx−ρ1 lL0Z 0ωt(x,0)φx(x,0)dx. (72) Integrating by parts with respect to xwe obtain ρ1 lTZ 0L0Z 0φtωtxdxdt +ρ1 lTZ 0L0Z 0ωtφtxdxdt=ρ1 lTZ 0φt(L0, t)ωt(L0, t)dt ≤ρ1L0 8TZ 0ω2 t(L0, t)dt+2ρ1 l2L0TZ 0φ2 t(L0, t)dt. (73) Taking into account (46) we get 27k1 lTZ 0L0Z 0(φx+ψ+lω)x(ωx−lφ)dxdt +σ1 lTZ 0L0Z 0(ωx−lφ)xφxdxdt =k1 lTZ 0(φx+ψ+lω)(L0, t)(ωx−lφ)(L0, t)dt−k1 lTZ 0(φx+ψ+lω)(0, t)(ωx−lφ)(0, t)dt +k1 lTZ 0L0Z 0ψx(ωx−lφ)dxdt+σ1TZ 0L0Z 0(ωx−lφ)2dxdt+σ1lTZ 0L0Z 0(ωx−lφ)φdxdt. (74) Using the estimates k1 lTZ 0(φx+ψ+lω)(L0, t)(ωx−lφ)(L0, t)dt ≤4k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+σ1L0 16TZ 0(ωx−lφ)2(L0, t)dt, k1 lTZ 0L0Z 0ψx(ωx−lφ)dxdt≤4k1 l2TZ 0L0Z 0ψ2 xdxdt +σ1 16TZ 0L0Z 0(ωx−lφ)2dxdt and (72)–(74) we infer 2815σ1 8TZ 0L0Z 0(ωx−lφ)2dxdt≤ρ1TZ 0L0Z 0φ2 tdxdt+ 2k1TZ 0L0Z 0(φx+ψ+lω)2dxdt +4k1 l2TZ 0L0Z 0ψ2 xdxdt+4k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+σ1L0 8TZ 0(ωx−lφ)2(L0, t)dt +4k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt+σ1L0 8TZ 0(ωx−lφ)2(0, t)dt ρ1L0 8TZ 0ω2 t(L0, t)dt+2ρ1 l2L0TZ 0φ2 t(L0, t)dt+C(R, T)lot+C(E(0) + E(T)). (75) Adding (75) to (71) we obtain σ1 4TZ 0L0Z 0(ωx−lφ)2dxdt+ρ1 2TZ 0L0Z 0ω2 tdxdt+ρ1L0 8TZ 0ω2 t(0, t)dt+σ1L0 16TZ 0(ωx−lφ)2(L0, t)dt +σ1L0 16TZ 0(ωx−lφ)2(L0, t)dt≤ρ1TZ 0L0Z 0φ2 tdxdt+k1(2+17 l2L2 0)TZ 0L0Z 0(φx+ψ+lω)2dxdt +4k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+4k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt +4k1 l2TZ 0L0Z 0ψ2 xdxdt +2ρ1 l2L0TZ 0φ2 t(L0, t)dt+C(R, T)lot+C(E(0) + E(T)). (76) Step 4. Now we multiply equation (49)by−16 l2L2 0xφxand−16 l2L2 0(x−L0)φx and sum up the results. After integration by parts with respect to twe get 2916ρ1 l2L2 0TZ 0L0Z 0φtxφtxdxdt +16ρ1 l2L2 0TZ 0L0Z 0φt(x−L0)φtxdxdt +16k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)xxφxdxdt+16k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)x(x−L0)φxdxdt +16σ1 lL2 0TZ 0L0Z 0(ωx−lφ)xφxdxdt +16σ1 lL2 0TZ 0L0Z 0(ωx−lφ)(x−L0)φxdxdt −16 l2L2 0TZ 0L0Z 0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))(2x−L0)φxdxdt =16ρ1 l2L2 0L0Z 0φt(x, T)(2x−L0)φx(x, T)dx−16ρ1 l2L2 0L0Z 0φt(x, T)(2x−L0)φx(x, T)dx. (77) It is easy to see that 16ρ1 l2L2 0TZ 0L0Z 0φtxφtxdxdt +16ρ1 l2L2 0TZ 0L0Z 0φt(x−L0)φtxdxdt =−16ρ1 l2L2 0TZ 0L0Z 0φ2 tdxdt+8ρ1 l2L0TZ 0φ2 t(L0, t)dt (78) and 3016k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)xxφxdxdt+16k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)x(x−L0)φxdxdt =−16k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt +8k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt +8k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt−16k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)xx(ψ+lω)dxdt −16k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)x(x−L0)(ψ+lω)dxdt =−16k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt +8k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt +8k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt−16k1 l2L0TZ 0(φx+ψ+lω)(L0, t)(ψ+lω)(L0, t)dt +32k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)(ψ+lω)dxdt++16k1 lL2 0TZ 0L0Z 0(φx+ψ+lω)(2x−L0)(ωx−lφ)dxdt +16k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)(2x−L0)ψxdxdt+16k1 L2 0TZ 0L0Z 0(φx+ψ+lω)(2x−L0)φdxdt. (79) Moreover, 16σ1 lL2 0TZ 0L0Z 0(ωx−lφ)xφxdxdt +16σ1 lL2 0TZ 0L0Z 0(ωx−lφ)(x−L0)φxdxdt =16σ1 lL2 0TZ 0L0Z 0(ωx−lφ)(2x−L0)(φx+ψ+lω)dxdt 31−16σ1 lL2 0TZ 0L0Z 0(ωx−lφ)(2x−L0)(ψ+lω)dxdt. (80) Collecting (77)–(80) and using the estimates 32k1 lL2 0TZ 0L0Z 0(φx+ψ+lω)(2x−L0)(ωx−lφ)dxdt ≤σ1 8TZ 0L0Z 0(ωx−lφ)2dxdt +2046k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt and 16k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)(2x−L0)ψxdxdt ≤k1 l2TZ 0L0Z 0ψ2 xdxdt +64k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt we come to 7k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+7k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt (81) +8ρ1 l2L0TZ 0φ2 t(L0, t)dt≤16ρ1 l2L2 0TZ 0L0Z 0φ2 tdxdt+2150k1 l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt +k1 l2TZ 0L0Z 0ψ2 xdxdt+3σ1 16TZ 0L0Z 0(ωx−lφ)2dxdt+C(R, T)lot+C(E(0)+E(T)). (82) Adding (82) to (76) we arrive at 32σ1 16TZ 0L0Z 0(ωx−lφ)2dxdt +ρ1 2TZ 0L0Z 0ω2 tdxdt +ρ1L0 8TZ 0ω2 t(L0, t)dt +σ1L0 16TZ 0(ωx−lφ)2(L0, t)dt+σ1L0 16TZ 0(ωx−lφ)2(0, t)dt +3k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+3k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt +6ρ1 l2L0TZ 0φ2 t(L0, t)dt≤ρ1 1 +16 l2L2 0TZ 0L0Z 0φ2 tdxdt +k1 2 + 17 l2L2 0+2150 l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt +5k1 l2TZ 0L0Z 0ψ2 xdxdt +C(R, T)lot+C(E(0) + E(T)).(83) Step 5. Next we multiply equation (49)by− 1 +18 l2L2 0 φand integrate by parts with respect to t ρ1 1 +18 l2L2 0TZ 0L0Z 0φ2 tdxdt +k1 1 +18 l2L2 0TZ 0L0Z 0(φx+ψ+lω)xφdxdt +lσ1 1 +18 l2L2 0TZ 0L0Z 0(ωx−lφ)φdxdt− 1 +18 l2L2 0TZ 0L0Z 0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))φdxdt = ρ1 1 +18 l2L2 0L0Z 0(φt(x, T)φ(x, T)−φt(x,0)φ(x,0))dx. (84) Since 33k1 1 +18 l2L2 0TZ 0L0Z 0(φx+ψ+lω)xφdxdt =−k1 1 +18 l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt +k1 1 +18 l2L2 0TZ 0(φx+ψ+lω)(L0, t)φ(L0, t)dt +k1 1 +18 l2L2 0TZ 0(φx+ψ+lω)(ψ+lω)dxdt (85) we obtain the estimate ρ1 1 +17 l2L2 0TZ 0L0Z 0φ2 tdxdt≤k1 2 +18 l2L2 0TZ 0L0Z 0(φx+ψ+lω)2dxdt +k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+σ1 32TZ 0L0Z 0(ωx−lφ)2dxdt +C(R, T)lot+C(E(0) + E(T)).(86) Summing up (83) and (86) we get 34σ1 32TZ 0L0Z 0(ωx−lφ)2dxdt +ρ1 2TZ 0L0Z 0ω2 tdxdt +ρ1L0 8TZ 0ω2 t(L0, t)dt +σ1L0 16TZ 0(ωx−lφ)2(L0, t)dt+σ1L0 16TZ 0(ωx−lφ)2(0, t)dt +2k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+2k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt +6ρ1 l2L0TZ 0φ2 t(L0, t)dt+1 l2L2 0TZ 0L0Z 0φ2 tdxdt ≤k1 4 + 17 l2L2 0+2200 l2L2 0TZ 0(φx+ψ+lω)2dxdt +6k1 l2TZ 0L0Z 0ψ2 xdxdt+C(R, T)lot+C(E(0)+E(T)). (87) Step 6. Next we multiply equation (50)byC1(φx+ψ+lω) and equation (49) byC1β1 ρ1ψx, where C1= 2(6 + 17 l2L2 0+2200 l2L2 0). Then we sum up the results and integrate by parts with respect to t. Taking into account (46),(47)we come to 35−β1C1TZ 0L0Z 0φtψtxdxdt−λ1C1TZ 0L0Z 0(φx+ψ+lω)xψxdxdt −lC1λ1TZ 0L0Z 0(ωx−lφ)ψxdxdt+C1β1 ρ1TZ 0L0Z 0(f1( ˜φ,˜ψ,˜ω)−f1( ˆφ,ˆψ,ˆω))ψxdxdt −β1C1TZ 0L0Z 0ψt(φxt+ψt+lωt)dxdt−λ1C1TZ 0L0Z 0ψxx(φx+ψ+lω)dxdt +k1C1TZ 0L0Z 0(φx+ψ+lω)2dxdt +C1TZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))(φx+ψ+lω)dxdt +C1TZ 0L0Z 0(h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω))(φx+ψ+lω)dxdt =β1C1L0Z 0φt(x,0)ψx(x,0)dx −β1C1L0Z 0φt(x, T)ψx(x, T)dx+β1C1L0Z 0ψt(x,0)(φx+ψ+lω)(x,0)dx −β1C1L0Z 0ψt(x, T)(φx+ψ+lω)(x, T)dx. (88) Integrating by parts with respect to xwe get β1C1TZ 0L0Z 0φtψtxdxdt +β1C1TZ 0L0Z 0ψt(φxt+lωt)dxdt ≤β1C1TZ 0φt(L0, t)ψt(L0, t)dt+β1C1lTZ 0L0Z 0ψtωtdxdt≤ρ1 l2L0TZ 0φ2 t(L0, t)dt +β2 1C2 1l2L0 4ρ1TZ 0ψ2 t(L0, t)dt+ρ1 4TZ 0L0Z 0ω2 tdxdt +β2 1C2 1l2 ρ1TZ 0L0Z 0ψ2 tdxdt (89) and 36λ1C1TZ 0L0Z 0(φx+ψ+lω)xψxdxdt +λ1C1TZ 0L0Z 0ψxx(φx+ψ+lω)dxdt =λ1C1TZ 0(φx+ψ+lω)(L0, t)ψx(L0, t)dt−λ1C1TZ 0(φx+ψ+lω)(0, t)ψx(0, t)dt ≤k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt +l2L0λ2 1C2 1 4k1TZ 0ψ2 x(L0, t)dt+l2L0λ2 1C2 1 4k1TZ 0ψ2 x(0, t)dt.(90) Moreover, lC1λ1TZ 0L0Z 0(ωx−lφ)ψxdxdt≤σ1 64TZ 0L0Z 0(ωx−lφ)2dxdt+16l2C2 1λ2 1 σ1TZ 0L0Z 0ψ2 xdxdt. (91) It follows from Lemma (6.6) with ε=k1C1 4 C1TZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))(φx+ψ+lω)dxdt ≤k1C1 4TZ 0L0Z 0(φx+ψ+lω)2dxdt +CTZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt (92) Consequently, collecting (88)–(92) we obtain 37C1k1 2TZ 0L0Z 0(φx+ψ+lω)2dxdt≤σ1 64TZ 0L0Z 0(ωx−lφ)2dxdt +20l2C2 1λ2 1 σ1TZ 0L0Z 0ψ2 xdxdt +C1 β1+β2 1l2 ρ1TZ 0L0Z 0ψ2 tdxdt+ k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt +l2L0λ2 1C2 1 4k1TZ 0ψ2 x(L0, t)dt+l2L0λ2 1C2 1 4k1TZ 0ψ2 x(0, t)dt +ρ1 l2L0TZ 0φ2 t(L0, t)dt+β2 1C2 1l2L0 4ρ1TZ 0ψ2 t(L0, t)dt+ρ1 4TZ 0L0Z 0ω2 tdxdt +CTZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(93) Combining (93) with (87) we get 38σ1 64TZ 0L0Z 0(ωx−lφ)2dxdt +ρ1 4TZ 0L0Z 0ω2 tdxdt +ρ1L0 8TZ 0ω2 t(L0, t)dt +σ1L0 16TZ 0(ωx−lφ)2(L0, t)dt+σ1L0 16TZ 0(ωx−lφ)2(0, t)dt +k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt +5ρ1 l2L0TZ 0φ2 t(L0, t)dt+1 l2L2 0TZ 0L0Z 0φ2 tdxdt + 2k1TZ 0L0Z 0(φx+ψ+lω)2dxdt ≤6k1 l2+20l2C2 1λ2 1 σ1TZ 0L0Z 0ψ2 xdxdt +C1 β1+β2 1l2 ρ1TZ 0L0Z 0ψ2 tdxdt +l2L0λ2 1C2 1 4k1TZ 0ψ2 x(L0, t)dt+l2L0λ2 1C2 1 4k1TZ 0ψ2 x(0, t)dt+β2 1C2 1l2L0 4TZ 0ψ2 t(L0, t)dt +CTZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(94) Step 7. Our next step is to multiply equation (50)by−C2xψx−C2(x−L0)ψx, where C2=l2λ1C2 1 k1. After integration by parts with respect to twe obtain β1C2TZ 0L0Z 0ψtxψxtdxdt+β1C2TZ 0L0Z 0ψt(x−L0)ψxtdxdt 39+λ1C2TZ 0L0Z 0ψxxxψxdxdt +λ1C2TZ 0L0Z 0ψxx(x−L0)ψxdxdt −k1C2TZ 0L0Z 0(φx+ψ+lω)(2x−L0)ψxdxdt−C2TZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))(2x−L0)ψxdxdt +TZ 0L0Z 0(h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω))(2x−L0)ψxdxdt =β1C2L0Z 0ψt(x, T)(2x−L0)ψx(x, T)dx−β1C2L0Z 0ψt(x,0)(2x−L0)ψx(x,0)dx. (95) After integration by parts with respect to xwe get β1C2TZ 0L0Z 0ψtxψxtdxdt +β1C2TZ 0L0Z 0ψt(x−L0)ψxtdxdt =−β1C2TZ 0L0Z 0ψ2 tdxdt +β1C2L0 2TZ 0ψ2 t(L0, t)dt(96) and λ1C2TZ 0L0Z 0ψxxxψxdxdt +λ1C2TZ 0L0Z 0ψxx(x−L0)ψxdxdt =λ1C2L0 2TZ 0ψ2 x(L0, t)dt+λ1C2L0 2TZ 0ψ2 x(0, t)dt−λ1C2TZ 0L0Z 0ψ2 xdxdt. (97) Furthermore, 40k1C2TZ 0L0Z 0(φx+ψ+lω)(2x−L0)ψxdxdt ≤k1TZ 0L0Z 0(φx+ψ+lω)2dxdt +k1C2 2L2 0 4TZ 0L0Z 0ψ2 xdxdt. (98) By Lemma (6.6) with ε=k1C22L2 0 4we have C2TZ 0L0Z 0ψt(2x−L0)ψxdxdt≤k1C2 2L2 0 4TZ 0L0Z 0ψ2 xdxdt+CTZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt. (99) As a result of (95)– (99) we obtain the estimate β1C2L0 2TZ 0ψ2 t(L0, t)dt+λ1C2L0 2TZ 0ψ2 x(L0, t)dt+λ1C2L0 2TZ 0ψ2 x(0, t)dt ≤k1TZ 0L0Z 0(φx+ψ+lω)2dxdt+ k1C2 2L2 0+λ1C2
TZ 0L0Z 0ψ2 xdxdt+β1C2TZ 0L0Z 0ψ2 tdxdt +CTZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(100) Summing up (94) and (100) and using (47) we infer 41σ1 64TZ 0L0Z 0(ωx−lφ)2dxdt +ρ1 4TZ 0L0Z 0ω2 tdxdt +ρ1L0 8TZ 0ω2 t(L0, t)dt +σ1L0 16TZ 0(ωx−lφ)2(L0, t)dt+σ1L0 16TZ 0(ωx−lφ)2(0, t)dt +k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt +5ρ1 l2L0TZ 0φ2 t(L0, t)dt+1 l2L2 0TZ 0L0Z 0φ2 tdxdt +k1TZ 0L0Z 0(φx+ψ+lω)2dxdt l2L0λ2 1C2 1 4k1TZ 0ψ2 x(L0, t)dt+l2L0λ2 1C2 1 4k1TZ 0ψ2 x(0, t)dt +β2 1C2 1l2L0 4ρ1TZ 0ψ2 t(L0, t)dt≤6k1 l2+20l2C2 1λ2 1 σ1+λ1C2+k1C2 2L2 0TZ 0L0Z 0ψ2 xdxdt + (C1+C2)β1+C1β2 1l2 ρ1TZ 0L0Z 0ψ2 tdxdt +CTZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(101) Step 8. Now we multiply equation (50)byC3ψ, where C3=2 λ1 6k1 l2+20l2C2 1λ2 1 σ1+λ1C2+k1C2 2L2 0 and integrate by parts with respect to t 42−C3β1TZ 0L0Z 0ψ2 tdxdt−λ1C3TZ 0L0Z 0ψxxψdxdt +k1C3TZ 0L0Z 0(φx+ψ+lω)ψdxdt +C3TZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψdxdt +C3TZ 0L0Z 0(h1( ˜φ,˜ψ,˜ω)−h1( ˆφ,ˆψ,ˆω))ψdxdt (102) =C3β1L0Z 0ψt(x,0)ψ(x,0)dx−C3β1L0Z 0ψt(x, T)ψ(x, T)dx (103) After integration by parts we infer the estimate λ1C3TZ 0L0Z 0ψ2 xdxdt≤k1 2TZ 0(φx+ψ+lω)2dxdt+C3β1TZ 0L0Z 0ψ2 tdxdt+l2L0λ2 1C2 1 8k1TZ 0ψ2 x(L0, t)dt +CTZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(104) Combining (104) with (101) we obtain σ1 64TZ 0L0Z 0(ωx−lφ)2dxdt +ρ1 4TZ 0L0Z 0ω2 tdxdt +ρ1L0 8TZ 0ω2 t(L0, t)dt +σ1L0 16TZ 0(ωx−lφ)2(L0, t)dt+σ1L0 16TZ 0(ωx−lφ)2(0, t)dt +k1 l2L0TZ 0(φx+ψ+lω)2(L0, t)dt+k1 l2L0TZ 0(φx+ψ+lω)2(0, t)dt 43+5ρ1 l2L0TZ 0φ2 t(L0, t)dt+1 l2L2 0TZ 0L0Z 0φ2 tdxdt +k1 2TZ 0L0Z 0(φx+ψ+lω)2dxdt l2L0λ2 1C2 1 8k1TZ 0ψ2 x(L0, t)dt+l2L0λ2 1C2 1 4k1TZ 0ψ2 x(0, t)dt+β2 1C2 1l2L0 4ρ1TZ 0ψ2 t(L0, t)dt +6k1 l2+20l2C2 1λ2 1 σ1+λ1C2+k1C2 2L2 0TZ 0L0Z 0ψ2 xdxdt ≤ (C1+C2)β1+C1β2 1l2 ρ1+C3β1TZ 0L0Z 0ψ2 tdxdt +CTZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt +C(R, T)lot+C(E(0) + E(T)).(105) Step 9. Consequently, it follows from (105) and assumption (D4) for any l >0 where exist constants Mi,i={1,3}(depending on l) such that TZ 0E1(t)dt+TZ 0B1(t)dt≤M1TZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt +M2(R, T)lot+M3(E(T) +E(0)),(106) where B1(t) =TZ 0(ωx−lφ)2(L0, t)dt+TZ 0(φx+ψ+lω)2(L0, t)dt+TZ 0ψ2 x(L0, t)dt +TZ 0ω2 t(L0, t)dt+TZ 0ψ2 t(L0, t)dt+TZ 0φ2 t(L0, t)dt.(107) Step 10. Finally, we multiply equation (52)by (x−L)ux, equation (53)by (x−L)vx, and (54)by (x−L)wx. Summing up the results and integrating 44by parts with respect to twe arrive at −ρ2TZ 0LZ L0ut(x−L)utxdxdt−k2TZ 0LZ L0(ux+v+lw)x(x−L)uxdxdt −lσ2TZ 0LZ L0(wx−lu)(x−L)uxdxdt+TZ 0LZ L0(f2(˜u,˜v,˜w)−f2(ˆu,ˆv,ˆw))(x−L)uxdxdt −β2TZ 0LZ L0vt(x−L)vxtdxdt−λ2TZ 0LZ L0vxx(x−L)vxdxdt +k2TZ 0LZ L0(ux+v+lw)(x−L)vxdxdt+TZ 0LZ L0(h2(˜u,˜v,˜w)−h2(ˆu,ˆv,ˆw))(x−L)vxdxdt −ρ2TZ 0LZ L0wt(x−L)wxtdxdt−σ2TZ 0LZ L0(wx−lu)x(x−L)wxdxdt +lk2TZ 0LZ L0(ux+v+lw)(x−L)wxdxdt+TZ 0LZ L0(g2(˜u,˜v,˜w)−g2(ˆu,ˆv,ˆw))(x−L)wxdxdt = −ρ2LZ L0(x−L)((utux)(x, T)−(utux)(x,0))dx−β2LZ L0(x−L)((vtvx)(x, T)−(vtvx)(x,0))dx −ρ2LZ L0(x−L)((wtwx)(x, T)−(wtwx)(x,0))dx. (108) After integration by parts with respect to xwe infer 45−ρ2TZ 0LZ L0ut(x−L)utxdx−β2TZ 0LZ L0vt(x−L)vxtdxdt−ρ2TZ 0LZ L0wt(x−L)wxtdxdt =ρ2 2TZ 0LZ L0u2 tdx+β2 2TZ 0LZ L0v2 tdxdt +ρ2 2TZ 0LZ L0w2 tdxdt −ρ2(L−L0) 2TZ 0u2 t(L0)dt−β2(L−L0) 2TZ 0v2 t(L0)dt−ρ2(L−L0) 2TZ 0w2 t(L0)dt (109) and −k2TZ 0LZ L0(ux+v+lw)x(x−L)uxdxdt−lσ2TZ 0LZ L0(wx−lu)(x−L)uxdxdt −λ2TZ 0LZ L0vxx(x−L)vxdxdt +k2TZ 0LZ L0(ux+v+lw)(x−L)vxdxdt 46−σ2TZ 0LZ L0(wx−lu)x(x−L)wxdxdt+lk2TZ 0LZ L0(ux+v+lw)(x−L)wxdxdt = −k2TZ 0LZ L0(ux+v+lw)x(x−L)(ux+v+lw)dxdt −σ2TZ 0LZ L0(wx−lu)x(x−L)(wx−lu)dxdt−λ2TZ 0LZ L0vxx(x−L)vxdxdt −lσ2(L−L0)TZ 0(wx−lu)(L0)u(L0)dt+k2(L−L0)TZ 0(ux+v+lw)(L0)v(L0)dt +lk2(L−L0)TZ 0(ux+v+lw)(L0)w(L0)dt= −k2(L−L0) 2TZ 0(ux+v+lw)2(L0)dt+k2 2TZ 0LZ L0(ux+v+lw)2dxdt +σ2 2TZ 0LZ L0(wx−lu)2dxdt−σ2(L−L0) 2TZ 0(wx−lu)2(L0)dt +λ2 2TZ 0LZ L0v2 xdxdt−λ2(L−L0) 2TZ 0v2 x(L0)dt−lσ2(L−L0)TZ 0(wx−lu)(L0)u(L0)dt +k2(L−L0)TZ 0(ux+v+lw)(L0)v(L0)dt +lk2(L−L0)TZ 0(ux+v+lw)(L0)w(L0)dt.(110) Consequently, it follows from (108) –(110) that for any l >0 where exist 47constants M4, M5, M6>0 such that TZ 0E2(t)dt≤M4TZ 0B2(t)dt+M5(R, T)lot+M6(E(T) +E(0)),(111) where B2(t) =TZ 0(wx−lu)2(L0, t)dt+TZ 0(ux+v+lw)2(L0, t)dt+TZ 0v2 x(L0, t)dt +TZ 0w2 t(L0, t)dt+TZ 0v2 t(L0, t)dt+TZ 0u2 t(L0, t)dt.(112) Then, due to transmission conditions (8)–(11) there exist δ, M 7, M8>0 (depending on l), such that TZ 0E(t)dt≤δTZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt+M7(R, T)lot+M8(E(T)+E(0)). (113) It follows from (55) that there exists C >0 such that TZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt≤C E(0) +TZ 0|H(ˆU(t),˜U(t))|dt .(114) By Lemma 5.6 we have that for any ε >0 there exists C(ε, R)>0 such that TZ 0|H(ˆU(t),˜U(t))|dt≤εTZ 0L0Z 0E(t)dxdt +C(ε, R, T )lot. (115) Combining (115) with (114) we arrive at TZ 0L0Z 0(γ(˜ψt)−γ(ˆψt))ψtdxdt≤CE(0) + C(R, T)lot. (116) 48Substituting (116) into (113) we obtain TZ 0E(t)dt≤C(R, T)lot+C(E(T) +E(0)) (117) for some C, C(R, T)>0. Our remaining task is to estimate the last term in (60). TZ 0TZ tH(ˆU(s),˜U(s))dsdt≤TZ 0E(t)dt+T3C(R)lot. (118) Then, it follows from (60) and (118) that TE(T)≤CTZ 0E(t)dt+C(T, R)lot. (119) Then the combination of (119) with (117) leads to TE(T)≤C(R, T)lot+C(E(T) +E(0)). (120) Choosing Tlarge enough one can obtain estimate (44)which together with Theorem 6.5 immediately leads to the asymptotic smoothness of the system. 6.3 Existence of attractors. The following statement collects criteria on existence and properties of attractors to gradient systems. Theorem 6.8 ([10,12]).Assume that (H, S t)is a gradient asymptotically smooth dynamical system. Assume its Lyapunov function L(y)is bounded from above on any bounded subset of Hand the set WR={y:L(y)≤R} is bounded for every R. If the set Nof stationary points of (H, S t)is bounded, then (St, H)possesses a compact global attractor. Moreover, the 49global attractor consists of full trajectories γ={U(t) :t∈R}such that lim t→−∞distH(U(t),N) = 0 and lim t→+∞distH(U(t),N) = 0 (121) and lim t→+∞distH(Stx,N) = 0 for any x∈H; (122) that is, any trajectory stabilizes to the set Nof stationary points. Now we state the result on the existence of an attractor. Theorem 6.9. Let assumptions of Theorems 6.2, 6.7, hold true, moreover, lim inf |s|→∞h1(s) s>0, (N5) ∇F2(u, v, w )(u, v, w )−a1F2(u, v, w )≥ −a2, a i≥0. Then, the dynamical system (H, S t)generated by (1)-(11)possesses a compact global attractor Apossessing properties (121) ,(122) . Proof. In view of Theorems 6.2, 6.7, 6.8 our remaining task is to show the boundedness of the set of stationary points and the set WR={Z:L(Z)≤R}, where Lis given by (30). The second statement follows immediately from the structure of function Land property (N5). The first statement can be easily shown by energy-like estimates for stationary solutions taking into account (N5). 7 Singular Limits on finite time intervals 7.1 Singular limit l→0 Let the nonlinearities fj, hj, gjare such that f1(φ, ψ, ω ) =f1(φ, ψ), h 1(φ, ψ, ω ) =h1(φ, ψ), g1(φ, ψ, ω ) =g1(ω), f2(u, v, w ) =f2(u, v), h 2(u, v, w ) =h2(u, v), g 2(u, v, w ) =g2(w).(N6) 50If we formally set l= 0 in (18)-(24), we obtain the contact problem for a straight Timoshenko beam ρ1φtt−k1(φx+ψ)x+f1(φ, ψ) =p1(x, t), (x, t)∈(0, L0)×(0, T), (123) β1ψtt−λ1ψxx+k1(φx+ψ) +γ(ψt) +h1(φ, ψ) =r1(x, t), (x, t)∈(0, L0)×(0, T), (124) ρ2utt−k2(ux+v)x+f2(u, v) =p2(x, t), (x, t)∈(L0, L)×(0, T), (125) β2vtt−λ2vxx+k2(ux+v) +h2(u, v) =r2(x, t), (x, t)∈(L0, L)×(0, T), (126) φ(0, t) =ψ(0, t) = 0 , u(L, t) =v(L, t) = 0 , (127) φ(L0, t) =u(L0, t), ψ(L0, t) =v(L0, t), (128) k1(φx+ψ)(L0, t) =k2(ux+v)(L0, t), λ1ψx(L0, t) =λ2vx(L0, t), (129) and an independent contact problem for wave equations ρ1ωtt−σ1ωxx+g1(ω) =q1(x, t), (x, t)∈(0, L0)×(0, T), (130) ρ2wtt−σ2wxx+g2(w) =q2(x, t), (x, t)∈(L0, L)×(0, T), (131) σ1ωx(L0, t) =σ2wx(L0, t), ω(L0, t) =w(L0, t), (132) w(L, t) = 0 , ω(0, t) = 0 . (133) The following theorem gives an answer, how close are solutions to (18)-(24) to the solution of decoupled system (123)-(133) when l→0. Theorem 7.1. Assume that the conditions of Theorem 5.4, (D3) and(N6) hold. Let Φ(l)be the solution to (18)-(24) with the fixed land the initial data Φ(x,0) = ( φ0, ψ0, ω0, u0, v0, w0)(x),Φt(x,0) = ( φ1, ψ1, ω1, u1, v1, w1)(x). 51Then for every T >0 Φ(l)∗⇀(φ, ψ, ω, u, v, w ) inL∞(0, T;Hd)asl→0, Φ(l) t∗⇀(φt, ψt, ωt, ut, vt, wt) inL∞(0, T;Hv)asl→0, where (φ, ψ, u, v )is the solution to (123) -(129) with the initial conditions (φ, ψ, u, v )(x,0) = ( φ0, ψ0, u0, v0)(x),(φt, ψt, ut, vt)(x,0) = ( φ1, ψ1, u1, v1)(x), and(ω, w)is the solution to (130) -(133) with the initial conditions (ω, w)(x,0) = ( ω0, w0)(x),(ωt, wt)(x,0) = ( ω1, w1)(x). The proof is similar to that of Theorem 3.1 [ 24] for the homogeneous Bresse beam with obvious changes, except for the limit transition in the nonlinear dissipation term. For the future use we formulate it as a lemma. Lemma 7.2. Let(D3) holds. Then ZT 0ZL0 0γ(ψ(l)(x, t))γ1(x, t)dxdt→ZT 0ZL0 0γ(ψ(x, t))γ1(x, t)dxdt asl→0 for every γ1∈L2(0, T;H1(0, L0)). Proof. Since (D1) and (D3) hold |γ(s)| ≤Ms, therefore ||γ(ψ(l))||L∞(0,T;L2(0,L0))≤C(||ψ(l)||L∞(0,T;L2(0,L0))). Thus, due to Lemmas 4.1, 5.6 the sequence RΦ(l) tt=AΦ(l)+ Γ(Φ(l) t) +F(Φ(l)) +P is bounded in L∞(0, T;H−1(0, L)) and we can extract from Φ(l) tta subse- quence, that converges ∗-weakly in L∞(0, T;H−1(0, L)). Thus, Φ(l) t→Φtstrongly in L2(0, T;H−ε(0, L)), ε > 0. 52Consequently, ZT 0ZL0 0(γ(ψ(l)(x, t))−γ(ψ(x, t)))γ1(x, t)dxdt≤ C(L)ZT 0ZL0 0|ψ(l)(x, t)−ψ(x, t)||γ1(x, t)|dxdt→0. We perform numerical modelling for the original problem with l= 1,1/3,1/10,1/30,1/100,1/300,1/1000 and the limiting problem ( l= 0) with the following values of constants ρ1=ρ2= 1, β1=β2= 2, σ1= 4, σ2= 2,λ1= 8,λ2= 4,L= 10, L0= 4 and the right-hand sides p1(x) = sin x, r 1(x) =x, q 1(x) = sin x, (134) p2(x) = cos x, r 2(x) =x+ 1, q 2(x) = cos x. (135) In this subsection we consider the nonlinearities with the potentials F1(φ, ψ, ω ) =|φ+ψ|4− |φ+ψ|2+|φψ|2+|ω|3, F2(u, v, w ) =|u+v|4− |u+v|2+|uv|2+|w|3. Consequently, the nonlinearities have the form f1(φ, ψ, ω ) = 4( φ+ψ)3−2(φ+ψ) + 2φψ2, f 2(u, v, w ) = 4( u+v)3−2(u+v) + 2uv2, h1(φ, ψ, ω ) = 4( φ+ψ)3−2(φ+ψ) + 2φ2ψ, h 2(u, v, w ) = 4( u+v)3−2(u+v) + 2u2v, g1(φ, ψ, ω ) = 3|ω|ω, g 2(u, v, w ) = 3|w|w. For modelling we choose the following (globally Lipschitz) dissipation γ(s) = 1 100s3,|s| ≤10, 10s, |s|>10 53and the following initial data: φ(x,0) =−3 16x2+3 4x, u (x,0) = 0 , ψ(x,0) =−1 12x2+7 12x, v (x,0) =−1 6x+5 3, ω(x,0) =1 16x2−1 4x, w (x,0) =−1 12x2+7 6x−10 3, φt(x,0) =x 4, u t(x,0) =−1 6(x−10), ψt(x,0) =x 4, v t(x,0) =−1 6(x−10), ωt(x,0) =x 4, w t(x,0) =−1 6(x−10). Figures 2-7 show the behavior of solutions when l→0 for the chosen cross-sections of the beam. 0 1 2 3 4 5 6 7 8-2-1.5-1-0.500.51 Figure 2: Transversal displacement of the beam, cross-section x= 2. 7.2 Singular limit ki→ ∞, l→0 The singular limit for the straight Timoshenko beam ( l= 0) as ki→+∞is the Euler-Bernoulli beam equation [ 19, Ch. 4]. We have a similar result for the Bresse composite beam when ki→ ∞ , l→0. Theorem 7.3. Let the assumptions of Theorem 5.4, (N6) and(D3) hold. 540 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. 0 1 2 3 4 5 6 7 8-2-1.5-1-0.500.51 Figure 4: Shear angle variation of the beam, cross-section x= 2. Moreover, (φ0, u0)∈ φ0∈H2(0, L0), u0∈H2(L0, L), φ0(0) = u0(L) = 0 , ∂xϕ0(0) = ∂xu0(L) = 0 , ∂xφ0(L0, t) =∂xu0(L0, t)};(I1) ψ0=−∂xφ0, v0=−∂xu0; (I2) (φ1, u1)∈ {φ1∈H1(0, L0), u1∈H1(L0, L), φ1(0) = u1(L) = 0 , φ1(L0, t) =u1(L0, t)}; (I3) ω0=w0= 0; (I4) h1, h2∈C1(R2); (N6) r1∈L∞(0, T;H1(0, L0)), r2∈L∞(0, T;H1(L0, L)), r1(L0, t) =r2(L0, t)for almost all t >0.(R3) 550 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. 0 1 2 3 4 5 6 7 800.20.40.60.81 Figure 6: Longitudinal displacement of the beam, cross-section x= 2. Letk(n) j→ ∞ ,l(n)→0asn→ ∞ , and Φ(n)be weak solutions to (18)-(24) with fixed k(n) j, l(n)and the same initial data Φ(x,0) = ( φ0, ψ0, ω0, u0, v0, w0)(x),Φt(x,0) = ( φ1, ψ1, ω1, u1, v1, w1). Then for every T >0 Φ(n)∗⇀(φ, ψ, ω, u, v, w ) inL∞(0, T;Hd)asn→ ∞ , Φ(n) t∗⇀(φt, ψt, ωt, ut, vt, wt) inL∞(0, T;Hv)asn→ ∞ , where 560 1 2 3 4 5 6 7 800.511.522.53Figure 7: Longitudinal displacement of the beam, cross-section x= 6. •(φ, u)is a weak solution to ρ1φtt−β1φttxx+λ1φxxxx−γ′(−φtx)φtxx+∂xh1(φ,−φx) +f1(φ,−φx) = p1(x, t) +∂xr1(x, t),(x, t)∈(0, L0)×(0, T), (136) ρ2utt−β2uttxx+λ2uxxxx+∂xh2(u,−ux) +f2(u,−ux) = p2(x, t) +∂xr2(x, t),(x, t)∈(L0, L)×(0, T), (137) φ(0, t) =φx(0, t) = 0 , u(L, t) =ux(L, t) = 0 , (138) φ(L0, t) =u(L0, t), φx(L0, t) =ux(L0, t), λ1φxx(L0, t) =λ2uxx(L0, t), (139) λ1φxxx(L0, t)−β1φttx(L0, t) +h1(φ(L0, t),−φx(L0, t)) +γ(−φtx(L0, t)) = λ2uxxx(L0, t)−β2uttx(L0, t) +h2(u(L0, t),−ux(L0, t)), (140) with the initial conditions (φ, u)(x,0) = ( φ0, u0)(x),(φt, ut)(x,0) = ( φ1, u1)(x). •ψ=−φx, v=−ux; 57•(ω, w)is the solution to ρ1ωtt−σ1ωxx+g1(ω) =q1(x, t),(x, t)∈(0, L0)×(0, T),(141) ρ2wtt−σ2wxx+g2(w) =q2(x, t),(x, t)∈(L0, L)×(0, T),(142) ω(0, t) = 0 , w(L, t) = 0 , (143) σ1ωx(L0, t) =σ2wx(L0, t), ω(L0, t) =w(L0, t) (144) with the initial conditions (ω, w)(x,0) = (0 ,0),(ωt, wt)(x,0) = ( ω1, w1)(x). Proof. The proof uses the idea from [ 19, Ch. 4.3] and differs from it mainly in transmission conditions. We skip the details of the proof, which coincide with [19]. Energy inequality (26) implies ∂t(φ(n), ψ(n), ω(n), u(n), v(n), w(n)) bounded in L∞(0, T;Hv),(145) ψ(n)bounded in L∞(0, T;H1(0, L0)),(146) v(n)bounded in L∞(0, T;H1(L0, L)) (147) ω(n) x−l(n)φ(n)bounded in L∞(0, T;L2(0, L0)),(148) w(n) x−l(n)u(n)bounded in L∞(0, T;L2(L0, L)),(149) k(n) 1(φ(n) x+ψ(n)+l(n)ω(n)) bounded in L∞(0, T;L2(0, L0)),(150) k(n) 2(u(n) x+v(n)+l(n)w(n)) bounded in L∞(0, T;L2(L0, L)),(151) Thus, we can extract subsequences which converge in corresponding spaces weak-∗. Similarly to [19] we have φ(n) x+ψ(n)+l(n)ω(n)∗⇀0 in L∞(0, T;L2(0, L0)), therefore φx=−ψ. 58Analogously, ux=−v. (146)-(151) imply ω(n)∗⇀ ω inL∞(0, T;H1(0, L0)), w(n)∗⇀ w inL∞(0, T;H1(L0, L)), (152) φ(n)∗⇀ φ inL∞(0, T;H1(0, L0)), u(n)∗⇀ u inL∞(0, T;H1(L0, L)). (153) Thus, the Aubin’s lemma gives that Φ(n)→Φ strongly in C(0, T; [H1−ε(0, L0)]3×[H1−ε(L0, L)]3) (154) for every ε >0 and then ∂xφ0+ψ0+l(n)ω0→0 strongly in H−ε(0, L0), This implies that ∂xφ0=−ψ0, ω 0= 0. Analogously, ∂xu0=−v0, w 0= 0. Let us choose a test function of the form B= (β1,−β1 x,0, β2,−β2 x,0)∈FT such that β1 x(L0, t) =β2 x(L0, t) for almost all t. Due to (152) -(154) and Lemma 7.2 we can pass to the limit in variational equality (25)asn→ ∞ . The same way as in [ 19, Ch. 4.3] we obtain that the limiting functions φ, u are of higher regularity and satisfy the following variational equality 59ZT 0ZL0 0 ρ1φtβ1 t−β1φtxβ1 tx
dxdt +ZT 0ZL L0 ρ2utβ2 t−β1utxβ2 tx
dxdt− ZL0 0 ρ1(φtβ1 t)(x,0)−β1(φtxβ1 tx)(x,0)
dx+ZL L0 ρ2(utβ2 t)(x,0)−β1(utxβ2 tx)(x,0)
dx+ ZT 0ZL0 0λ1φxxβ1 xxdxdt+ZT 0ZL L0λ2uxxβ2 xxdxdt−ZT 0ZL0 0γ′(−φxt)φtxxβ1dxdt+ ZT 0ZL0 0 f1(φ,−φx)β1−h1(φ,−φx)β1 x
dxdt+ZT 0ZL L0 f2(u,−ux)β2−h2(u,−ux)β2 x
dxdt = ZT 0ZL0 0 p1β1−r1β1 x
dxdt +ZT 0ZL L0 p2β2−r2β2 x
dxdt. (155) Provided φ, uare smooth enough, we can integrate (155) by parts with respect to x, tand obtain ZT 0ZL0 0(ρ1−β1∂xx)φttβ1dxdt +ZT 0ZL L0(ρ2−β2∂xx)uttβ2dxdt+ ZT 0[β1φttx(t, L0)−β2uttx(t, L0)]β1(t, L0)dt+ ZT 0ZL0 0λ1φxxxxβ1dxdt +ZT 0ZL L0λ2uxxxxβ2dxdt+ ZT 0[λ1φxx−λ2uxx] (t, L0)β1 x(t, L0)dt−ZT 0[λ1φxxx−λ2uxxx] (t, L0)β1(t, L0)dt− ZT 0ZL0 0γ′(−φxt)φxxtβ1dxdt−ZT 0γ(−φxt(L0, t))β1(L0, t)+ ZT 0ZL0 0(f1(φ,−φx) +∂xh1(φ,−φx))β1dxdt+ZT 0ZL L0(f2(u,−ux) +∂xh2(u,−ux))β2dxdt+ ZT 0(h2(u(L0, t),−ux(L0, T))−h1(φ(L0, t),−φx(L0, T)))β1(L0, t)dt= ZT 0ZL0 0(p1+∂xr1)β1dxdt+ZT 0ZL L0(p2+∂xr2)β2dxdt+ZT 0[r2(t, L0)−r1(t, L0)]β1(t, L0)dt. (156) Requiring all the terms containing β1(L0, t),β1 x(L0, t) to be zero, we get transmission conditions (139) -(137) . Equations (136) -(137) are recovered 60from the variational equality (156). Problem (141)-(144) can be obtained in the same way. We perform numerical modelling for the original problem with the initial parameters l(1)= 1, k(1) 1= 4, k(1) 2= 1; We model the simultaneous convergence l→0 and k1, k2→ ∞ in the following way: we divide lby the factor χand multiply k1, k2by the factor χ. Calculations performed for the original problem with χ= 1, χ= 3, χ= 10, χ= 30, χ= 100 , χ= 300 and the limiting problem (136) -(140) . Other constants in the original problem are the same as in the previous subsection and we choose the functions in the right-hand side of (134)-(135) as follows: r1(x) =x+ 4, r2(x) = 2 x. The nonlinear feedbacks are f1(φ, ψ, ω ) = 4 φ3−2φ, f 2(u, v, w ) = 4 u3−8u, h1(φ, ψ, ω ) = 0 , h 2(u, v, w ) = 0 , g1(φ, ψ, ω ) = 3|ω|ω, g 2(u, v, w ) = 6|w|w. We use linear dissipation γ(s) =sand choose the following initial displace- ment and shear angle variation φ0(x) =−13 640x4+6 40x2−23 40x2, u0(x) =41 2160x4−68 135x3+823 180x2−439 27x+520 27. ψ0(x) =− −13 160x3+27 40x2−23 20x , 61v0(x) =−41 540x3−68 45x2+823 90x−439 27 . and set ω0(x) =w0(x) = 0 . We choose the following initial velocities φ1(x) =−1 32x3+3 16x2, u 1(x) =1 108x3−7 36x2+10 9x−25 27, ω1(x) =ψ1(x) =3 5x, w1(x) =v1(x) =−2 5x+ 4. The double limit case appeared to be more challenging from the point of view of numerics, then the case l→0. The numerical simulations of the coupled system in equations (1)-(7)including the interface conditions in(8)-(11)were done by a semidiscretization of the functions ϕ, ψ, ω, u, v, w with respect to the position xand by using an explicit scheme for the time integration. That allows to choose the discretized values at grid points near the interface in a separate step so that they obey the transmission conditions. It was necessary to solve a nonlinear system of equations for the six functions at three grid points (at the interface, and left and right of the interface) in each time step. Any attempt to use a full implicit numerical scheme led to extremely time-expensive computations due to the large nonlinear system over all discretized values which was to solve in each time step. On the other hand, increasing k1, k2increase the stiffness of the system of ordinary differential equations which results from the semidiscretization, and the CFL- conditions requires small time steps — otherwise numerical oscillations occur. Figures 8-13 present smoothed numerical solutions, particularly necessary for large factors χ, e. g. χ= 300. When the parameters k1, k2are large, the material of the beam gets stiff, and so does the discretized system of differential equations. Nevertheless the oscillations are still noticeable in the graph. The observation that the factor χcannot be arbitrarily enlarged, underlines the importance of having the limit problem for χ→ ∞ in(1)-(15). 620 1 2 3 4 5 6 7 8-1-0.500.511.52Figure 8: Transversal displacement of the beam, cross-section x= 2. 0 1 2 3 4 5 6 7 8-4-2024 Figure 9: Transversal displacement of the beam, cross-section x= 6. 8 Discussion There is a number of papers devoted to long-time behaviour of linear ho- mogeneous Bresse beams (with various boundary conditions and dissipation nature). If damping is present in all three equations, it appears to be sufficient for the exponential stability of the system without additional assumptions on the parameters of the problems (see, e.g., [2]). The situation is different if we have a dissipation of any kind in one or two equations only. First of all, it matters in which equations the dissipation is present. There are results on the Timoshenko beams [ 25] and the Bresse 630 1 2 3 4 5 6 7 8-20246Figure 10: Shear angle variation of the beam, cross-section x= 2. 0 1 2 3 4 5 6 7 80510 Figure 11: Shear angle variation of the beam, cross-section x= 6. beams [ 13] that damping in only one of the equations does not guarantee the exponential stability of the whole system. It seems that for the Bresse system the presence of the dissipation in the shear angle equation is necessary for the stability of any kind. To get the exponential stability, one needs additional assumptions on the coefficients of the problem, usually, the equality of the propagation speeds: k1=σ1,ρ1 k1=β1 λ1. Otherwise, only polynomial (non-uniform) stability holds (see, e.g., [ 1] for mechanical dissipation and [ 13] for thermal dissipation). In [ 6] analogous 640 1 2 3 4 5 6 7 8-3-2-10123Figure 12: Longitudinal displacement of the beam, cross-section x= 2. 0 1 2 3 4 5 6 7 8-3-2-1012 Figure 13: Longitudinal displacement of the beam, cross-section x= 6. results are established in case of nonlinear damping. If dissipation is present in all three equations of the Bresse system, corresponding problems with nonlinear source forces of local nature possesses global attractors under the standard assumptions for nonlinear terms (see, e.g., [ 24]). Otherwise, nonlinear source forces create technical difficulties and may cause instability of the system. To the best of our knowledge, there is no literature on such cases. In the present paper we study a transmission problem for the Bresse system. Transmission problems for various equation types have already had 65some history of investigations. One can find a number of papers concern- ing their well-posedness, long-time behaviour and other aspects (see, e.g., [26] for a nonlinear thermoelastic/isothermal plate, or [ 14] for the Euler- Bernoulli/Timoshenko beam and [ 15] for the full von Karman beam). Prob- lems with localized damping are close to transmission problems. In the recent years a number of such problems for the Bresse beams were studied in, e.g., [24,6]. To prove the existence of attractors in this case a unique continuation property is an important tool, as well as the frequency method. The only paper we know on a transmission problem for the Bresse system is [28]. The beam in this work consists of a thermoelastic (damped) and elastic (undamped) parts, both purely linear. Despite the presence of dissipation in all three equations for the damped part, the corresponding semigroup is not exponentially stable for any set of parameters, but only polynomially (non-uniformly) stable. In contrast to [ 28], we consider mechanical damping only in the equation for the shear angle for the damped part. However, we can establish exponential stability for the linear problem and existence of an attractor for the nonlinear one under restrictions on the coefficients in the damped part only. The assumption on the nonlinearities can be simplified in 1D case (cf. e.g. [16]). Conflict of Interest Statement The research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Acknowledgements The research is supported by the Volkswagen Foundation project ”From Modeling and Analysis to Approximation”. The first and the third authors were also successively supported by the Volkswagen Foundation project ”Dynamic Phenomena in Elasticity Problems” at Humboldt-Universit¨ at zu Berlin, Funding for Refugee Scholars and Scientists from Ukraine. 66References [1]Fatiha Alabau-Boussouira and Jaime E. Munoz-Rivera and Dilberto da S. Almeida Junior, Stability to weak dissipative Bresse system, Journal of Mathematical Analysis and Applications, 374 (2011), 481- 498, https://doi.org/10.1016/j.jmaa.2010.07.046. [2]Dilberto da S. Almeida Junior and M L Santos, Numerical exponential decay to dissipative Bresse system, Journal of Applied Mathematics, 2010, Art. ID 848620, 17 pp. [3]Viorel Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Nordhof, 1976. [4]J. A. C. Bresse, Cours de Mechanique Appliquee, Mallet Bachelier, Paris, 1859. [5] R.P. Kanwal, Generalized functions. Theory and applications, Birkh¨ auser, 2004. [6]Wenden Charles and J.A. Soriano and Fl´ avio A. Falc˜ ao Nascimento and J.H. Rodrigues, Decay rates for Bresse system with arbitrary nonlinear localized damping, Journal of Differential Equations, 255 (2013), 2267-2290. [7]Igor Chueshov, Strong solutions and attractor of the von Karman equations (in Russian), Mathematicheskii Sbornik, 181 (1990) , 25–36. [8]Igor Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkiv, 2002. [9]Igor Chueshov and Matthias Eller and Irena Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Communications in Partial Differential Equations, 27 (2002), 1901–1951. 67[10] Igor Chueshov and Tamara Fastovska and Iryna Ryzhkova, Quasistabil- ity method in study of asymptotical behaviour of dynamical systems, J. Math. Phys. Anal. Geom., 15 (2019), 448-501.10.15407/mag15.04.448. [11] Igor Chueshov and Irena Lasiecka, Long-time dynamics of von Kar- man semi-flows with non-linear boundary/interior damping, Journal of differential equation, 233 (2007), 42–86. [12] Igor Chueshov and Irena Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS ,912, AMS, Providence, RI, 2008. [13] F. Dell’Oro, Asymptotic stability of thermoelastic systems of Bresse type, J. Differential Equations, 258 (2015), 3902-3927. [14] Tamara Fastovska, Decay rates for Kirchhoff-Timoshenko transmission problems, Communications on Pure and Applied Analysis, 12 (2013), 2645-2667, 10.3934/cpaa.2013.12.2645. [15] Tamara Fastovska, Global attractors for a full von Karman beam transmission problem” Communications on Pure and Applied Analysis, 22 (2023), 1120-1158. [16] T. Fastovska, Attractor for a composite system of nonlinear wave and thermoelastic plate equations, Visnyk of Kharkiv National University 70 (2014), 4-35. [17] AKh. Khanmamedov, Global attractors for von Karman equations with nonlinear dissipation, Journal of Mathematical Analysis and Applications 318 (2016), 92-101, 10.1016/j.jmaa.2005.05.031. [18] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Prog. Nonlinear Differ. Equ. Appl. vol.50, Birkh¨ auser, Basel, 2002, 197-216. [19] J.E. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadel- phia, PA, 1989. 68[20]J. E. Lagnese and G. Leugering and E. J. P. G. Schmidt, Modeling, Anal- ysis and Control of Dynamic Elastic Multi-Link Structures, Birkh auser, Boston, 1994. [21] Irena Lasiecka and Roberto Triggiani, Regularity of Hyperbolic Equa- tions Under L2(0, T;L2(Γ))-Dirichlet Boundary Terms, Appl. Math. Optim., 10 (1983), 275-286. [22] J.-L. Lions and E. Magenes, Probl´ emes aux limites non homog´ enes et applications, Vol 1, Dunod, Paris, 1968. [23] Weijiu Liu and Graham H. Williams, Exact Controllability for Prob- lems of Transmission of the Plate Equation with Lower-order Terms, Quarterly of Applied Math., 58 (2000), 37-68. [24] To Fu Ma and Rodrigo Nunes Monteiro, Singular Limit and Long-Time Dynamics of Bresse Systems, SIAM Journal on Mathematical Analysis, 49 (2017), 2468-2495, 10.1137/15M1039894. [25] Jaime E. Mu˜ noz Rivera, Reinhard Racke, Mildly dissipative nonlinear Timoshenko systems—global existence and exponential stability, Journal of Mathematical Analysis and Applications, 276 (2002), 10.1016/S0022- 247X(02)00436-5. [26] M. Potomkin, A nonlinear transmission problem for a compound plate with thermoelastic part, Mathematical Methods in the Applied Sciences, 35 (2012), 530-546. [27] Roberto Triggiani and P. F. Yao, Carleman Estimates with No Lower- Order Terms for General Riemann Wave Equations. Global Uniqueness and Observability in One Shot, Appl. Math. Optim., 46 (2002), 331-375. [28]W. Youssef, Asymptotic behavior of the transmission problem of the Bresse beam in thermoelasticity, Z. Angew. Math. Phys., 73 (2022), 10.1007/s00033-022-01797-7. 69

2023-09-26

We consider a nonlinear transmission problem for a Bresse beam, which consists of two parts, damped and undamped. The mechanical damping in the damped part is present in the shear angle equation only, and the damped part may be of arbitrary positive length. We prove well-posedness of the corresponding PDE system in energy space and establish existence of a regular global attractor under certain conditions on nonlinearities and coefficients of the damped part only. Moreover, we study singular limits of the problem when $l\to 0$ or $l\to 0$ simultaneously with $k_i\to +\infty$ and perform numerical modelling for these processes.

Qualitative properties of solutions to a nonlinear transmission problem for an elastic Bresse beam

2309.15171v2

Spin transport and dynamics in all-oxide perovskite La 2=3Sr1=3MnO 3/SrRuO 3bilayers probed by ferromagnetic resonance Satoru Emori,1,Urusa S. Alaan,1, 2Matthew T. Gray,1, 2Volker Sluka,3Yizhang Chen,3Andrew D. Kent,3and Yuri Suzuki1, 4 1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305 USA 2Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 USA 3Department of Physics, New York University, New York, NY 10003, USA 4Department of Applied Physics, Stanford University, Stanford, CA 94305 USA (Dated: November 11, 2021) Thin lms of perovskite oxides oer the possibility of combining emerging concepts of strongly correlated electron phenomena and spin current in magnetic devices. However, spin transport and magnetization dynamics in these complex oxide materials are not well understood. Here, we ex- perimentally quantify spin transport parameters and magnetization damping in epitaxial perovskite ferromagnet/paramagnet bilayers of La 2=3Sr1=3MnO 3/SrRuO 3(LSMO/SRO) by broadband ferro- magnetic resonance spectroscopy. From the SRO thickness dependence of Gilbert damping, we estimate a short spin diusion length of <1 nm in SRO and an interfacial spin-mixing conductance comparable to other ferromagnet/paramagnetic-metal bilayers. Moreover, we nd that anisotropic non-Gilbert damping due to two-magnon scattering also increases with the addition of SRO. Our results demonstrate LSMO/SRO as a spin-source/spin-sink system that may be a foundation for examining spin-current transport in various perovskite heterostructures. I. INTRODUCTION Manipulation and transmission of information by spin current is a promising route toward energy-ecient mem- ory and computation devices1. Such spintronic devices may consist of ferromagnets interfaced with nonmagnetic conductors that exhibit spin-Hall and related spin-orbit eects2{4. The direct spin-Hall eect in the conductor can convert a charge current to a spin current, which ex- erts torques on the adjacent magnetization and modies the state of the device5,6. Conversely, the inverse spin- Hall eect in the conductor can convert a propagating spin current in the magnetic medium to an electric signal to read spin-based information packets7. For these device schemes, it is essential to understand the transmission of spin current between the ferromagnet and the conductor, which is parameterized by the spin-mixing conductance and spin diusion length. These spin transport parame- ters can be estimated by spin pumping at ferromagnetic resonance (FMR), in which a spin current is resonantly generated in the ferromagnet and absorbed in the adja- cent conductor8,9. Spin pumping has been demonstrated in various combinations of materials, where the magnetic layer may be an alloy (e.g., permalloy) or insulator (e.g., yttrium iron garnet) and the nonmagnetic conductor may be a transition metal, semiconductor, conductive poly- mer, or topological insulator10{16. Transition metal oxides, particularly those with the perovskite structure, oer the intriguing prospect of integrating a wide variety of strongly correlated elec- tron phenomena17,18with spintronic functionalities19,20. Among these complex oxides, La 2=3Sr1=3MnO 3(LSMO) and SrRuO 3(SRO) are attractive materials for epitaxial, lattice-matched spin-source/spin-sink heterostructures. LSMO, a metallic ferromagnet known for its colossalmagnetoresistance and Curie temperature of >300 K, can be an excellent resonantly-excited spin source be- cause of its low magnetization damping21{26. SRO, a room-temperature metallic paramagnet with relatively high conductivity27, exhibits strong spin-orbit coupling28 that may be useful for emerging spintronic applications that leverage spin-orbit eects2{4. A few recent studies have reported dc voltages at FMR in LSMO/SRO bilayers that are attributed to the in- verse spin-Hall eect in SRO generated by spin pump- ing24{26. However, it is generally a challenge to separate the inverse spin-Hall signal from the spin rectication sig- nal, which is caused by an oscillating magnetoresistance mixing with a microwave current in the conductive mag- netic layer29{31. Moreover, while the spin-mixing con- ductance is typically estimated from the enhancement in the Gilbert damping parameter , the quantication of  is not necessarily straightforward in epitaxial thin lms that exhibit pronounced anisotropic non-Gilbert damp- ing23,32{37. It has also been unclear how the Gilbert and non-Gilbert components of damping in LSMO are each modied by an adjacent SRO layer. These points above highlight the need for an alternative experimental ap- proach for characterizing spin transport and magnetiza- tion dynamics in LSMO/SRO. In this work, we quantify spin transport parameters and magnetization damping in epitaxial LSMO/SRO bi- layers by broadband FMR spectroscopy with out-of-plane andin-plane external magnetic elds. Out-of-plane FMR enables straightforward extraction of Gilbert damping as a function of SRO overlayer thickness, which is repro- duced by a simple \spin circuit" model based on diusive spin transport38,39. We nd that the spin-mixing conduc- tance at the LSMO/SRO interface is comparable to other ferromagnet/conductor interfaces and that spin current is absorbed within a short length scale of <1 nm in thearXiv:1610.06661v1 [cond-mat.mtrl-sci] 21 Oct 20162 42 44 46 48 50LSMO(10) /SRO(18)LSMO(10) log(intensity) (a.u.) 2 (deg.)LSAT(002) LSMO(002) SRO(002) Figure 1. 2 -!x-ray diraction scans of a single-layer LSMO(10 nm) lm and LSMO(10 nm)/SRO(18 nm) bilayer. conductive SRO layer. From in-plane FMR, we observe pronounced non-Gilbert damping that is anisotropic and scales nonlinearly with excitation frequency, which is ac- counted for by an existing model of two-magnon scat- tering40. This two-magnon scattering is also enhanced with the addition of the SRO overlayer possibly due to spin pumping. Our ndings reveal key features of spin dynamics and transport in the prototypical perovskite ferromagnet/conductor bilayer of LSMO/SRO and pro- vide a foundation for future all-oxide spintronic devices. II. SAMPLE AND EXPERIMENTAL DETAILS Epitaxial lms of LSMO(/SRO) were grown on as-received (001)-oriented single-crystal (LaAlO 3)0:3 (Sr2AlTaO 6)0:7(LSAT) substrates using pulsed laser de- position. LSAT exhibits a lower dielectric constant than the commonly used SrTiO 3substrate and is therefore better suited for high-frequency FMR measurements. The lattice parameter of LSAT (3.87 A) is also closely matched to the pseudocubic lattice parameter of LSMO (3.88 A). By using deposition parameters similar to those in previous studies from our group41,42, all lms were deposited at a substrate temperature of 750C with a target-to-substrate separation of 75 mm, laser uence of1 J/cm2, and repetition rate of 1 Hz. LSMO was de- posited in 320 mTorr O 2, followed by SRO in 100 mTorr O2. After deposition, the samples were held at 600C for 15 minutes in 150 Torr O 2and then the substrate heater was switched o to cool to room temperature. The deposition rates were calibrated by x-ray re ectivity mea- surements. The thickness of LSMO, tLSMO , in this study is xed at 10 nm, which is close to the minimum thickness at which the near-bulk saturation magnetization can be attained. X-ray diraction results indicate that both the LSMO lms and LSMO/SRO bilayers are highly crystalline and epitaxial with the LSAT(001) substrate, with high- resolution 2 -!scans showing distinct Laue fringes around the (002) Bragg re ection (Fig. 1). In this study, the maximum thickness of the LSMO and SRO layers 770 780 790 800 810 main mode sec. mode dIFMR/dH (a.u.) 0H (mT) data fit(a) (b) 770 780 790 800 810 data fit dIFMR/dH (a.u.) 0H (mT)Figure 2. Exemplary FMR spectra and tting curves: (a) one mode of Lorentzian derivative; (b) superposition of a main mode and a small secondary mode due to slight sample inho- mogeneity. combined is less than 30 nm and below the threshold thickness for the onset of structural relaxation by mist dislocation formation41,42. The typical surface roughness of LSMO and SRO measured by atomic force microscopy is<4A, comparable to the roughness of the LSAT sub- strate surface. SQUID magnetometry conrms that the Curie tem- perature of the LSMO layer is 350 K and the room- temperature saturation magnetization is Ms300 kA/m for 10-nm thick LSMO lms. The small LSMO thickness is desirable for maximizing the spin-pumping-induced en- hancement in damping, since spin pumping scales in- versely with the ferromagnetic layer thickness8,9. More- over, the thickness of 10 nm is within a factor of 2 of the characteristic exchange lengthp 2Aex=0M2s5 nm, assuming an exchange constant of Aex2 pJ/m in LSMO (Ref. 43), so standing spin-wave modes are not expected. Broadband FMR measurements were performed at room temperature. The lm sample was placed face- down on a coplanar waveguide with a center conductor width of 250 m. Each FMR spectrum was acquired at a constant excitation frequency while sweeping the exter- nal magnetic eld H. The eld derivative of the FMR absorption intensity (e.g., Fig. 2) was acquired using an rf diode combined with an ac (700 Hz) modulation eld. Each FMR spectrum was tted with the derivative of the sum of the symmetric and antisymmetric Lorentzians, as shown in Fig. 2, from which the resonance eld HFMR and half-width-at-half-maximum linewidth
Hwere ex- tracted. In some spectra (e.g., Fig. 2(b)), a small sec- ondary mode in addition to the main FMR mode was observed. We t such a spectrum to a superposition of two modes, each represented by a generalized Lorentzian derivative, and analyze only the HFMR and
Hof the larger-amplitude main FMR mode. The secondary mode is not a standing spin-wave mode because it appears above or below the resonance eld of the main mode HFMR with no systematic trend in eld spacing. We attribute the secondary mode to regions in the lm with3 0.360.400.440.48 0Meff (T) 0 5 10 15 201.952.002.05 tSRO (nm) gop 0 5 10 15 200.00.20.40.60.81.01.2 LSMO/SRO LSMO 0HFMR (T) f (GHz)(a) (b) (c) Figure 3. (a) Out-of-plane resonance eld HFMR versus excitation frequency ffor a single-layer LSMO(10 nm) lm and a LSMO(10 nm)/SRO(3 nm) bilayer. The solid lines indicate ts to the data using Eq. 1. (b,c) SRO-thickness dependence of the out-of-plane Land e g-factor (b) and ef- fective saturation magnetization Me(c). The dashed lines indicate the values aver- aged over all the data shown. slightly dierent Msor magnetic anisotropy. More pro- nounced inhomogeneity-induced secondary FMR modes have been observed in epitaxial magnetic lms in prior reports22,44. III. OUT-OF-PLANE FMR AND ESTIMATION OF SPIN TRANSPORT PARAMETERS Out-of-plane FMR allows for conceptually simpler ex- traction of the static and dynamic magnetic properties of a thin-lm sample. For tting the frequency depen- dence ofHFMR, the Land e g-factor gopand eective sat- uration magnetization Meare the only adjustable pa- rameters in the out-of-plane Kittel equation. The fre- quency dependence of
Hfor out-of-plane FMR arises solely from Gilbert damping, so that the conventional model of spin pumping8,9,38,39can be used to analyze the data without complications from non-Gilbert damping. This consideration is particularly important because the linewidths of our LSMO(/SRO) lms in in-plane FMR measurements are dominated by highly anisotropic non- Gilbert damping (as shown in Sec. IV). Furthermore, a simple one-dimensional, time-independent model of spin pumping outlined by Boone et al.38is applicable in the out-of-plane conguration, since the precessional orbit of the magnetization is circular to a good approximation. This is in contrast with the in-plane conguration with a highly elliptical orbit from a large shape anisotropy eld. By taking advantage of the simplicity in out-of- plane FMR, we nd that the Gilbert damping parame- ter in LSMO is approximately doubled with the addition of a suciently thick SRO overlayer due to spin pump- ing. Our results indicate that spin-current transmission at the LSMO/SRO interface is comparable to previously reported ferromagnet/conductor bilayers and that spin diusion length in SRO is <1 nm. We rst quantify the static magnetic properties of LSMO(/SRO) from the frequency dependence of HFMR. The Kittel equation for FMR in the out-of-plane cong- uration takes a simple linear form, f=gopB h0(HFMRMe); (1) where0is the permeability of free space, Bis the Bohrmagneton, and his the Planck constant. As shown in Fig. 3(a), we only t data points where 0HFMR is at least 0.2 T above 0Meto ensure that the lm is sat- urated out-of-plane. Figures 3(b) and (c) plot the ex- tractedMeandgop, respectively, each exhibiting no signicant dependence on SRO thickness tSROto within experimental uncertainty. The SRO overlayer therefore evidently does not modify the bulk magnetic proper- ties of LSMO, and signicant interdiusion across the SRO/LSMO interface can be ruled out. The averaged Meof 33010 kA/m (0Me= 0:420:01 T) is close toMsobtained from static magnetometery and implies negligible out-of-plane magnetic anisotropy; we thus as- sumeMs=Mein all subsequent analyses. The SRO- thickness independence of gop, averaging to 2 :010:01, implies that the SRO overlayer does not generate a signif- icant orbital contribution to magnetism in LSMO. More- over, the absence of detectable change in gopwith in- creasingtSRO may indicate that the imaginary compo- nent of the spin-mixing conductance8,9is negligible at the LSMO/SRO interface. The Gilbert damping parameter is extracted from the frequency dependence of
H(e.g., Figure 4(a)) by tting the data with the standard linear relation,
H=
H0+h gopBf: (2) The zero-frequency linewidth
H0is typically attributed to sample inhomogeneity. We observe sample-to-sample variation of 0
H0in the range14 mT with no systematic correlation with tSRO or the slope in Eq. 2. Moreover, similar to the analysis of HFMR, we only t data obtained at 0.2 T above 0Meto minimize spu- rious broadening of
Hat low elds. The linear slope of
Hplotted against frequency up to 20 GHz is therefore a reliable measure of decoupled from
H0in Eq. 2. Figure 4(a) shows an LSMO single-layer lm and an LSMO/SRO bilayer with similar
H0. The slope, which is proportional to , is approximately a factor of 2 greater for LSMO/SRO. Figure 4(b) summarizes the dependence ofon SRO-thickness, tSRO. For LSMO single-layer lms we nd = (0:90:2)103, which is on the same order as previous reports of LSMO thin lms21{23,26. This low damping is also comparable to the values re-4 0 5 10 15 200123 (10-3) tSRO (nm) 1/Gext 1/G↑↓m LSMO SRO 0 5 10 15 201.01.52.02.53.0 LSMO 0H (mT) f (GHz)LSMO/SRO(a) (b) (c) Figure 4. (a) Out-of-plane FMR linewidth
Hversus excitation frequency for LSMO(10 nm) and LSMO(10 nm)/SRO(3 nm). The solid lines indicate ts to the data using Eq. 2. (b) Gilbert damping parameter versus SRO thickness tSRO. The solid curve shows a t to the diusive spin pumping model (Eq. 5). (c) Schematic of out-of-plane spin pumping and the equivalent \spin circuit." ported in Heusler alloy thin lms45,46and may arise from the half-metal-like band structure of LSMO (Ref. 47). LSMO can thus be an ecient source of spin current generated resonantly by microwave excitation. With a few-nanometer thick overlayer of SRO, in- creases to2103(Fig. 4(b)). This enhanced damping with the addition of SRO overlayer may arise from (1) spin scattering48,49at the LSMO/SRO interface or (2) spin pumping8,9where nonequilibrium spins from LSMO are absorbed in the bulk of the SRO layer. Here, we assume that interfacial spin scattering is negligible, since <1 nm of SRO overlayer does not enhance signicantly (Fig. 4(b)). This is in contrast with the pronounced in- terfacial eect in ferromagnet/Pt bilayers48,49, in which even<1 nm of Pt can increase by as much as a fac- tor of2 (Refs. 50{52). In the following analysis and discussion, we show that spin pumping alone is sucient for explaining the enhanced damping in LSMO with an SRO overlayer. We now analyze the data in Fig. 4(b) using a one- dimensional model of spin pumping based on diusive spin transport38,39. The resonantly-excited magnetiza- tion precession in LSMO generates non-equilibrium spins polarized along ^ md ^m=dt, which is transverse to the magnetization unit vector ^ m. This non-equilibrium spin accumulation diuses out to the adjacent SRO layer and depolarizes exponentially on the characteristic length scales. The spin current density ~jsat the LSMO/SRO interface can be written as38,53 ~jsjinterface =~2 2e2^md^m dt 1 G"#+1 Gext; (3) where ~is the reduced Planck constant, G"#is the inter- facial spin-mixing conductance per unit area, and Gextis the spin conductance per unit area in the bulk of SRO. In Eq. 3, 1/ G"#and 1/Gextconstitute spin resistors in series such that the spin transport from LSMO to SRO can be regarded analogously as a \spin circuit," as il- lustrated in Fig. 4(c). In literature, these interfacialand bulk spin conductances are sometimes lumped to- gether as an \eective spin-mixing conductance" Ge "#= (1=G"#+ 1=Gext)1(Refs. 10{13, 16, 20, 23, 26, 44). We also note that the alternative form of the (eective) spin- mixing conductance g(e) e, with units of m2, is related to G(e) "#, with units of 1m2, byg(e) e= (h=e2)G(e) "# 26 k G(e) "#. The functional form of Gextis obtained by solving the spin diusion equation with appropriate boundary condi- tions38,39,53. In the case of a ferromagnet/nonmagnetic- metal bilayer, we obtain Gext=1 2SROstanhtSRO s ; (4) whereSROis the resistivity of SRO, tSROis the thick- ness of the SRO layer, and sis the diusion length of pumped spins in SRO. Finally, the out ow of spin cur- rent (Eq. 3) is equivalent to an enhancement of Gilbert damping9with respect to 0of LSMO with tSRO = 0 such that =0+gopB~ 2e2MstLSMO1 G"#+ 2SROscothtSRO s1 : (5) Thus, two essential parameters governing spin transport G"#andscan be estimated by tting the SRO-thickness dependence of (Fig. 4(b)) with Eq. 5. In carrying out the t, we x 0= 0:9103. We note thatSROincreases by an order of magnitude compared to the bulk value of 2106 m astSROis reduced to a few nm; also, at thicknesses of 3 monolayers ( 1.2 nm) or below, SRO is known to be insulating54. We there- fore use the tSRO-dependent SRO shown in Appendix A while assuming sis constant. An alternative tting model that assumes a constant SRO, which is a common approach in literature, is discussed in Appendix A. The curve in Fig. 4(b) is generated by Eq. 5 with G"#= 1:61014 1m2ands= 0:5 nm. Given the scatter of5 170175180185 170 175 180 185[010] [110] [100] 0HFMR (mT) LSMO LSMO/SRO 0 5 10 15 200100200300400500 0HFMR (mT) f (GHz)H||[100] H||[110](a) (b) (c) 0 5 10 15 201.952.002.05 tSRO (nm) gip -6-4-20 0H||,4 (mT) (d) 14 15330360 Figure 5. (a) Angular dependence of HFMR at 9 GHz for LSMO(10 nm) and LSMO(10 nm)/SRO(7 nm). The solid curves indicate ts to the data using Eq. 6. (b) Frequency dependence of HFMR for LSMO(10 nm)/SRO(7 nm) with eld applied in the lm plane along the [100] and [110] directions. Inset: close-up of HFMR versus frequency around 14-15 GHz. In (a) and (b), the solid curves show ts to the Kittel equation (Eq. 6). (c,d) SRO-thickness dependence of the in-plane cubic magnetocrystalline anisotropy eld (c) and in-plane Land e g-factor (d). The dashed lines indicate the values averaged over all the data shown. the experimental data, acceptable ts are obtained with G"#(1:22:5)1014 1m2ands0:30:9 nm. The estimated ranges of G"#andsalso depend strongly on the assumptions behind the tting model. For example, as shown in Appendix A, the constant- SRO model yields G"#>31014 1m2ands2:5 nm. Nevertheless, we nd that the estimated G"# is on the same order of magnitude as those of various ferromagnet/transition-metal heterostruc- tures39,55,56, signifying that the LSMO/SRO interface is reasonably transparent to spin current. More impor- tantly, the short simplies the presence of strong spin- orbit coupling that causes rapid spin scattering within SRO. This nding is consistent with a previous study on SRO at low temperature in the ferromagnetic state show- ing extremely fast spin relaxation with Gilbert damping 1 (Ref. 28). The short sindicates that SRO may be suitable as a spin sink or detector in all-oxide spintronic devices. IV. IN-PLANE FMR AND ANISOTROPIC TWO-MAGNON SCATTERING In epitaxial thin lms, the analysis of in-plane FMR is generally more complicated than that of out-of-plane FMR. High crystallinity of the lm gives rise to a non- negligible in-plane magnetocrystalline anisotropy eld, which manifests in an in-plane angular dependence of HFMR and introduces another adjustable parameter in the nonlinear Kittel equation for in-plane FMR. More- over,
Hin in-plane FMR of epitaxial thin lms often depends strongly on the magnetization orientation and exhibits nonlinear scaling with respect to frequency due to two-magnon scattering, a non-Gilbert mechanism for damping23,32{37. We indeed nd that damping of LSMO in the in-plane conguration is anisotropic and domi- nated by two-magnon scattering. We also observe ev-idence of enhanced two-magnon scattering with added SRO layers, which may be due to spin pumping from nonuniform magnetization precession. Figure 5(a) plots HFMR of a single-layer LSMO lm and an LSMO/SRO bilayer as a function of applied eld angle within the lm plane. For both samples, we observe clear four-fold symmetry, which is as expected based on the epitaxial growth of LSMO on the cubic LSAT(001) substrate. Similar to previous FMR studies of LSMO on SrTiO 3(001)57,58, the magnetic hard axes (corresponding to the axes of higher HFMR) are alongh100i. The in- plane Kittel equation for thin lms with in-plane cubic magnetic anisotropy is59, f=gipB h0 HFMR +Hjj;4cos(4)1 2  HFMR +Me+1 4Hjj;4(3 + cos(4))1 2 ; (6) wheregipis the Land e g-factor that is obtained from in- plane FMR data, Hjj;4is the eective cubic anisotropy eld, andis the in-plane eld angle with respect to the [100] direction. Given that LSMO is magnetically very soft (coercivity on the order of 0.1 mT) at room temperature, we assume that the magnetization is par- allel to the eld direction, particularly with 0H10 mT. In tting the angular dependence (e.g., Fig. 5(a)) and frequency dependence (e.g., Fig. 5(b)) of HFMR to Eq. 6, we x Meat the values obtained from out-of- plane FMR (Fig. 3(b)) so that Hjj;4andgipare the only tting parameters. For the two samples shown in Fig. 5(a), the ts to the angular dependence and fre- quency dependence data yield consistent values of Hjj;4 andgip. For the rest of the LSMO(/SRO) samples, we use the frequency dependence data with Hjj[100] and Hjj[110] to extract these parameters. Figures 5(c) and (d) show that Hjj;4andgip, respectively, exhibit no sys- tematic dependence on tSRO, similar to the ndings from6 out-of-plane FMR (Figs. 3(b),(c)). The in-plane cubic magnetocrystalline anisotropy in LSMO(/SRO) is rela- tively small, with 0Hjj;4averaging to2.5 mT.gipav- erages out to 1 :990:02, which is consistent with gop found from out-of-plane FMR. While the magnetocrytalline anisotropy in LSMO(/SRO) is found to be modest and indepen- dent oftSRO, we observe much more pronounced in-plane anisotropy and tSRO dependence in linewidth
H, as shown in Figs. 6(a) and (b). Figure 6(a) indicates that the in-plane dependence of
His four- fold symmetric for both LSMO(10 nm) and LSMO(10 nm)/SRO(7 nm).
His approximately a factor of 2 larger when the sample is magnetized along h100icom- pared to when it is magnetized along h110i. One might attribute this pronounced anisotropy to anisotropic Gilbert damping60, such that the sample magnetized along the hard axes h100imay lead to stronger damp- ing. However, we nd no general correlation between magnetocrystalline anisotropy and anisotropic
H: As we show in Appendix B, LSMO grown on NdGaO 3(110) with pronounced uniaxial magnetocrystalline anisotropy exhibits identical
Hwhen magnetized along the easy and hard axes. Moreover, whereas Gilbert damping should lead to a linear frequency dependence of
H, for LSMO(/SRO) the observed frequency dependence of
His clearly nonlinear as evidenced in Fig. 6(b). The pronounced anisotropy and nonlinear frequency dependence of
Htogether suggest the presence of a dierent damping mechanism. A well-known non-Gilbert damping mechanism in highly crystalline ultrathin magnetic lms is two-magnon scattering23,32{37,40,61,62, in which uniformly precessing magnetic moments (a spin wave, or magnon mode, with wavevector k= 0) dephase to a k6= 0 magnon mode with adjacent moments precessing with a nite phase dier- ence. By considering both exchange coupling (which re- sults in magnon energy proportional to k2) and dipolar coupling (magnon energy proportional to jkj) among precessing magnetic moments, the k= 0 andk6= 0 modes become degenerate in the magnon dispersion re- lation61as illustrated in Fig. 6(c). The transition from k= 0 tok6= 0 is activated by defects that break the translational symmetry of the magnetic system by localized dipolar elds40,61,62. In LSMO(/SRO), the activating defects may be faceted such that two-magnon scattering is more pronounced when the magnetization is oriented along h100i. One possibil- ity is that LSMO thin lms naturally form pits or islands faceted alongh100iduring growth. However, we are un- able to consistently observe signs of such faceted defects in LSMO(/SRO) samples with an atomic force micro- scope (AFM). It is possible that these crystalline defects are smaller than the lateral resolution of our AFM setup (<10 nm) or that these defects are not manifested in sur- face topography. Such defects may be point defects or nanoscale clusters of distinct phases that are known to exist intrinsically even in high-quality crystals of LSMO(Ref. 63). Although the denitive identication of defects that drive two-magnon scattering would require further in- vestigation, we can rule out (1) atomic step terraces and (2) mist dislocations as sources of anisotropic two- magnon scattering. (1) AFM shows that the orienta- tion and density of atomic step terraces dier randomly from sample to sample, whereas the anisotropy in
H is consistently cubic with larger
HforHjjh100ithan Hjjh110i. This is in agreement with the recent study by Lee et al. , which shows anisotropic two-magnon scat- tering in LSMO to be independent of regularly-spaced parallel step terraces on a buered-oxide etched SrTiO 3 substrate23. (2) Although Woltersdorf and Heinrich have found that mist dislocations in Fe/Pd grown on GaAs are responsible for two-magnon scattering33, such dis- locations are expected to be virtually nonexistent in fully strained LSMO(/SRO) lms on the closely-latticed matched LSAT substrates41,42. We assume that the in-plane four-fold anisotropy and nonlinear frequency dependence of
Hare entirely due to two-magnon scattering. For a sample magnetized along a given in-plane crystallographic axis hhk0i=h100i orh110i, the two-magnon scattering contribution to
H is given by40
Hhhk0i 2m = hhk0i 2m sin1sp f2+ (fM=2)2fM=2p f2+ (fM=2)2+fM=2;(7) wherefM= (gipB=h)0Msand hhk0i 2m is the two- magnon scattering parameter. The angular dependence of
His tted with33
H=
H0+h gipBf +
Hh100i 2m cos2(2) +
Hh110i 2m cos2(2[ 4]):(8) Similarly, the frequency dependence of
Hwith the sam- ple magnetized along [100] or [110], i.e., = 0 or=4, is well described by Eqs. 7 and 8. In principle, it should be possible to t the linewidth data with
H0,, and 2mas adjustable parameters. In practice, the t car- ried out this way is overspecied such that wide ranges of these parameters appear to t the data. We there- fore impose a constraint on by assuming that Gilbert damping for LSMO(/SRO) is isotropic: For each SRO thicknesstSRO,is xed to the value estimated from the t curve in Fig. 4(c) showing out-of-plane FMR data. (This assumption is likely justied, since the damping for LSMO(10 nm) on NdGdO 3(110) with strong uniaxial magnetic anisotropy is identical for the easy and hard directions, as shown in Appendix B.) To account for the uncertainty in the Gilbert damping in Fig. 4(c), we vary  by25% for tting the frequency dependence of in-plane
H. Examples of ts using Eqs. 7 and 8 are shown in Fig. 6(a),(b). Figure 6(d) shows that the SRO overlayer enhances the two-magnon scattering parameter 2mby up to a7 0 5 10 15 2004812 LSMOLSMO/SRO 0H (mT) f (GHz)(a) (b) (c) (d) FMR freq. k f k=0 k≠0 0 5 10 15 200102030 02m (mT) tSRO (nm)H||[100] H||[110] 0510 0 5 10 LSMO LSMO/SRO[110][010] [100] 0H (mT) Figure 6. (a) In-plane angular dependence of linewidth
H at 9 GHz for LSMO(10 nm) and LSMO(10 nm)/SRO(7 nm). The solid curves indicate ts to Eq. 8. (b) Frequency depen- dence of
H for LSMO(10 nm) and LSMO(10 nm)/SRO(7 nm) with Happlied along the [100] direction. The solid curves indicate ts to Eq. 7. The dashed and dotted curves indicate estimated two-magnon and Gilbert damping contributions, respectively. (c) Schematic of a spin wave dispersion curve (when the magne- tization is in-plane and has a nite component parallel to the spin wave wavevector k) and two- magnon scattering. (d) Two-magnon scattering coecient 2m, estimated for the cases with H applied along the [100] and [110] axes, plotted against SRO thickness tSRO. The dashed curve is the same as that in Fig. 4(c) scaled to serve as a guide for the eye for 2mwith H along [100]. factor of2 forHjj[100]. By contrast, for Hjj[110], al- though LSMO/SRO exhibits enhanced
Hcompared to LSMO, the enhancement in 2mis obscured by the un- certainty in Gilbert damping. In Table I, we summa- rize the Gilbert and two-magnon contributions to
H for LSMO single layers and LSMO/SRO (averaged val- ues for samples with tSRO>4 nm) with Hjj[100] and Hjj[110]. Comparing the eective spin relaxation rates, (gipB=h)0Msand (gipB=h)02m, reveals that two- magnon scattering dominates over Gilbert damping. We now speculate on the mechanisms behind the enhancement in 2min LSMO/SRO, particularly for Hjj[100]. One possibility is that SRO interfaced with LSMO directly increases the rate of two-magnon scat- tering, perhaps due to formation of additional defects at the surface of LSMO. If this were the case we might ex- pect a signicant increase and saturation of 2mat small tSRO. However, in reality, 2mincreases for tSRO>1 nm (Fig. 6(d)), which suggests spin scattering in the bulk of SRO. We thus speculate another mechanism, where k6= 0 magnons in LSMO are scattered by spin pump- Table I. Spin relaxation rates extracted from in-plane FMR (106s1) LSMO LSMO/SRO* Gilbert:gipB h0Ms 112 234 two-magnon:gipB h02m(Hjj[100]) 29050 550100 two-magnon:gipB h02m(Hjj[110]) 14060 25060 * Averaged over samples with tSRO>4 nm.ing into SRO. As shown by the guide-for-the-eye curve in Fig. 6(d), the tSROdependence of 2m(forHjj[100]) may be qualitatively similar to the tSROdependence of measured from out-of-plane FMR (Fig. 4(c)); this cor- respondence would imply that the same spin pumping mechanism, which is conventionally modeled to act on thek= 0 mode, is also operative in the degenerate k6= 0 magnon mode in epitaxial LSMO. Indeed, previous stud- ies have electrically detected the presence of spin pump- ing fromk6= 0 magnons by the inverse spin-Hall eect in Y3Fe5O12/Pt bilayers64{66. However, we cannot conclu- sively attribute the observed FMR linewidth broadening in LSMO/SRO to such k6= 0 spin pumping, since it is unclear whether faster relaxation of k6= 0 magnons should necessarily cause faster relaxation of the k= 0 FMR mode. Regardless of its origin, the pronounced anisotropic two-magnon scattering introduces additional complexity to the analysis of damping in LSMO/SRO and possibly in other similar ultrathin epitaxial magnetic heterostructures. V. SUMMARY We have demonstrated all-oxide perovskite bilayers of LSMO/SRO that form spin-source/spin-sink systems. From out-of-plane FMR, we deduce a low Gilbert damp- ing parameter of 1103for LSMO. The two-fold en- hancement in Gilbert damping with an SRO overlayer is adequately described by the standard model of spin pumping based on diusive spin transport. We ar- rive at an estimated spin-mixing conductance G"# (12)1014 1m2and spin diusion length s<1 nm, which indicate reasonable spin-current transparency at the LSMO/SRO interface and strong spin scattering8 within SRO. From in-plane FMR, we reveal pronounced non-Gilbert damping, attributed to two-magnon scatter- ing, which results in a nonlinear frequency dependence and anisotropy in linewidth. The magnitude of two- magnon scattering increases with the addition of an SRO overlayer, pointing to the presence of spin pumping from nonuniform spin wave modes. Our ndings lay the foun- dation for understanding spin transport and magneti- zation dynamics in epitaxial complex oxide heterostruc- tures. ACKNOWLEDGEMENTS We thank Di Yi, Sam Crossley, Adrian Schwartz, Han- kyu Lee, and Igor Barsukov for helpful discussions, and Tianxiang Nan and Nian Sun for the design of the copla- nar waveguide. This work was funded by the National Security Science and Engineering Faculty Fellowship of the Department of Defense under Contract No. N00014- 15-1-0045. APPENDIX A: SPIN PUMPING AND SRO RESISTIVITY When tting the dependence of the Gilbert damping parameteron spin-sink thickness, a constant bulk re- sistivity for the spin sink layer is often assumed in lit- erature. By setting the resistivity of SRO to the bulk valueSRO= 2106 m and tting the -versus-tSRO data (Fig. 4(c) and reproduced in Fig. 7(a)) to Eq. 5, we arrive at G"#>31014 1m2ands2:5 nm. The t curve is insensitive to larger values of G"#because the bulk spin resistance 1/ Gext, with the relatively large resistivity of SRO, dominates over the interfacial spin re- sistance 1/G"#(see Eqs. 4 and 5). As shown by the dot- ted curve in Fig. 7, this simple constant- SROmodel ap- pears to mostly capture the tSRO-dependence of . This model of course indicates nite spin pumping at even very small SRO thickness <1 nm, which is likely non- physical since SRO should be insulating in this thickness regime54. Indeed,sestimated with this model should probably be considered a phenomenological parameter: As pointed out by recent studies, strictly speaking, a physically meaningful estimation of sshould take into account the thickness dependence of the resistivity of the spin sink layer39,56,67, especially for SRO whose thickness dependence of resistivity is quite pronounced. Figure 7(b) plots the SRO-thickness dependence of the resistivity of SRO lms deposited on LSAT(001) mea- sured in the four-point van der Pauw geometry. The trend can be described empirically by SRO=b+s tSROtth; (9) whereb= 2106 m is the resistivity of SRO in the bulk limit,s= 1:41014 m2is the surface resistivity 0 5 10 15 200123 (10-3) tSRO (nm) 0 10 20 3010-61x10-51x10-4 SRO (m) tSRO (nm)(a) (b) Figure 7. (a) Gilbert damping parameter versus SRO thicknesstSRO. The solid curve is a t taking into account thetSROdependence of SRO resistivity, whereas the dotted curve is a t assuming a constant bulk-like SRO resistivity. (b) Resistivity of SrRuO 3lms on LSAT(001) as a function of thickness. 0 5 10 15 20012345 hard easy 0H (mT) f (GHz) 0 5 10 15 200123450H (mT) f (GHz)hard easy(b) (a) Figure 8. Frequency dependence of in-plane FMR linewidth
Hof LSMO(10 nm) on (a) LSAT(001) and (b) NdGaO 3(110), with the magnetization along the magnetic easy and hard axes. The solid curves are ts to Eq. 7 with the Gilbert damping parameter xed to 0:9103. coecient, and tth= 1 nm is the threshold thickness below which the SRO layer is essentially insulating. The value oftthagrees with literature reporting that SRO is insulating at thickness of 3 monolayers ( 1.2 nm) or below54. Given the large deviation of SROfrom the bulk value, especially at small tSRO, the trend in Fig. 7(b) suggests that taking into account the tSRO dependence ofSROis a sensible approach. APPENDIX B: IN-PLANE DAMPING OF LSMO ON DIFFERENT SUBSTRATES In Fig. 8, we compare the frequency dependence of
H for 10-nm thick LSMO lms deposited on dierent sub- strates: LSAT(001) and NdGaO 3(110). (NdGaO 3is an orthorhombic crystal and has ap 2-pseudocubic param- eter of3.86 A, such that (001)-oriented LSMO grows on the (110)-oriented surface of NdGaO 3.) As shown in Sec. IV, LSMO on LSAT(001) exhibits cubic mag- netic anisotropy within the lm plane with the h110iand h100ias the easy and hard axes, respectively. LSMO on NdGaO 3(110) exhibits uniaxial magnetic anisotropy within the lm plane with [1 10] and [001] of NdGaO 3as9 the easy and hard axes, respectively.68Whereas LSMO on LSAT(001) shows distinct magnitudes of damping when the lm is magnetized along the easy and hard axes (Figs. 8(a) and 6(a)), in LSMO on NdGaO 3(110)damping is identical for both the easy and hard axes (Figs. 8(b)). These results demonstrate that higher damping (wider linewidth) is in general not linked to the magnetic hard axis of LSMO. satorue@stanford.edu 1R. L. Stamps, S. Breitkreutz, J. Akerman, A. V. 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2016-10-21

Thin films of perovskite oxides offer the possibility of combining emerging concepts of strongly correlated electron phenomena and spin current in magnetic devices. However, spin transport and magnetization dynamics in these complex oxide materials are not well understood. Here, we experimentally quantify spin transport parameters and magnetization damping in epitaxial perovskite ferromagnet/paramagnet bilayers of La$_{2/3}$Sr$_{1/3}$MnO$_3$/SrRuO$_3$ (LSMO/SRO) by broadband ferromagnetic resonance spectroscopy. From the SRO thickness dependence of Gilbert damping, we estimate a short spin diffusion length of $\lesssim$1 nm in SRO and an interfacial spin-mixing conductance comparable to other ferromagnet/paramagnetic-metal bilayers. Moreover, we find that anisotropic non-Gilbert damping due to two-magnon scattering also increases with the addition of SRO. Our results demonstrate LSMO/SRO as a spin-source/spin-sink system that may be a foundation for examining spin-current transport in various perovskite heterostructures.

Spin transport and dynamics in all-oxide perovskite La$_{2/3}$Sr$_{1/3}$MnO$_3$/SrRuO$_3$ bilayers probed by ferromagnetic resonance

1610.06661v1

Brownian thermophoresis of an antiferromagnetic soliton Se Kwon Kim,1Oleg Tchernyshyov,2and Yaroslav Tserkovnyak1 1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 2Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218, USA (Dated: September 25, 2021) We study dynamics of an antiferromagnetic soliton under a temperature gradient. To this end, we start by phenomenologically constructing the stochastic Landau-Lifshitz-Gilbert equation for an antiferromagnet with the aid of the uctuation-dissipation theorem. We then derive the Langevin equation for the soliton's center of mass by the collective coordinate approach. An antiferromagentic soliton behaves as a classical massive particle immersed in a viscous medium. By considering a thermodynamic ensemble of solitons, we obtain the Fokker-Planck equation, from which we extract the average drift velocity of a soliton. The diusion coecient is inversely proportional to a small damping constant , which can yield a drift velocity of tens of m/s under a temperature gradient of 1 K/mm for a domain wall in an easy-axis antiferromagnetic wire with 104. PACS numbers: 75.78.-n, 66.30.Lw, 75.10.Hk Introduction. |Ordered magnetic materials exhibit solitons and defects that are stable for topological rea- sons [1]. Well-known examples are a domain wall (DW) in an easy-axis magnet or a vortex in a thin lm. Their dynamics have been extensively studied because of fun- damental interest as well as practical considerations such as the racetrack memory [2]. A ferromagnetic (FM) soli- ton can be driven by various means, e.g., an external magnetic eld [3] or a spin-polarized electric current [4]. Recently, the motion of an FM soliton under a temper- ature gradient has attracted a lot of attention owing to its applicability in an FM insulator [5{8]. A temperature gradient of 20 K/mm has been demonstrated to drive a DW at a velocity of 200 m/s in an yttrium iron garnet lm [9]. An antiferromagnet (AFM) is of a great current inter- est in the eld of spintronics [10{12] due to a few advan- tages over an FM. First, the characteristic frequency of an AFM is several orders higher than that of a typical FM, e.g., a timescale of optical magnetization switching is an order of ps for AFM NiO [13] and ns for FM CrO 2 [14], which can be exploited to develop faster spintronic devices. Second, absence of net magnetization renders the interaction between AFM particles weak, and, thus, leads us to prospect for high-density AFM-based devices. Dynamics of an AFM soliton can be induced by an elec- tric current or a spin wave [15{17]. A particle immersed in a viscous medium exhibits a Brownian motion due to a random force that is required to exist to comply with the uctuation-dissipation theo- rem (FDT) [18, 19]. An externally applied temperature gradient can also be a driving force, engendering a phe- nomenon known as thermophoresis [20]. Dynamics of an FM and an AFM includes spin damping, and, thus, in- volves thermal uctuations at a nite temperature [21]. The corresponding thermal stochastic eld in uences dy- namics of a magnetic soliton [8, 22, 23], e.g., by assisting a current-induced motion of an FM DW [24]. FIG. 1. (Color online) A thermal stochastic force caused by a temperature gradient pushes an antiferromagnetic domain wall to a colder region. The diusion coecient of the domain wall is inversely proportional to a small damping constant, which may give rise to a sizable drift velocity. In this Rapid Communication, we study the Brown- ian motion of a soliton in an AFM under a tempera- ture gradient. We derive the stochastic Landau-Lifshitz- Gilbert (LLG) equation for an AFM with the aid of the FDT, which relates the uctuation of the staggered and net magnetization to spin damping. We then derive the Langevin equation for the soliton's center of mass by em- ploying the collective coordinate approach [16, 25]. We develop the Hamiltonian mechanics for collective coordi- nates and conjugate momenta of a soliton, which sheds light on stochastic dynamics of an AFM soliton; it can be considered as a classical massive particle moving in a vis- cous medium. By considering a thermodynamic ensemble of solitons, we obtain the Fokker-Planck equation, from which we extract the average drift velocity. As a case study, we compute the drift velocity of a DW in a quasi one-dimensional easy-axis AFM. Thermophoresis of a Brownian particle is a multi- faceted phenomenon, which involves several competing mechanisms. As a result, a motion of a particle depends on properties of its environment such as a medium or a temperature T[26]. For example, particles in pro- tein (e.g., lysozyme) solutions move to a colder region forT > 294 K and otherwise to a hotter region [20, 27].arXiv:1503.07854v2 [cond-mat.mes-hall] 8 Jul 20152 Thermophoresis of an AFM soliton would be at least as complex as that of a Brownian particle. We focus on one aspect of it in this Rapid Communication; the eect of thermal stochastic force on dynamics of the soliton. We discuss two other possible mechanisms, the eects of a thermal magnon current and an entropic force [5], later in the Rapid Communication. Main results .|Before pursuing details of derivations, we rst outline our three main results. Let us consider a bipartite AFM with two sublattices that can be trans- formed into each other by a symmetry transformation of the crystal. Its low-energy dynamics can be devel- oped in terms of two elds: the unit staggered spin eldn(m1m2)=2 and the small net spin eld m(m1+m2)=2 perpendicular to n. Here, m1and m2are unit vectors along the directions of spin angular momentum in the sublattices. Starting from the standard Lagrangian description of the antiferromagnetic dynamics [28], we will show below that the appropriate theory of dissipative dynamics of antiferromagnets at a nite temperature is captured by the stochastic LLG equation s(_n+n_m) =n(h+hth); (1a) s(_m+m_m+n_n) =n(g+gth) +m(h+hth);(1b) in conjunction with the correlators of the thermal stochastic elds gthandhth, hgth i(r;t)gth j(r0;t0)i= 2kBTs ij(rr0)(tt0);(2a) hhth i(r;t)hth j(r0;t0)i= 2kBTs ij(rr0)(tt0);(2b) which are independent of each other [29]. This is our rst main result. Here, andare the damping constants associated with _nand _m,gU=nandhU=m are the eective elds conjugate to nandm,U[n;m] U[n]+R dVjmj2=2is the potential energy ( represents the magnetic susceptibility), and s~S=Vis the spin angular momentum density ( Vis the volume per spin) per each sublattice. The potential energy U[n(r;t)] is a general functional of n, which includes the exchange energyR dVA ij@in
@jnat a minimum [28]. Slow dynamics of stable magnetic solitons can often be expressed in terms of a few collective coordinates parametrizing slow modes of the system. The center of massRrepresents the proper slow modes of a rigid soli- ton when the translational symmetry is weakly broken. Translation of the stochastic LLG equation (1) into the language of the collective coordinates results in our sec- ond main result, a Langevin equation for the soliton's center of mass R: MR+ _R=@U=@R+Fth; (3) which adds the stochastic force Fthto Eq. (5) of Tveten et al. [17]. The mass and dissipation tensors are symmet- ric and proportional to each other: MijR dV(@in
@jn) and ijMij=;where=s is the relaxation time,s2is the inertia of the staggered spin eld n. The correlator of the stochastic eld Fthobeys the Einstein relation hFth i(t)Fth j(t0)i= 2kBTij(tt0): (4) A temperature gradient causes a Brownian motion of an AFM soliton toward a colder region. In the absence of a deterministic force, the average drift velocity is pro- portional to a temperature gradient V/kBrTin the linear response regime. The form of the proportional- ity constant can be obtained by a dimensional analy- sis. Let us suppose that the mass and dissipation ten- sors are isotropic. The Langevin equation (3) is, then, characterized by three scalar quantities: the mass M, the viscous coecient , and the temperature T, which dene the unique set of natural scales of time M=, lengthlpkBTM= , and energy kBT. Using these scales to match the dimension of a velocity yields V=c(kBrT);where1is the mobility of an AFM soliton and cis a numerical constant. The explicit solution of the Fokker-Planck equation, indeed, shows c= 1. This simple case illustrates our last main result; a drift velocity of an AFM soliton under a temperature gradient in the presence of a deterministic force Fis given by V=F(kBrT): (5) For a DW in an easy-axis one-dimensional AFM, the mobility is ==2s, whereis the width of the wall andis the cross-sectional area of the AFM. For a numerical estimate, let us take an angular momen- tum density s= 2~nm1, a width= 100 nm, and a damping constant = 104following the previous stud- ies [17, 30]. For these parameters, the AFM DW moves at a velocity V= 32 m/s for the temperature gradient of rT= 1 K/mm. Stochastic LLG equation. |Long-wave dynamics of an AFM on a bipartite lattice at zero temperature can de- scribed by the Lagrangian [28] L=sZ dVm
(n_n)U[n;m]: (6) We use the potential energy U[n;m]R dVjmj2=2+ U[n] throughout the Rapid Communication, which re- spects the sublattice exchange symmetry ( n!n;m! m). Minimization of the action subject to nonlinear con- straintsjnj= 1 and n
m= 0 yields the equations of mo- tion for the elds nandm. Damping terms that break the time reversal symmetry can be added to the equa- tions of motion to the lowest order, which are rst order in time derivative and zeroth order in spatial derivative. The resultant phenomenological LLG equations are given3 by s(_n+n_m) =nh; (7a) s(_m+m_m+n_n) =ng+mh (7b) [16, 30, 31]. The damping terms can be derived from the Rayleigh dissipation function R=Z dV(sj_nj2+sj_mj2)=2; (8) which is related to the energy dissipation rate by _U= 2R. The microscopic origin of damping terms does not concern us here but it could be, e.g., caused by thermal phonons that deform the exchange and anisotropy inter- action. At a nite temperature, thermal agitation causes uc- tuations of the spin elds nandm. These thermal uc- tuations can be considered to be caused by the stochas- tic elds gthandhthwith zero mean, which are con- jugate to nandm, respectively; their noise correlators are then related to the damping coecients by the FDT. The standard procedure to construct the noise sources yields the stochastic LLG equation (1). The correlators of the stochastic elds are obtained in the following way [18, 32]. Casting the linearized LLG equation (7) into the formfh;gg= ^ f_n;_mgprovides the kinetic coecients ^ . Symmetrizing the kinetic coecients ^ produces the correlators (2) of the stochastic elds consistent with the FDT. Langevin equation. |For slow dynamics of an AFM, the energy is mostly dissipated through the temporal variation of the staggered spin eld ndue toj_mj2' ()2jnj2j_nj2(from Eq. (7)), which allows us to set = 0 to study long-term dynamics of the magnetic soli- ton [17]. At this point, we switch to the Hamiltonian for- malism of an AFM [33], which sheds light on the stochas- tic dynamics of a soliton. The canonical momentum eld conjugate to the staggered spin eld nis L= _n=smn: (9) The stochastic LLG equations (1) can be interpreted as Hamilton's equations, _n=H= ==; _=H=nR= _n+gth;(10) with the Hamiltonian HZ dV
_nL=Z dVjj2 2+U[n]: (11) Long-time dynamics of magnetic texture can often be captured by focusing on a small subset of slow modes, which are parametrized by the collective coordinates q=fq1;q2;


g. A classical example is a DW in a one- dimensional easy-axis magnet described by the position of the wall Xand the azimuthal angle  [3, 33]. An- other example is a skyrmion in an easy-axis AFM lm,which is described by the position R= (X;Y ) [34, 35]. Translation from the eld language into that of collective coordinates can be done as follows. If the staggered spin eldnis encoded by coordinates qasn(r;t) =n[r;q(t)], time dependence of nre ects evolution of the coordi- nates: _n= _qi@n=@qi. With the canonical momenta p dened by pi@L @_qi=Z dV@n @qi
; (12) Hamilton's equations (10) translate into M_q=p;_p+ _q=F+Fth; (13) where F@U=@qis the deterministic force and Fth iR dV@ qin
gthis the stochastic force. Hamilton's equa- tions (13) can be derived from the Hamiltonian in the collective coordinates and conjugate momenta, HpTM1p=2 +U(q); (14) with the Poisson brackets fqi;pjg=ij;fqi;qjg= fpi;pjg= 0. An AFM soliton, thus, behaves as a classi- cal particle moving in a viscous medium. We focus on a translational motion of a rigid AFM soliton by choosing its center of mass as the collective coordinates q=R;n(r;t) =n(rR(t)). Eliminat- ing momenta from Hamilton's equations (13) yields the Langevin equation for the soliton's center of mass: R+_R=F+; (15) where Fthis the stochastic velocity. Here the mo- bility tensor of the soliton 1relates a deterministic force to a drift velocity h_Ri=Fat a constant temper- ature [36]. The mobility is inversely proportional to a damping constant, which can be a small number for an AFM, e.g., 104for NiO [37]. The correlator (2) of thermal stochastic elds is translated into the correlator of the stochastic velocity, hi(t)j(t0)i= 2kBTij(tt0)2Dij(tt0):(16) From Eq. (16), we see that diusion coecient and the mobility of the soliton respect the Einstein-Smoluchowski relation:D=kBT, which is expected on general grounds. It can also be explicitly veried as follows. A system of an ensemble of magnetic solitons at ther- mal equilibrium is described by the partition function ZR i[dpidxi=2~] exp(H=k BT), which provides the autocorrelation of the velocity, h_xi_xji=2 =M1 ijkBT=2 (the equipartition theorem). In the absence of an ex- ternal force, multiplying xi+ _xi=i(15) byxjand symmetrizing it with respect to indices iandjgive the equation,d2hxixji=dt2+dhxixji=dt= 2h_xi_xji, where the rst term can be neglected for long-term dynamics t. This equation in conjunction with the autocorre- lation of the velocity allows us to obtain the diusion co- ecientDijin Eq. (16),hxixji= 2kBTM1 ijt= 2Dijt,4 without prior knowledge about the correlator (2) of the stochastic elds. Average dynamics. |An AFM soliton exhibits Brown- ian motion at a nite temperature. The following Fokker- Planck equation for an ensemble of solitons in an inho- mogeneous medium describes the evolution of the density (R;t) at timet: @ @t+r
j= 0;withjFDrDT(kBrT);(17) whereDTis the thermophoretic mobility (also known as the thermal diusion coecient) [20, 38]. A steady-state current density j=F0DT(kBrT) with a constant soliton density (r;t) =0solves the Fokker- Planck equation (17), from which the average drift veloc- ity of a soliton can be extracted [39]: V=F(kBrT): (18) Let us take an example of a DW in a quasi one- dimensional easy-axis AFM with the energy U[n] =R dV(Aj@xnj2Kn2 z)=2. A DW in the equilibrium isn(0)= (sincos ;sinsin ;cos) with cos = tanh[(xX)=], wherep A=K is the width of the wall. The position Xand the azimuthal angle  parametrize zero-energy modes of the DW, which are en- gendered by the translational and spin-rotational symme- try of the system. Their dynamics are decoupled, X= 0, which allows us to study the dynamics of Xseparately from . The mobility of the DW is ==2s, where is the cross-sectional area of the AFM. The average drift velocity (18) is given by V=1 2kBrT s: (19) Discussion |The deterministic force Fon an AFM soliton can be extended to include the eect of an elec- tric current, an external eld, and a spin wave [15{17]. It depends on details of interaction between the soliton and the external degrees of freedom, whose thorough under- standing would be necessary for a quantitative theory for the deterministic drift velocity F. The Brownian drift velocity V(18) is, however, determined by local property of the soliton. We have focused on the thermal stochastic force as a trigger of thermophoresis of an AFM soliton in this Rapid Communication. There are two other possi- ble ingredients of thermophoresis of a magnetic soliton. One is a thermal magnon current, scattering with which could exert a force on a soliton [40]. The other is an entropic force, which originates from thermal softening of the order-parameter stiness [5]. Eects of these two mechanisms have not been studied for an AFM soliton; full understanding of its thermophoresis is an open prob- lem. In order to compare dierent mechanisms of thermally- driven magnetic soliton motion, let us address a closelyrelated problem of thermophoresis of a DW in a quasi one-dimensional FM wire with an easy- xz-plane easy- z-axis [3], which has attracted a considerable scrutiny recently. To that end, we have adapted the approach de- veloped in this Rapid Communication to the FM case, which leads to the conclusion that a DW drifts to a colder region by a Brownian stochastic force at the ve- locity given by the same expression for an AFM DW, VB=kBrT=2s [41]. A thermal magnon cur- rent pushes a DW to a hotter region at the velocity VM=kBrT=62sm, wheremp ~A=sT is the thermal-magnon wavelength [7]. According to Schlick- eiser et al. [5], an entropic force drives a DW to a hotter region at the velocity VE=kBrT=4sa, whereais the lattice constant. The Brownian stochastic force, there- fore, dominates the other forces for a thin wire, a= (supposing rigid motion) [42]. Within the framework of the LLG equations that are rst order in time derivative, the thermal noise is white as long as slow dynamics of a soliton is concerned, i.e., the highest characteristic frequency of the natural modes parametrized by the collective coordinates is much smaller than the temperature scale, ~!kBT. The thermal noise could be colored in general [26], e.g., for fast excitations of magnetic systems, which may be ex- amined in the future. In addition, local energy dissi- pation (8) allowed us to invoke the standard FDT at the equilibrium to derive the stochastic elds. It would be worth pursuing to understand dissipative dynamics of general magnetic systems, e.g., with nonlocal energy dissipation with the aid of generalized FDTs at the out- of-equilibrium [43]. We have studied dynamics of an AFM soliton in the Hamiltonian formalism. Hamiltonian's equations (13) for the collective coordinates and conjugate momenta can be derived from the Hamiltonian (14) with the conventional Poisson bracket structure. By replacing Poisson brackets with commutators, the coordinates and conjugate mo- menta can be promoted to quantum operators. This may provide a one route to study the eect of quantum uc- tuations on dynamics of an AFM soliton [44]. After the completion of this work, we became aware of two recent reports. One is on thermophoresis of an AFM skyrmion [35], whose numerical simulations sup- port our result on diusion coecient. The other is on thermophoresis of an FM DW by a thermodynamic magnon recoil [45]. We are grateful for useful comments on the manuscript to Joseph Barker as well as insightful discussions with Scott Bender, So Takei, Gen Tatara, Oleg Tretiakov, and Jiadong Zang. This work was supported by the US DOE- BES under Award No. DE-SC0012190 and in part by the ARO under Contract No. 911NF-14-1-0016 (S.K.K. and Y.T.) and by the US DOE-BES under Award No. DE- FG02-08ER46544 (O.T.).5 [1] A. Kosevich, B. Ivanov, and A. Kovalev, Phys. Rep. 194, 117 (1990). [2] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). [3] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). [4] L. Berger, Phys. Rev. B 54, 9353 (1996); J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [5] D. Hinzke and U. Nowak, Phys. Rev. Lett. 107, 027205 (2011); F. Schlickeiser, U. Ritzmann, D. Hinzke, and U. Nowak, Phys. Rev. Lett. 113, 097201 (2014). [6] G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012). [7] P. Yan, X. S. Wang, and X. R. Wang, Phys. Rev. Lett. 107, 177207 (2011); A. A. Kovalev and Y. Tserkovnyak, Europhys. Lett. 97, 67002 (2012). [8] L. Kong and J. Zang, Phys. Rev. Lett. 111, 067203 (2013); A. A. Kovalev, Phys. Rev. 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Qaiumzadeh, O. A. Tretiakov, and A. Brataas, Phys. Rev. Lett. 110, 127208 (2013). [17] E. G. Tveten, A. Qaiumzadeh, and A. Brataas, Phys. Rev. Lett. 112, 147204 (2014); S. K. Kim, Y. Tserkovnyak, and O. Tchernyshyov, Phys. Rev. B 90, 104406 (2014). [18] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Sta- tistical Physics, Part 1 , 3rd ed. (Pergamon Press, New York, 1980). [19] M. Ogata and Y. Wada, J. Phys. Soc. Jpn. 55, 1252 (1986). [20] R. Piazza and A. Parola, J. Phys.: Condens. Matter 20, 153102 (2008). [21] W. F. Brown, Phys. Rev. 130, 1677 (1963); R. Kubo and N. Hashitsume, Prog. Theor. Phys. Suppl. 46, 210 (1970); J. L. Garc a-Palacios and F. J. L azaro, Phys. Rev. B 58, 14937 (1998); J. Foros, A. Brataas, G. E. W. Bauer, and Y. Tserkovnyak, Phys. Rev. B 79, 214407 (2009); S. Homan, K. Sato, and Y. Tserkovnyak, Phys. Rev. B 88, 064408 (2013); U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko, Phys. Rev. B 86, 104414 (2012). [22] B. A. Ivanov, A. K. Kolezhuk, and E. V. Tartakovskaya, J. Phys.: Condens. Matter 5, 7737 (1993).[23] C. Sch utte, J. Iwasaki, A. Rosch, and N. Nagaosa, Phys. Rev. B 90, 174434 (2014). [24] R. A. Duine, A. S. N u~ nez, and A. H. MacDonald, Phys. Rev. Lett. 98, 056605 (2007). [25] O. A. Tretiakov, D. Clarke, G.-W. Chern, Y. B. Baza- liy, and O. Tchernyshyov, Phys. Rev. Lett. 100, 127204 (2008). [26] S. Hottovy, G. Volpe, and J. Wehr, Europhys. Lett. 99, 60002 (2012). [27] S. Iacopini and R. Piazza, Europhys. Lett. 63, 247 (2003). [28] A. F. Andreev and V. I. Marchenko, Sov. Phys. Usp. 23, 21 (1980). [29] Employing the quantum FDT would change the corre- lator of the stochastic elds to hhth i(r;!)hth j(r0;!0)i= [2ijs~!=tanh( ~!=2kBT)](rr0)(!!0) in the fre- quency space [18]. We focus on slow dynamics of an AFM soliton in the manuscript, ~!kBT, which allows us to replace ~!=tanh( ~!=2kBT) with 2kBT, yielding Eq. (2). [30] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak, Phys. Rev. B 90, 094408 (2014). [31] B. A. Ivanov and D. D. Sheka, Phys. Rev. Lett. 72, 404 (1994); N. Papanicolaou, Phys. Rev. B 51, 15062 (1995); H. V. Gomonay and V. M. Loktev, Phys. Rev. B 81, 144427 (2010); A. C. Swaving and R. A. Duine, Phys. Rev. B 83, 054428 (2011). [32] Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79, 014402 (2009). [33] H. J. Mikeska, J. Phys. C: Solid St. Phys. 13, 2913 (1980); F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 (1983). [34] I. Rai cevi c, D. Popovi c, C. Panagopoulos, L. Benfatto, M. B. Silva Neto, E. S. Choi, and T. Sasagawa, Phys. Rev. Lett. 106, 227206 (2011); X. Zhang, Y. Zhou, and M. Ezawa, arXiv:1504.01198. [35] J. Barker and O. A. Tretiakov, arXiv:1505.06156. [36] The relation between the mobility and dissipation tensor can also be understood by equating the (twice) Rayleigh function, 2R=VTV, to the dissipation governed by the mobility,_E=V
F=VT1V. [37] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and R. Huber, Nature Photon. 5, 31 (2011). [38] N. van Kampen, IBM J. Res. Dev. 32, 107 (1988); T. Kuroiwa and K. Miyazaki, J. Phys. A: Math. Theor. 47, 012001 (2014). [39] M. Braibanti, D. Vigolo, and R. Piazza, Phys. Rev. Lett. 100, 108303 (2008). [40] The eect is, however, negligible at a temperature lower than the magnon gap ( 40 K for NiO [37]). Also for our example|a DW in a 1D easy-axis AFM|the conserva- tive force and torque exerted by magnons vanish [17]. [41] Unlike 1D domain walls, Brownian motions are drasti- cally distinct between 2D FM and AFM solitons due to the gyrotropic force, which signicantly slows down fer- romagnetic diusion [23, 35]. [42] A DW in a wire with a large crosssection a2forms a 2D membrane. Its uctuations foment additional soft modes of the dynamics, which needs to be taken into account to understand the dynamics of such a DW [25]. [43] M. Baiesi, C. Maes, and B. Wynants, Phys. Rev. Lett. 103, 010602 (2009); U. Seifert and T. Speck, Europhys. Lett. 89, 10007 (2010). [44] S.-Z. Lin and L. N. Bulaevskii, Phys. Rev. B 88, 060404 (2013). [45] P. Yan, Y. Cao, and J. Sinova, arXiv:1504.00651.

2015-03-26

We study dynamics of an antiferromagnetic soliton under a temperature gradient. To this end, we start by phenomenologically constructing the stochastic Landau-Lifshitz-Gilbert equation for an antiferromagnet with the aid of the fluctuation-dissipation theorem. We then derive the Langevin equation for the soliton's center of mass by the collective coordinate approach. An antiferromagentic soliton behaves as a classical massive particle immersed in a viscous medium. By considering a thermodynamic ensemble of solitons, we obtain the Fokker-Planck equation, from which we extract the average drift velocity of a soliton. The diffusion coefficient is inversely proportional to a small damping constant $\alpha$, which can yield a drift velocity of tens of m/s under a temperature gradient of $1$ K/mm for a domain wall in an easy-axis antiferromagnetic wire with $\alpha \sim 10^{-4}$.

Thermophoresis of an Antiferromagnetic Soliton

1503.07854v2

Tunable damping, saturation magnetization, and exchange stiness of half-Heusler NiMnSb thin lms P. D urrenfeld,1F. Gerhard,2J. Chico,3R. K. Dumas,1, 4M. Ranjbar,1A. Bergman,3 L. Bergqvist,5, 6A. Delin,3, 5, 6C. Gould,2L. W. Molenkamp,2and J. Akerman1, 4, 5 1Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden 2Physikalisches Institut (EP3), Universit at W urzburg, 97074 W urzburg, Germany 3Department of Physics and Astronomy, Uppsala University, Box 520, 752 20 Uppsala, Sweden 4NanOsc AB, 164 40 Kista, Sweden 5Materials and Nano Physics, School of ICT, KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden 6Swedish e-Science Research Centre (SeRC), 100 44 Stockholm, Sweden The half-metallic half-Heusler alloy NiMnSb is a promising candidate for applications in spin- tronic devices due to its low magnetic damping and its rich anisotropies. Here we use ferromagnetic resonance (FMR) measurements and calculations from rst principles to investigate how the com- position of the epitaxially grown NiMnSb in uences the magnetodynamic properties of saturation magnetization MS, Gilbert damping , and exchange stiness A.MSandAare shown to have a maximum for stoichiometric composition, while the Gilbert damping is minimum. We nd excellent quantitative agreement between theory and experiment for MSand. The calculated Ashows the same trend as the experimental data, but has a larger magnitude. Additionally to the unique in- plane anisotropy of the material, these tunabilities of the magnetodynamic properties can be taken advantage of when employing NiMnSb lms in magnonic devices. I. INTRODUCTION Interest in the use of half-metallic Heusler and half- Heusler alloys in spintronic and magnonic devices is steadily increasing,1{3as these materials typically exhibit both a very high spin polarization4{8and very low spin- wave damping.9{12One such material is the epitaxially grown half-Heusler alloy NiMnSb,13,14which not only has one of the lowest known spin-wave damping values of any magnetic metal, but also exhibits an interesting and tunable combination of two-fold in-plane anisotropy15 and moderate out-of-plane anisotropy,10all potentially interesting properties for use in both nanocontact- based spin-torque oscillators16{22and spin Hall nano- oscillators23{27. To successfully employ NiMnSb in such devices, it is crucial to understand, control, and tailor both its magnetostatic and magnetodynamic properties, such as its Gilbert damping ( ), saturation magnetiza- tion (MS), and exchange stiness ( A). Here we investigate these properties in Ni 1-xMn1+xSb lms using ferromagnetic resonance (FMR) measure- ments and calculations from rst principles for compo- sitions of -0.1x0.4.MSandAare shown experi- mentally to have a maximum for stoichiometric compo- sition, while the Gilbert damping is minimum; this is in excellent quantitative agreement with calculations of and experiment on MSand. The calculated Ashows the same trend as the experimental data, but with an overall larger magnitude. We also demonstrate that the exchange stiness can be easily tuned over a wide range in NiMnSb through Mn doping, and that the ultra-low damping persists over a wide range of exchange sti- nesses. This unique behavior makes NiMnSb ideal for tailored spintronic and magnonic devices. Finally, by comparing the experimental results with rst-principlescalculations, we also conclude that the excess Mn mainly occupies Ni sites and that interstitial doping plays only a minor role. II. METHODS A. Thin Film Growth The NiMnSb lms were grown by molecular beam epitaxy onto InP(001) substrates after deposition of a 200 nm thick (In,Ga)As buer layer.15The lms were subsequently covered in situ by a 10 nm thick mag- netron sputtered metal cap to avoid oxidation and sur- face relaxation.28The Mn content was controlled dur- ing growth via the temperature, and hence the ux, of the Mn eusion cell. Six dierent samples (see table I) were grown with increasing Mn concentration, sample 1 having the lowest and sample 6 the highest concentra- tion of Mn. High-resolution x-ray diraction (HRXRD) measurements give information on the structural proper- ties of these samples, conrming the extremely high crys- talline quality of all samples with dierent Mn concentra- Samplevertical lattice constant ( A)thickness (nm)uniaxial easy axis2K1 MS(Oe) 1 5.94 38 [110] 170 2 5.97 38 [110] 8.4 3 5.99 40 [110] 0 4 6.02 45 [1 10] 9.0 5 6.06 45 [1 10] 14.2 6 6.09 38 [1 10] 25.5 Table I. Overview of NiMnSb lms investigated in this study.arXiv:1510.01894v1 [cond-mat.mtrl-sci] 7 Oct 20152 tion, even in the far from stoichiometric cases (samples 1 and 6).15The vertical lattice constant is found to increase with increasing Mn concentration and, assuming a linear increase,29we estimate the dierence in Mn concentra- tion across the whole set of samples to be about 40 at. %. We will thus represent the Mn concentration in the fol- lowing experimental results by the measured vertical lat- tice constant. Stoichiometric NiMnSb exhibits vertical lattice constants in the range of 5.96{6.00 A, leading to the expectation of stoichiometric NiMnSb in samples 2 and 3.15Finally, the layer thicknesses are also determined from the HRXRD measurements, giving an accuracy of 1 nm. B. Ferromagnetic Resonance Broadband eld-swept FMR spectroscopy was per- formed using a NanOsc Instruments PhaseFMR system with a coplanar waveguide for microwave eld excitation. Microwave elds hrfwith frequencies of up to 16 GHz were applied in the lm plane, perpendicularly oriented to an in-plane dc magnetic eld H. The derivative of the FMR absorption signal was measured using a lock- in technique, in which an additional low-frequency mod- ulation eld Hmod<1 Oe was applied using a pair of Helmholtz coils parallel to the dc magnetic eld. The eld directions are shown schematically in Fig. 1(a) and a typical spectrum measured at 13.6 GHz is given in the inset of Fig. 1(b). In addition to the zero wave vector uniform FMR mode seen at about H=2.1 kOe, an addi- tional weaker resonance is observed at a much lower eld of about 500 Oe, and is identied as the rst exchange- dominated perpendicular standing spin wave (PSSW) mode. The PSSW mode has a nonzero wave vector point- ing perpendicular to the thin lm plane and a thickness- dependent spin-wave amplitude and phase.30,31This can be eciently excited in the coplanar waveguide geome- try due to the nonuniform strength of the microwave eld across the lm thickness.32 The eld dependence of the absorption spectra (inset of Fig. 1(b)) can be t well (red line) by the sum of a sym- metric and an antisymmetric Lorentzian derivative:33,34 dP dH(H) =8C1
H(HH0) h
H2+ 4 (HH0)2i2 +2C2
H24(HH0)2
h
H2+ 4 (HH0)2i2; (1) whereH0is the resonance eld,
Hthe full width at half maximum (FWHM), and C1andC2tted param- eters representing the amplitude of the symmetric and antisymmetric Lorentzian derivatives, respectively. Both the FMR and the PSSW peaks can be tted indepen- dently, as they are well separated by the exchange eld 0Hex/(=d)2, wheredis the thickness of the layer. easy axis H, Hmod hard axis hrf p = 0 (FMR)p = 1 (PSSW)z Hexx y FMRPSSW(a) (b) 0 500 1000 1500 2000 2500 0 5 10 15 f(GHz) Field (Oe) 400 600 2000 2200 -0.5 0.0 0.5 1.0 1.5 dP/dH (a. u.) Field (Oe) f= 13.6 GHzFigure 1. (a) Schematic diagram of the FMR measurement showing eld directions. In our setup, the FMR mode and the rst PSSW mode are excited. (b) Frequency vs. resonance elds of the PSSW (red) and uniform FMR (black) mode for sample 2. The solid lines are ts to the Kittel equation, and both modes are oset horizontally by Hex. Inset: Resonance curves forf=13.6 GHz. The rst PSSW mode on the left and the FMR mode on the right were t with Eq. 1 For our chosen sample thicknesses, the dierences in res- onance elds are always much larger than the resonance linewidths. The eld dependence of both resonances is shown in Fig. 1(b) and can now be used to extract information about the magnetodynamic properties and anisotropies of the lms. The curves are ts to the Kittel equation, including internal elds from the anisotropy and the ex- change eld for the PSSW excitation:15,35 f= 0 2 H0+2KU MS2K1 MS+Hex  H0+2KU MS+K1 MS+Hex+Me1=2 ;(2) whereH0is the resonance eld, =2the gyromagnetic ratio, and0the permeability of free space. Meis the ef- fective magnetization, which has a value close to the satu- ration magnetization MS. 2KU=MSand 2K1=MSstands for the internal anisotropy elds coming from the uniaxial (KU) and biaxial ( K1) anisotropy energy densities in the half-Heusler material. The eective magnetic eld also includes an exchange eld 0Hex= (2A=M S)(p=d )2,3 which is related to the exchange stiness A, the lm thicknessd, and the integer order of the PSSW mode p, wherep= 0 denotes the uniform FMR excitation and p= 1 the rst PSSW mode. This mode numbering re- ects the boundary conditions with no surface pinning of the spins, which is expected for the in-plane measurement geometry.36 We stress that the expression for the anisotropy con- tribution in Eq. 2 is only valid for the case in which the magnetization direction is parallel to the uniaxial easy axis and also parallel to the applied eld. A full angular- dependent formulation of the FMR condition is described in Ref. 15. To fulll the condition of parallel alignment for all resonances, we perform the FMR measurements with the dc magnetic eld being applied along the domi- nant uniaxial easy axis of each lm, which changes from the [110] crystallographic direction to the [1 10]-direction with increasing Mn concentration (see Table I). The values of the biaxial anisotropy2K1 MShave been de- termined in a previous study by xed-frequency in-plane angular dependent FMR measurements,15and were thus taken as constant values in the tting process for Eq. 2; a simultaneous t of both contributions can yield arbi- trary combinations of anisotropy elds due to their great interdependence. The values for the uniaxial anisotropy 2KU MSobtained from the frequency-dependent tting are in very good agreement with the previously obtained val- ues in Ref. 15. The gyromagnetic ratio was measured to be =2= (28.590.20) GHz/T for all investigated samples, and was therefore xed for all samples to allow better comparison of the eective magnetization values. The Gilbert damping of the lms is obtained by tting the FMR linewidths
Hwith the linear depen- dence:37 0
H=0
H0+4 f; (3) where
H0is the inhomogeneous linewidth broaden- ing of the lm. The parallel alignment between mag- netization and external magnetic eld ensures that the linewidth is determined by the Gilbert damping process only.38 C. Calculations from First Principles The electronic and magnetic properties of the NiMnSb half-Heusler system were studied via rst-principles cal- culations. The material was assumed to be ordered in a face-centered tetragonal structure with an in-plane lat- tice parameter ak lat= 5.88 A, close to the lattice con- stant of the InP substrate, and an out-of-plane lattice constant of a? lat= 5.99 A, matching the value for the stoichiometric composition. Fixed values for the lattice parameters were chosen since an exact relation between the o-stoichiometric composition and the experimen- tally measured vertical lattice constants cannot be es- tablished. Moreover, calculations with a varying verticallattice parameter for a constant composition showed only a negligible eect on M S,A, and. The calculations were performed using the multiple scattering Korringa- Kohn-Rostocker (KKR) Green's function formalism as implemented in the SPRKKR package.39Relativistic ef- fects were fully taken into account by solving the Dirac equation for the electronic states, the shape of the poten- tial was considered via the Atomic Sphere Approximation (ASA), and the local spin density approximation (LSDA) was used for the exchange correlation potential. The co- herent potential approximation (CPA) was used for the chemical disorder of the system. The Gilbert damping of the material was calculated using linear response theory40, including the temperature eects from interatomic displacements and spin uctua- tions.41,42 The exchange interactions Jijbetween the atomic magnetic moments were calculated using the magnetic force theorem, as considered in the LKAG formalism.43,44 The interactions were calculated for up to 4.5 times the lattice constant in order to take into account any long- range interactions. Given the interatomic exchange in- teractions, the spin-wave stiness Dcan be calculated. Due to possible oscillations in the exchange interactions as a function of the distance, it becomes necessary to in- troduce a damping parameter, , to assure convergence of the summation. Dcan then be obtained by evaluating the limit!0 of D=2 3X ijJijp MiMjr2 ijexp rij alat ; (4) as described in [45]. Here, MiandMjare the local mag- netic moments at sites iandj,Jijis the exchange cou- pling between the magnetic moments at sites iandj, andrijis the distance between the atoms iandj. This formalism can be extended to a multisublattice system46. To calculate the eect of chemical disorder on the ex- change stiness of the system, the obtained exchange in- teractions were summed over a supercell with a random distribution of atoms in the chemically disordered sub- lattice. The eect that distinct chemical congurations can have over the calculation of the exchange stiness was treated by taking 200 dierent supercells. The re- sults were then averaged and the standard deviation was calculated. The cells were obtained using the atomistic spin dynamics package UppASD.47 Finally, with the spin-wave stiness determined as de- scribed above, the exchange stiness Acan be calculated from:48 A=DM S(T) 2gB: (5) Here,gis the Land e g-factor of the electron, Bthe Bohr magneton, and MS(T) the magnetization density of the system for a given temperature T, which for T= 0 K corresponds to the saturation magnetization. From the rst-principles calculations, the magnetic properties for ordered NiMnSb and chemically disordered4 0.60.70.80.95 .956 .006 .056 .100123(b) m0MS m0Meffm0M (T)(a)4 µB/u.f.KS (mJ/m2)v ertical lattice constant (Å) Figure 2. (a) MSandMeas functions of vertical lattice constant. The theoretical value of 4.0 B=u.f. is shown by the blue dashed line. (b) The calculated surface anisotropy density follows from the dierence between MSandMe. Ni1-xMn1+xSb were studied. To obtain the values of the exchange stiness AforT= 300 K, the exchange interac- tions from the ab initio calculations were used in conjunc- tion with the value of the magnetization at T= 300 K obtained from Monte Carlo simulations. III. RESULTS A. Magnetization The values of 0Meare plotted in Fig. 2(a) as red dots. The eective magnetization is considerably lower than the saturation magnetization 0MS, which was in- dependently assessed using SQUID measurements and al- ternating gradient magnetometry (AGM). The values for 0MScorrespond to a saturation magnetization between 3.5B=unit formula and 3.9 B=u.f., with the latter value being within the error bars of the theoretically ex- pected value of 4.0 B=u.f. for stoichiometric NiMnSb.49 A reduction of MSis expected in Mn-rich NiMnSb alloys, due to the antiferromagnetic coupling of the Mn Nidefects to the Mn lattice in the C1 bstructure of the half-Heusler material.29An even stronger reduction is observed for the Ni-rich sample 1, which is in accordance with the formation of Ni Mnantisites.50 While the measurement error for MSis comparatively large due to uncertainties in the volume determination, the error bars for Me, as obtained from ferromagnetic resonance, are negligible. NiMnSb lms have been shown to possess a small but substantial perpendicular mag- netic anisotropy, which can arise from either interfacial anisotropy or lattice strain.10,12To quantify the dier- ence observed between MSandMe, we assume a uniax- ial perpendicular anisotropy due to a surface anisotropy 2 4 6 8 10 5.95 6.00 6.05 6.10 0 2 4 (b) A (pJ/m) (a) α(10-3) vertical lattice constant (Å) 2 4 6 8 10 -0.1 0.0 0.1 0.2 0.3 0.4 0 2 4 (d) (c) xantisitesFigure 3. (a) and (b) show respectively the exchange stiness and Gilbert damping constant obtained from FMR measure- ments, plotted as a function of the vertical lattice constant. (c) and (d) show the corresponding values obtained from rst- principle calculations for T= 300 K. Negative values for x imply the introduction of Ni Mnantisites and positive values are related to Mn Niantisite defects. The error bars in (c) are the standard deviations from repeated rst-principles calcu- lations with 200 randomized supercells. energy density KS, which is known to follow the rela- tion:51 0Me=0MS2KS MSd: (6) TheKScalculated in this way has values between 0.5mJ=m2and 1.5mJ=m2, as shown in Fig. 2(b); these are comparable to the surface anisotropies obtained in other crystalline thin lm systems.52. Although the lm thicknesses in our set vary unsystematically, we can ob- serve systematic behavior of KSwith the vertical lat- tice constant, with an apparent minimum under the con- ditions where stoichiometric NiMnSb is expected|that is, for samples 2 and 3. The increasing values for o- stoichiometric NiMnSb can be thus attributed to the con- comitant increase in lattice defects, and thus of surface defects, in these lms. B. Exchange Stiness and Gilbert Damping The experimentally determined exchange stiness, as a function of the vertical lattice constant, and the Gilbert damping parameter are shown in Fig. 3(a) and (b), re- spectively. The minimum damping observed in our mea- surements is 1 :0103for sample 3, and so within sto- ichiometric composition. Sample 1, with a deciency of Mn atoms, showed nonlinear linewidth behavior at low frequencies, which vanished for out-of-plane measure- ments (not shown). This is typical with the presence of two-magnon scattering processes.52However, the damp- ing is considerably lower in all samples than in a permal- loy lm of comparable thickness. The exchange stiness and Gilbert damping ob- tained from the rst-principles calculations are shown in5 Fig. 3(c) and (d), respectively. For both parameters, the experimental trends are reproduced quantitatively, with Ahaving a maximum and a minimum value at stoi- chiometry. As the concentration of both Mn or Ni antisites in- creases, the exchange stiness decreases. This behavior can be explained by analyzing the terms in the expres- sion for the spin-wave stiness, Eq. 4. It turns out that the new exchange couplings Jij, which appear when an- tisites are present, play a major role, whereas changes in the atomic magnetic moments or the saturation magne- tization appear to be relatively unimportant. Mn anti- sites in the Ni sublattice (i.e., excess Mn) have a strong (2 mRy) antiferromagnetic coupling to the Mn atoms in the adjacent Mn layers. This results in a negative contri- bution toDcompared to the stoichiometric case, where this interaction is not present. On the other hand, Ni antisites in the Mn sublattice have a negative in-plane exchange coupling of 0.3 mRy to their nearest-neighbor Mn atoms, with a frustrated antiferromagnetic coupling to the Ni atoms in the adjacent Ni plane. The net eect is a decreasing spin-wave stiness as the composition moves away from stoichiometry. The calculated values of Aare around 30 % larger than the experimental results, which is the same degree of overestimation we recently observed in a study of doped permalloy lms53. It thus seems to be inherent in our calculations from rst principles. The calculated Gilbert damping also agrees well with the experimental values. The damping has its minimum value of 1.0103at stoichiometry and increases with a surplus of Ni faster than with the same surplus of Mn. Both Mn and Ni antisites will act as impurities and it is thus reasonable to attribute the observed increase in damping at o-stoichiometry to impurity scattering. While the damping at stoichiometry also agrees quanti- tatively, the increase in damping is underestimated in the calculations compared to the experimental values. Despite the fact that the calculations here focus purely on the formation of Mn Nior Ni Mnantisites, they are nonetheless capable of reproducing the experimental trends well. However, interstitials|that is, Mn or Ni sur- plus atoms in the vacant sublattice|may also be a possi- ble o-stoichiometric defect in our system.50We have cal- culated their eects and can therefore discuss about the existence of interstitials in our samples. A large fraction of Mn interstitials seems unlikely, as an increase in the saturation magnetization can be predicted through calcu- lations, contrary to the experimental trend; see Fig. 2(a).On the other hand, the existence of Ni interstitials may be compatible with the observed experimental trend, as they decrease the saturation magnetization|albeit at a slower rate than Ni antisites and slower than experimen- tally observed. Judging from the measured data, it is therefore likely that excess Ni exists in the samples as both antisites and interstitials. IV. CONCLUSIONS In summary, we have found that o-stoichiometry in the epitaxially grown half-Heusler alloy NiMnSb has a signicant impact on the material's magnetodynamic properties. In particular, the exchange stiness can be altered by a factor of about 2 while keeping the Gilbert damping very low ( 5 times lower than in permalloy lms). This is a unique combination of properties and opens up for the use of NiMnSb in, e.g., magnonic cir- cuits, where a small spin wave damping is desired. At the stoichiometric composition, the saturation magnetization and exchange stiness take on their maximum values, whereas the Gilbert damping parameter is at its mini- mum. These experimentally observed results are repro- duced by calculations from rst principles. Using these calculations, we can also explain the microscopic mecha- nisms behind the observed trends. We also conclude that interstitial Mn is unlikely to be present in the samples. The observed eects can be used to ne-tune the mag- netic properties of NiMnSb lms towards their specic requirements in spintronic devices. ACKNOWLEDGMENTS We acknowledge nancial support from the G oran Gustafsson Foundation, the Swedish Research Council (VR), Energimyndigheten (STEM), the Knut and Alice Wallenberg Foundation (KAW), the Carl Tryggers Foun- dation (CTS), and the Swedish Foundation for Strate- gic Research (SSF). F.G. acknowledges nancial support from the University of W urzburg's \Equal opportunities for women in research and teaching" program. This work was also supported initially by the European Commission FP7 Contract ICT-257159 \MACALO". A.B acknowl- edges eSSENCE. The computer simulations were per- formed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Su- percomputer Centre (NSC) and High Performance Com- puting Center North (HPC2N). 1R. Okura, Y. Sakuraba, T. Seki, K. Izumi, M. Mizuguchi, and K. 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2015-10-07

The half-metallic half-Heusler alloy NiMnSb is a promising candidate for applications in spintronic devices due to its low magnetic damping and its rich anisotropies. Here we use ferromagnetic resonance (FMR) measurements and calculations from first principles to investigate how the composition of the epitaxially grown NiMnSb influences the magnetodynamic properties of saturation magnetization $M_S$, Gilbert damping $\alpha$, and exchange stiffness $A$. $M_S$ and $A$ are shown to have a maximum for stoichiometric composition, while the Gilbert damping is minimum. We find excellent quantitative agreement between theory and experiment for $M_S$ and $\alpha$. The calculated $A$ shows the same trend as the experimental data, but has a larger magnitude. Additionally to the unique in-plane anisotropy of the material, these tunabilities of the magnetodynamic properties can be taken advantage of when employing NiMnSb films in magnonic devices.

Tunable damping, saturation magnetization, and exchange stiffness of half-Heusler NiMnSb thin films

1510.01894v1

arXiv:1602.06201v2 [cond-mat.mtrl-sci] 23 Jun 2016A systematic study of magnetodynamic properties at finite te mperatures in doped permalloy from first principles calculations Fan Pan,1,2,∗Jonathan Chico,3Johan Hellsvik,1Anna Delin,1,2,3Anders Bergman,3and Lars Bergqvist1,2 1Department of Materials and Nano Physics, School of Informa tion and Communication Technology, KTH Royal Institute of Technology, Electrum 229, SE-16440 K ista, Sweden 2Swedish e-Science Research Center (SeRC), KTH Royal Instit ute of Technology, SE-10044 Stockholm, Sweden 3Department of Physics and Astronomy, Materials Theory Divi sion, Uppsala University, Box 516, SE-75120 Uppsala, Sweden (Dated: June 24, 2018) By means of first principles calculations, we have systemati cally investigated how the magnetody- namicproperties Gilbert damping, magnetization andexcha ngestiffness areaffected whenpermalloy (Py) (Fe 0.19Ni0.81) is doped with 4d or 5d transition metal impurities. We find th at the trends in the Gilbert damping can be understood from relatively few ba sic parameters such as the density of states at the Fermi level, the spin-orbit coupling and the im purity concentration. The temperature dependence of the Gilbert damping is found to be very weak whi ch we relate to the lack of intraband transitions in alloys. Doping with 4 delements has no major impact on the studied Gilbert damping, apart from diluting the host. However, the 5 delements have a profound effect on the damping and allows it to be tuned over a large interval while maintaining the magnetization and exchange stiff- ness. As regards spin stiffness, doping with early transitio n metals results in considerable softening, whereas late transition metals have a minor impact. Our resu lt agree well with earlier calculations where available. In comparison to experiments, the compute d Gilbert damping appears slightly underestimated while the spin stiffness show good general ag reement. I. INTRODUCTION Spintronics and magnonic applications have attracted a large degree of attention due to the potential of cre- ating devices with reduced energy consumption and im- provedperformancecomparedtotraditionalsemiconduc- tor devices1–3. An important ingredient for understand- ing and improving the performance of these devices is a good knowledge of the magnetic properties. In this study, we focus on the saturation magnetization Ms, the exchange stiffness Aand the Gilbert damping α4. The latter is related to the energy dissipation rate of which a magnetic system returns to its equilibrium state from an excited state, e.g. after the system has been subjected to anexternalstimuliisuchasanelectricalcurrentwhichal- tersits magneticstate. The threeparameters, Ms,Aand αdescribe the magnetodynamical properties of the sys- tem of interest. Ultimately one would like to have com- plete independent control and tunability of these proper- ties. In this study, the magnetodynamical properties of Permalloy (Py) doped with transition metal impurities are systematically investigated within the same compu- tational framework. The capability of tuning the damping for a material with such a technological importance as Py is important for the development of possible new devices in spintron- ics and magnonics. The understanding of how transi- tion metals or rare earth dopants can affect the prop- erties of Py has been the focus of a number of recent experimental studies5–9. Typically in these studies, the ferromagnetic resonance10(FMR) technique is employed andαandMsare extracted from the linewidth of the uniform precession mode while Ais extracted from the first perpendicularstandingspin-wavemode11,12. On thetheory side, calculations of Gilbert damping from first- principles density functional theory methods have only recently become possible due to the complexity of such calculations. Two main approaches have emerged, the breathing Fermi surface model13,14and the torque corre- lation models15,16. Common to both approaches is that spin-orbit coupling along with the density of states at the Fermi level are the main driving forces behind the damping. The breathing Fermi surface model only takes only into account intraband transitions while torque cor- relation model also includes interband transitions. The torque correlation model in its original form contains a free parameter, namely the scattering relaxation time. Brataaset. al17later lifted this restriction by employing scattering theory and linear response theory. The result- ing formalism provides a firm foundation of calculating αquantitative from first principles methods and allows further investigations of the source of damping. Gilbert damping in pure Py as well as doping with selected el- ements have been calculated in the past9,18–20, however no systematic study of the magnetodynamic properties within the same computational framework has been con- ducted which the present paper aims to address. The paper is outlined as follows: In Section II we present the formalism and details of the calculations, in Section III we present the results of our study and in Section IV we summarize our findings and provide an outlook.2 II. THEORY A. Crystal structure of permalloy and treatment of disorder in the first-principles calculations Pure Permalloy (Py), an alloy consisting of iron (Fe) andnickel(Ni) withcompositionFe 0.19Ni0.81, crystallizes in the face centered cubic (fcc) crystal structure, where Fe and Ni atoms are randomly distributed. Additional doping with 4 dand 5dimpurities (M) substitutes Fe (or Ni) sothat it becomes a threecomponent alloywith com- position Py 1−xMx, wherexis the concentration of the dopant. All first principles calculations in this study were performed using the spin polarized relativistic (SPR) Korringa-Kohn-Rostoker (KKR)21Green’s function (GF) approach as implemented in the SPR-KKR software22. The generalized gradient approximation (GGA)23wasusedintheparametrizationoftheexchange correlation potential and both the core and valence elec- trons were solved using the fully relativistic Dirac equa- tion. The broken symmetry associated with the chemical substitution in the system was treated using the coherent potential approximation (CPA)24,25. B. Calculation of magnetodynamical properties of alloys: Gilbert damping within linear response theory and spin stiffness One of the merits with the KKR-CPA method is that it has a natural way of incorporating calculations of re- sponse properties using linear response formalism17,19,20. The formalism for calculating Gilbert damping in the present first principles method has been derived in Refs. [17] and [20], here we only give a brief outline of the most important points. The damping can be related as the dissipation rate of the magnetic energy which in turn can be associated to the Landau-Lifshitz-Gilbert (LLG) equation4, leading to the expression ˙E=Heff·dM dτ=1 γ2˙ˆm[˜G(m)˙ˆm], (1) whereˆm=M/Msdenotesthe normalizedmagnetization vector,Msthe saturation magnetization, γthe gyromag- netic ratio and ˜G(m) the Gilbert relaxation rate tensor. Perturbing a magnetic moment from its equilibrium state by a small deviation, ˆm(τ) =ˆm0+u(τ), gives an alternative expression of the dissipation rate by employ- ing linear response theory ˙Edis=π/planckover2pi1/summationdisplay ij/summationdisplay µν˙uµ˙uν∝angbracketleftψi|∂ˆH ∂uµ|ψj∝angbracketright∝angbracketleftψj|∂ˆH ∂uν|ψi∝angbracketright× δ(EF−Ei)δ(EF−Ej),(2)where the δ-functions restrict the summation over eigenstates to the Fermi level which can be rewrit- ten in terms of Green’s function as Im G+(EF) = −π/summationtext i|ψi∝angbracketright∝angbracketleftψi|δ(EF−Ei). By comparing Eqs. (1) and (2), the Gilbert damping parameter αis obtained, which is dimensionless and is related to the Gilbert relaxation tensorα=˜G/(γMs). This can be expressed as a trans- port Kubo-Greenwood-like equation26,27in terms of the retarded single-particle Green’s functions αµν=−/planckover2pi1γ πMsTrace/angbracketleftBig∂ˆH ∂uµImG+(EF)∂ˆH ∂uνImG+(EF)/angbracketrightBig c, (3) where∝angbracketleft...∝angbracketrightcdenotes a configurational average. For the cubic systems treated in this study, the tensorial form of the damping can with no loss of generality be replaced with a scalar damping parameter. Thermal effects from atomicdisplacementsandspinfluctuationswereincluded using the alloy-analogy model28within CPA. The spin-wave stiffness Dis defined as the curvature of the spin wave dispersion spectrum at small wave vec- tors (ω(q)≈Dq2).Din turn is directly related to the exchange interactions in the Heisenberg model which are obtained using the LKAG formalism29,30such that D=2 3/summationdisplay ijJijR2 ij√mimj, (4) whereJijis the interatomic exchangeparameterbetween thei-thandj-thmagneticmoment, Rijthe distancecon- necting the atomic sites with index iandjandmi(mj) the magneticmoment atsite i(j). It is worthnotingthat Eq. (4) only holds for cubic systems as treated here, for lower symmetries the relation needs modifications. The exchange couplings in metallic systems are typically long ranged and could have oscillations of ferromagnetic and antiferromagnetic character, such as present in RKKY type interactions. Due to the oscillations in exchange in- teractions, care is needed to reach numerical convergence of the series in Eq. (4) and it is achieved following the methodology as outlined in Refs. 31 and 32. C. Calculation of finite temperature magnetic properties Once the exchange interactions within the Heisen- berg model have been calculated, we obtained finite temperature properties from Metropolis33Monte Carlo simulations as implemented in the UppASD software package34,35. In particular, the temperature dependent magnetization was obtained, and enters the expression for micromagnetic exchange stiffness A, defined as36–39 A(T) =DM(T) 2gµB, (5)3 whereµBis the Bohr magneton, gis the Land´ e g-factor andM(T) the magnetization at temperature T. D. Details of the calculations ForeachconcentrationofthedifferentimpuritiesinPy, the lattice parameter was optimized by varying the vol- ume and finding the energy minimum. The k-point mesh for the self consistent calculations and exchange interac- tions was set to 223giving around 800 k-points in the ir- reducible wedge of the Brillouin zone (IBZ). The Gilbert damping calculation requires a very fine mesh to resolve allthe Fermisurfacefeaturesandthereforeasignificantly denser k-point mesh of 2283(∼1.0×106k-points in IBZ) was employed in these calculations to ensure nu- mericalconvergence. Moreover,vertexcorrections40were included in the damping calculationssince it has been re- vealedtobeimportantinpreviousstudies20forobtaining quantitative results. III. RESULTS A. Equilibrium volumes and induced magnetic moments 6.656.706.756.806.856.906.957.00 NbMoTcRuRhPdAgTaWReOsIrPtAuLattice constant (Bohr) 10%M 15%M Py FIG. 1. Calculated equilibrium volumes of Py-M, where M stands for a 4 d(left) or 5 d(right) transition metal. Values for 10% and 15% doping concentrations are shown. Reference value of pure Py is diplayed with a dashed line. Figure 1 shows the calculated equilibrium volume of doped Pyfor twodifferent concentrations(10%and 15%) ofimpuritiesfrom the 4 dand5dseriesofthe PeriodicTa- ble. Firstofall, itis notedthat thevolumeincreaseswith the concentration, and the volume within a series (4 dor 5d) has a parabolicshape with minimum in the middle of the series. This is expected since bonding states are con-secutivelyfilledand maximizedinthe middle ofthe series andthusthebondingstrengthreachesamaximum. Mov- ing further through the series, anti-bonding states start to fill, giving rise to weaker bonding and larger equilib- riumvolumes. Thisisconsistentwiththeatomicvolumes within the two series41. 0.500.600.700.800.901.001.10Total moment ( µΒ) M 5% 10% 15% 20% Py -0.4-0.20.00.20.40.6 NbMoTcRuRhPdAgTaWReOsIrPtAuLocal moment ( µΒ)M FIG.2. (Upper)Totalmagneticmoment(spinandorbital)for different impurities and concentrations. Reference value f or pure Py marked with a dashed line. (Lower) Local impurity magnetic moment for Py 0.95M0.05 The local moments of the host atoms are only weakly dependent on the type of impurity atom present. More- over, the magnetic moments are dominated by the spin momentµSwhile the orbital moments µLare much smaller. As an example, in pure Py without additional doping, the spin (orbital) moments of Fe is calculated to ≈2.64 (0.05) µBand for Ni ≈0.64 (0.05) µB, respec- tively. This adds up to an average spin (orbital) moment of≈1.04(0.05)µBby taking into account the concentra- tion of Fe and Ni in Py. The total moment is analyzed in more detail in Fig. 2 (upper panel). As mentioned above, one would like to achieve tunable and indepen- dent control of the saturation magnetization. Reducing the magnetization reduces the radiative extrinsic damp- ing but could at the same time affect the other properties in an unwanted manner. In many situations, one strive for keeping the value of the total moment (saturation magnetization) at least similar to pure Py, even for the doped systems. It is immediately clear from Fig. 2 that doping elements late in the series are the most preferable in that respect, for instance Rh and Pd in the 4 dseries and Ir, Pt and Au in the 5 dseries. In Fig. 2 (lower panel) we show the local impurity magnetic moments for 5% impurities in Py. In the be- ginning of the 4 d(5d) series, the impurity atoms have an antiferromagnetic coupling, reflected in the negative mo- ments compared to the host (Fe and Ni) atoms while lat-4 ter in the series couples ferromagnetically (positive mo- ments). The antiferromagnetic coupling may not be pre- ferred since it will tend to soften the magnetic properties and maybe even cause more complicated non-collinear magnetic configurations to occur. B. Band structure Since Py and doped-Py are random alloys, they lack translational symmetry and calculations using normal band structure methods are more challenging due to the need for large supercells. However, employing CPA re- stores the translational symmetry and more importantly, the band structure of disordered systems can be ana- lyzedthroughtheBlochspectralfunction (BSF) A(E,k), which can be seen as a wave vector k-dependent density of states (DOS) function. For ordered systems the BSF is aδ-like function at energy E( k) while for disordered systems each peak has an associated broadening with a linewidth proportional to the amount of disorder scatter- ing. In the upper panel of Fig. 3 the calculated BSF for pure Py is displayed. Despite being a disordered system, the electron bands are rather sharp below the Fermi level while in the vicinity ofthe Fermi level the bands becomes much more diffuse indicating that most of the disorder scattering takes place around these energies. If Py is doped with 20% Pt impurities, the positions of the electron bands do not change much as shown by the BSF in the lower panel of Fig. 3. The most strik- ing change is the large increase of the disorder scatter- ing compared to than Py causing diffuse electron bands throughout the Brillouin zone and energies. However, exactly at the Fermi level the differences between the doped and undoped system is not very pronounced and thesestatesarethe mostimportantforthe determination of the Gilbert damping, as seen from Eq. 3. C. Gilbert damping: effect of doping The calculated Gilbert damping of the doped Py sys- tems fordifferent concentrationsofimpurities isshownin Fig. 4 (upper panel). The 4 dimpurities only marginally influence the damping while the 5 dimpurities dramati- cally change the damping. The first observation is that weobtainverygoodagreementasinthe previousstudy19 for the 5dseries with 10% impurities, howevernot so sur- prising since we use same methodology. Secondly, the most dramatic effect on damping upon doping is for the case of Py doped with 20% Os impurities in which the damping increases with approximately 800% compared to pure Py, as previously reported in Ref. [20]. How- ever, in the present study we have systematically var- ied the impurity elements and concentrations and tried to identify trends over a large interval. Compared to experiments5, the calculated values of the Gilbert damp- ing are consistently underestimated. However it is worth FIG. 3. The Bloch spectral function A(E,k) of Py (upper panel) and Py doped with 20% Pt impurities (lower panel). The Fermi level is indicated with a horizontal black line at zero energy. remembering that calculations only shows the intrinsic part of the damping while experiments may still have some additional portion of extrinsic damping left such as Eddy current damping and radiation damping, since it is difficult to fully separate the different contributions. Moreover,incalculationsacompleterandomdistribution of atoms is assumed while there may be sample inhomo- geneities such as clustering in the real samples. From most theoretical models, the two main material properties that determine the damping are the density of states (DOS) at the Fermi level and the strength of the spin-orbit coupling. In the following, we first inves- tigate separately how these properties affect the damp- ing and later the combination of the two. In the lower panel of Fig. 4, the total DOS and the impurity-DOS are displayed for 10% impurity concentration of 4 dand 5d series transition metals. In the both 4 dand 5dseries the impurity-DOS exhibits a maximum in the middle of the series. However, the value of the DOS are similar for the 4dand 5dseries and therefore cannot solely explain the large difference in damping found between the two series.5 For the 4dseries, the calculated damping is not directly proportional to the DOS while there is a significant cor- relation of the DOS and damping in the 5 dseries. 0.000.010.020.030.04Gilbert damping αM 5% 10% 15% 20% Py Exp.5% 0.200.400.600.801.001.201.40 NbMoTcRuRhPdAgTaWReOsIrPtAun(EF) (sts./eV) M Py-M FIG. 4. (Upper) Calculated Gilbert damping parameter for Py+M in different concentrations of 4 dand 5dtransition metal M at low temperatures ( T= 10K). Experimental re- sults from Ref. [5] measured at room temperature are dis- played by solid squares and dashed line indicate reference value for pure Py. (Lower) Total (blue) and impurity (black) density of states at the Fermi level EFfor 10% impurities in Py. In order to analyze the separate influence of spin-orbit coupling on the damping, we show in upper panel of Fig. 5 the spin-orbit parameter ξ∝1 rdV(r) dr, where V(r) is the radial potential, of the impurity d-states. The calculations include all relativistic effects by solving the Dirac equation but here we have specifically extracted the main contribution from the spin-orbit coupling. As expected, the spin-orbit parameter increases with atomic numberZ, and is therefore considerably larger in the 5dseries compared to the 4 dseries. This is the most likely explanation why the damping is found to be larger in the 5dseries than the 4 dseries. However, within a single element in either the 4d or 5d series, the damp- ing is quadratically dependent on the relatve strength of the spin-orbit strength20. The calculated values of the spin-orbit parameter are in good agreement with previ- ous calculations42,43and reaches large values of 0.6-0.9 eV 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 across elements would only be proportional to the spin- orbit coupling, then the damping would monotonously increase with atomic number and since this is not what happens, we conclude that there is a delicate balance be- tween spin-orbit coupling and DOS that determines the damping which is further highlighted through a qualita- tive analysis of the involved scattering processes. 0.000.300.600.90ξ (eV)Spin-orbit coupling 0.000.010.02 NbMoTcRuRhPdAgTaWReOsIrPtAuα (Norm.)TC model Calculation FIG. 5. Upper: the spin-orbit parameter of d-electrons of the impurity atoms. Lower: qualitative comparison between calculations and torque correlation (TC) model for damping with 10% impurity concentration. In the torque correlation model, the dominant con- tribution to damping is through the scattering44,45and takes the following form α=1 γMs(γ 2)2n(EF)ξ2(g−2)2/τ, (6) whereτis the relaxation time between scattering events, andgthe Lande g-factor, for small orbital contributions, can be related as46g= 2(1 +µL µS). We assume that τ is the same for all impurities, which is clearly an ap- proximation but calculating τis beyond the scope of the present study. By normalizing the damping from Eq. (6) such that the value for Os (10% concentration) coincides with the first principles calculations, we obtain a quali- tatively comparison between the model and calculations, as illustrated in lower panel of Fig. 5. It confirms the6 trend in which 5 dseries lead to a larger damping than the4dseriesandcapturesqualitativelythemainfeatures. However, the peak value of the damping within the 5 d series in the TC model occurs for Ir while calculations give Os as in experiment. Another model developed for low dimensional magnetic systems such as adatoms and clusters suggests that the damping is proportional to the product of majority and minority density of states at the Fermi level47. It produces a parabolic trend but with maximum at incorrect position and fails to capture the increased damping of the 5 delements. To further analyse the role of impurity atoms on the damping we also performed calculations where instead of impurities we added vacancies in the system, i.e. void atoms. The results are shown in Fig. 6 where damping as a function of concentrationofAg (4 d), Os (5d) and va- cancies are compared to each other along with Os results from experimental5and previous calculations. Surpris- ingly, vacancies have more or less the same effect as Ag with the damping practically constant when increasing concentration. Since Ag has a zero moment, small spin- orbit coupling and small density of states at the Fermi level, the net effect of Ag from a damping (or scatter- ing) point of view is mainly diluting the host similar to adding vacancies. In contrast, in the Os case, being a 5dmetal, there is a strong dependence on the concentra- tion that was previously analyzed in terms of density of states and Os having a strong spin-orbit coupling. Our results from Os is slightly lower than the previous re- ported values19,20, despite using same software. How- ever, the most likely reason for the small discrepancy is the use of different exchange-correlationpotentials in the two cases. 0.000.010.020.030.040.050.060.070.08 0 5 10 15 20Gilbert damping α (%) of MPyOs PyAg PyVac 10%Os ref(theo.1) 15%Os ref(theo.2) PyOs ref(expt.) FIG. 6. Calculated Gilbert damping as a function of Os, Ag and vacancy (Vac) concentration in Py. Open red circle: calculation from Ref. [19], solid red circle: calculation f rom Ref. [20] and red solid square: experimental data from Ref. [ 5]D. Gilbert damping: effect of temperature Intheprevioussectionwestudiedhowthedampingde- pends on the electronic structure and spin-orbit coupling at low temperatures. However, with increasing tempera- ture additional scattering mechanisms contribute to the damping, most importantly phonon and magnon scatter- ing. The phonon scattering is indirectly taken into ac- count by including a number of independent atomic dis- placementsbringingtheatomsoutfromtheirequilibrium positions and magnon scattering is indirectly included by reducing the magnetic moment for a few configurations and then average over all atomic and magnetic configu- rations within CPA. It should be noted that the present methodology using the alloy-analogymodel28has limita- tions for pure systems at very low temperatures where the damping diverges, but we are far from that situation in this study since all systems have intrinsic chemical disorder. However, the limitations for pure systems can be lifted using a more advanced treatment using explicit calculation of the dynamical susceptibility48. 0.000.010.02α(T) 1.001.25 0 50 100 150 200 250 300 350 400α(T)pho+mag/α(T)pho Temperature (K) Py+20% Mo Py+20% Rh Py+20% W Py+20% Pt FIG. 7. Gilbert damping parameter including temperature effects from both atomic displacements and spin fluctuations (upper panel). The effect of spin fluctuations on the Gilbert damping (lower panel), see text. The temperature dependence of damping for a few se- lected systems is displayed in Fig. 7 where both atomic displacements and spin fluctuations are taken into ac- count. From the 4 d(5d) series, we choose to show results7 for Mo and Rh (W and Pt), where Mo (W) has a small antiferromagnetic moment and Rh (Pt) a sizeable ferro- magnetic moment, from Fig. 2. All systems display an overall weak temperature dependence on damping which only marginally increases with temperature, as shown in upper panel of Fig. 7. However, in order to sepa- rate the temperature contributions from atomic displace- ments and spin fluctuations, we show the ratio between the total damping and damping where only atomic dis- placements are taken into account in the lower panel of Fig. 7. The two systems with sizable moments (Rh and Pt), clearly have a dominant contribution from spin fluc- tuations when the moments are reduced upon increased scattering due to temperature. In contrast, the two sys- tems with (small) antiferromagnetic moments (Mo and W), the effect of the spin fluctuations on the damping is negligible and atomic displacements are solely responsi- ble. The weak temperature dependence found in these doped Py systems is somewhat surprising since in pure metals like Fe and Ni, a strong temperature dependence has been both measured and calculated20, however data for other random alloy systems is scarce. The temperature dependence of damping from the band structure is often attributed to interband and intraband transitions which arises from the torque- correlation model. Intraband transitions has conduc- tivity like dependence on temperature while interband shows resistivty-like dependence. The weak overall de- pendence found in the systems in Fig. 7 suggests lack of intraband transitions but a more detailed analysis of the band structure and thermal disorder are left for a future study. E. Spin-wave stiffness and exchange stiffness In the previous sections, we investigated saturation magnetization and damping and we are therefore left with the exchange stiffness. The calculated spin-wave stiffnessDatT= 0 K, from Eq. 4, is displayed in the up- perpanelofFig.8. Dcanbedirectlymeasuredfromneu- tron scattering experiments but as far as we are aware, no such data exist. For the late elements in the 4 dand 5dseries, the spin wave stiffness is maximized and have values rather similar to pure Py, however with a reduc- tion of approximately 20%. In micromagnetic modelling, it is common to use the exchange stiffness Ainstead of D.Ais proportional to D, from Eq. 5, and the sole temperature dependence of Atherefore comes from the magnetization. In the lower panel of Fig. 8, we show the calculated room temperature ( T= 300 K) values of A, together with values for pure Py and available experi- mental data. In the beginning of the 4 d(5d) series, the exchangestiffnessbecomes smalluponincreasingconcen- tration of impurities and the systems are magnetically very soft. It follows from the fact that magnetization is small because the systems are close to their ordering temperature. Contrary, for the late elements in the 4 d100200300400500Spin−wave stiffness D (meV Å2)5% M 10% M15% M Py 5 10 15 NbMoTcRuRhPdAgTaWReOsIrPtAuExchange stiffness A (pJ/M) Py(expt.) 15%(expt.) FIG. 8. Spin-wave stiffness Dof Py-M in the ground state (top) and exchange stiffness constant Aat room temperature T= 300K (bottom) as a function of doping concentration. The strict dashed lines show the reference value of pure Py from calculation and experiments. The scattered dots indi- cate the experimental data for Py+15%M (Ag/Pt/Au) from Ref.9 (5d)series,themagnetizationhasalargefinitevalueeven at room temperature and thereforethe exchangestiffness also has a large value, howeverreduced by approximately 15% compared to pure Py. IV. SUMMARY AND CONCLUSIONS Asystematic study ofthe intrinsicmagnetic properties of transition metal doped Py has been presented. It is found that the Gilbert damping is strongly dependent on the spin-orbit coupling of the impurity atoms and more weakly dependent on the density of states that deter- mines disorder scattering. The strong influence of the spin-orbit coupling makes the 5 delements much more effective to change the Gilbert damping and more sen- sible to the concentration. As a result, the damping can be increased by an order of magnitude compared to undoped Py. Overall, the damping features are quali- tatively rather well explained by the torque correlation model, yet it misses some quantitative predictive power thatonlyfirstprinciplesresultscanprovide. Moreover,it8 isfoundthatthedampingoverallhasaweaktemperature dependence, howeverit is slightly enhanced with temper- ature due to increased scattering caused by atomic dis- placements and spin fluctuations. Elements in the begin- ning of the 4 dor 5dseries are found to strongly influence the magnetization and exchange stiffness due to antifer- romagnetic coupling between impurity and host atoms. Incontrast,elementsinthe endofthe4 dor5dserieskeep the magnetization and exchange stiffness rather similar to undoped Py. More specifically, doping of the 5 dele- ments Os, Ir and Pt are found to be excellent candidates for influencing the magnetodynamical properties of Py. Recently, there have been an increasing attention for finding metallic materials with small intrinsic damping, for instance half metallic Heusler materials and FeCoalloys32,49. Controlling and varying the magnetodynam- ical properties in these systems through doping or by other means, like defects, are very relevant and left for a future study. ACKNOWLEDGMENTS The work was financed through VR (the Swedish Re- search Council) and GGS (G¨ oran Gustafssons Founda- tion). A.B. acknowledge support from eSSENCE. 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2016-02-19

By means of first principles calculations, we have systematically investigated how the magnetodynamic properties Gilbert damping, magnetization and exchange stiffness are affected when permalloy (Py) (Fe$_{0.19}$Ni$_{0.81}$) is doped with 4d or 5d transition metal impurities. We find that the trends in the Gilbert damping can be understood from relatively few basic parameters such as the density of states at the Fermi level, the spin-orbit coupling and the impurity concentration. % The temperature dependence of the Gilbert damping is found to be very weak which we relate to the lack of intraband transitions in alloys. % Doping with $4d$ elements has no major impact on the studied Gilbert damping, apart from diluting the host. However, the $5d$ elements have a profound effect on the damping and allows it to be tuned over a large interval while maintaining the magnetization and exchange stiffness. % As regards spin stiffness, doping with early transition metals results in considerable softening, whereas late transition metals have a minor impact. % Our result agree well with earlier calculations where available. In comparison to experiments, the computed Gilbert damping appears slightly underestimated while the spin stiffness show good general agreement.

A systematic study of magnetodynamic properties at finite temperatures in doped permalloy from first principles calculations

1602.06201v2

Damping's eect on the magnetodynamics of spin Hall nano-oscillators Yuli Yin,1, 2,Philipp D urrenfeld,2Mykola Dvornik,2Martina Ahlberg,2Afshin Houshang,2Ya Zhai,1and Johan Akerman2, 3 1Department of Physics, Southeast University, 211189 Nanjing, China 2Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden 3Department of Materials and Nano Physics, School of Information and Communication Technology, KTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden (Dated: July 27, 2021) We study the impact of spin wave damping ( ) on the auto-oscillation properties of nano- constriction based spin Hall nano-oscillators (SHNOs). The SHNOs are based on a 5 nm Pt layer interfaced to a 5 nm Py 100xyPtxAgymagnetic layer, where the Pt and Ag contents are co-varied to keep the saturation magnetization constant (within 10 %), while varies close to a factor of three. We systematically investigate the in uence of the Gilbert damping on the magnetodynamics of these SHNOs by means of electrical microwave measurements. Under the condition of a constant eld, the threshold current scales with the damping in the magnetic layer. The threshold current as a function of eld shows a parabolic-like behavior, which we attribute to the evolution of the spatial prole of the auto-oscillation mode. The signal linewidth is smaller for the high-damping materials in low magnetic elds, although the lowest observed linewidth was measured for the alloy with least damping. PACS numbers: 75.70.-i, 76.50.+g, 75.78.-n INTRODUCTION Spin Hall nano-oscillators (SHNO) are spintronic de- vices in which magnetization oscillations are induced by pure spin currents [1]. These pure spin currents can be experimentally realized via the spin Hall eect (SHE) in an adjacent heavy metal layer [2{4] or by non-local spin injection [5, 6]. SHNOs, which use the SHE in a heavy metal layer, have been fabricated in a vari- ety of device layouts, which all utilize the focusing of charge current into a region with a lateral size of tens to hundreds of nanometers. This focusing is commonly done via a nano-gap between two highly conductive elec- trodes [3, 7, 8], with a nanoconstriction [9{13], or with a nanowire [14, 15]. Most recently, nanoconstriction- SHNOs have attracted large interest, due to their rel- ative ease of fabrication, their direct optical access to the magnetization oscillation area, and their potential for large scale and large distance synchronization of multiple SHNOs [16, 17]. Nanoconstriction-SHNOs consist of a bilayer of a fer- romagnetic free layer and a SHE inducing heavy metal layer. Since the SHE and the concomitant spin accumu- lation at the bilayer interface are only in uenced by the current density in the heavy metal layer, magnetization oscillations of the device under a constant current can be directly linked to the magnetodynamic properties of the magnetic free layer. Until now, the variety of materi- als from which SHNOs has been fabricated is limited to a few standards like permalloy (Py, Ni 80Fe20), (Co,Fe)B, or yttrium iron garnet (YIG). However, these materials are dierent from each other in every one of the key magneto- dynamic parameters, such as magnetization ( M), Gilbertdamping (), or exchange constant ( A). In a recent study, we have shown how the magneto- dynamic properties of Py can be engineered by alloying with the noble metals Pt, Au, and Ag [18]. While alloy- ing with Pt leads to a large increase in damping but only a small decrease in magnetization, alloying with Ag has only a weak eect on the damping but reduces the mag- netization relatively strongly. Co-alloying with both ele- ments Pt and Ag thus results in Py 100xyPtxAgylms, whoseMandcan be tuned independently, e.g. the magnetization can be kept constant, while the damping is strongly increased with increasing Pt concentration. Here, we employ a series of alloyed Py 100xyPtxAgy thin lms in nanoconstriction-SHNOs, where we vary the eective damping of the free layer by a factor of three, while we keep the magnetization of the lms constant. Based on these lms, we fabricate geometrically identical nanoconstriction-SHNOs and compare their microwave auto-oscillation characteristics. This allows us to directly analyze the in uence of one single magnetodynamic prop- erty, namely the Gilbert damping, on the spectral char- acteristics, i.e. the onset current ( Ith DC), the output power (P), and the linewidth (
f). SPIN HALL NANO-OSCILLATOR DEVICES Bilayers of 5 nm Py 100xyPtxAgyand 5 nm Pt were sputter-deposited onto sapphire substrates in a high- vacuum chamber with a base pressure of less than 3108Torr. The deposition was carried out with 3 mTorr argon gas at a ow rate of 30 sccm. The alloyed layers were co-sputtered from up to 3 targets, and the Py target power was kept constant at 350 W, while thearXiv:1802.05548v1 [cond-mat.mes-hall] 15 Feb 20182 150 nm 200 nm (a) (c) (b) HIPCurrent FIG. 1. (a) Schematic representation of the sputtered bilayer structure. (b) SEM micrograph of a nanoconstriction-SHNO showing the relative orientations of current and eld. (c) Op- tical micrograph showing the microwave wave guide used for contacting the SHNOs. noble metal sputtering powers and the sputtering time was adjusted for composition and thickness, respectively. The top Pt layer was magnetron sputtered with a dc power of 50 W. The alloy compositions are Py 84Ag16 (S01), Py 77:5Pt10Ag12:5(S02), Py 75Pt15Ag10(S03), and Py73Pt19Ag8(S04), chosen to result in a constant satura- tion magnetization throughout the series of SHNOs [18]. Devices for electrical measurements were fabricated from these bilayers by electron beam lithography and ar- gon ion beam etching, using the negative resist as an etch- ing mask. Nanoconstrictions were formed by two sym- metrical indentations with a 50 nm tip radius into 4 µm wide stripes, see Fig. 1(b). The width of the nanocon- strictions is 150 nm. Finally, 1 µm thick copper waveg- uides with a 150 µm pitch were fabricated by optical lithography and lift-o, see Fig. 1(c). FILM CHARACTERIZATION Characterization of the extended bilayer samples was performed by ferromagnetic resonance (FMR), and two- point anisotropic magneotresistance (AMR) measure- ments. The FMR was carried out with in-plane applied elds using a NanOsc Instruments PhaseFMR with a 200µm wide coplanar waveguide (CPW). An asymmet- ric Lorentzian was t to the absorption peaks. The fre- quency dependence of the determined resonance elds and linewidths was subsequently used to extract the ef- fective magnetization ( 0Me) and the damping param- 0.0 0.2 0.4 0.6 0.8 μ0M eff α 0.02 0.03 0.04 0.05 0.06 α 0 5 10 15 20 0.3 0.4 0.5 AMR (%) xPt(%)φ (°) 0 90 180 270 360 0.0 0.1 0.2 0.3 0.4 MR (%) S01μ0Meff(T) μ0Meff = 0.617 TFIG. 2. (a) Magnetization and damping of the alloyed lms in the bilayer as measured by CPW-based FMR. (b) AMR of the extended layer structure. The inset shows the angular- dependent relative resistance of the Py 84Ag16/Pt (S01) bi- layer, together with a t to a cos2-function. eter (), respectively [18]. Figure 2(a) shows the two parameters, 0Meand, as a function of Pt concen- tration. The magnetization is constant throughout the sample series ( 0Me= 0.617(34) T), while the damp- ing increases linearly from 0 :023(1) to 0 :058(3) as the Pt concentration increases from 0 (Py 84Ag16) to 19 % (Py73Pt19Ag8). The small layer thickness compared to the lms in Ref. 18 results in a slightly lower magnetiza- tion, whereas the damping is enhanced as a consequence of spin pumping into the adjacent Pt layer [19{21]. The AMR was determined by probing the resistance of 4µm wide stripes in a rotating 90 mT in-plane mag- netic eld. A representative AMR measurement is pre- sented in the inset of Fig. 2(b), together with a t of a cos2-function to the data. The angle '= 0denotes a perpendicular orientation between current and eld, and the AMR (Fig. 2(b)) is calculated by the dierence in resistance at perpendicular and parallel alignments via AMR =RkR? R?. The AMR is below 1 %, which is a re- sult of the majority of the current owing through the nonmagnetic platinum layer, which has a higher conduc- tivity than the Py alloys. The AMR reduces by 30 % across the samples series, but the absolute resistance of the bilayers decreases by less than 5 %. The AMR magni- tude is therefore most likely governed by the alloy compo- sition, since the amount of current in the magnetic layer does not change signicantly.3 f(GHz) Current (mA) 2.2 2.4 2.6 2.8 3.0 3.2 3.45.96.06.1 2468S01, H = 500 mT 5.85 5.95 6.05 6.150246 f(GHz)PSD (nV2/Hz) FIG. 3. Power spectral density (PSD) of the Py 84Ag16/Pt (S01) SHNO as a function of current in an external eld of 0Hext= 0:5 T, tilted 80OOP. The inset shows the PSD at IDC= 3:26 mA and the solid line is a Lorentzian t resulting in
f= 5:98 MHz and P= 1:02 pW. MICROWAVE EMISSION MEASUREMENTS AND DISCUSSION The microwave measurements were conducted with the devices placed in a magnetic eld oriented at an out-of- plane (OOP) angle of 80from the lm plane, and an in-plane angle of '= 0. The in-plane component of the magnetic eld ( HIP ext) was thus perpendicular to the cur- rent ow direction, as sketched in Fig. 1(b). The relative orientation of the current and HIP extyields a spin-torque caused by the spin current from the Pt layer, which re- duces the damping in the Py layer and leads to auto- oscillations for suciently large positive applied dc cur- rents (IDC) [22]. The current was applied to the samples via the dc port of a bias-tee and the resulting microwave signals from the devices were extracted from the rf port of the bias-tee. The microwave signals were then amplied by a broadband (0 :1 to 40 GHz) low-noise amplier with a gain of +32 dB before being recorded by a spectrum an- alyzer (Rohde&Schwarz FSV-40) with a resolution band- width of 500 kHz. All measurements were carried out at room temperature. A typical microwave measurement of a Py 84Ag16/Pt (S01) device in a constant eld of 0Hext= 0:5 T and a varying current is displayed in Fig. 3. The peak fre- quency rst decreases slightly after the oscillation onset atIth DC= 2:26 mA, then reaches a minimum at 2:6 mA, and nally increases up to the maximum applied current of 3:4 mA. A Lorentzian peak function ts well to the auto-oscillation signal, see inset of Fig. 3, allowing for de- termination of the full-width at half-maximum linewidth (
f) and the integrated output power ( P). Besides the 10 100 Δf(MHz) 5.9 6.0 6.1 S01 S02 S03 S04 f(GHz) 2.2 2.4 2.6 2.8 3.0 3.2 3.4 0.1 1 Power (pW) I (mA)(a) (b) (c)FIG. 4. (a) Frequency, (b) linewidth, and (c) integrated power of the microwave auto-oscillations as a function of current for four dierent SHNOs with increasing damping. The applied eld is0Hext= 0:5 T, tilted 80OOP. highly coherent auto-oscillation mode, no other modes are excited under these eld conditions. Figure 4 shows the determined auto-oscillation char- acteristics of SHNOs with dierent alloy composition and damping. The measurements were again made in a constant eld of 0 :5 T. The oscillation frequencies in Fig. 4(a) lie around 6.0 0.1 GHz for all samples, and the current-frequency dependence is virtually identical above the individual threshold currents. However, the current range where fdecreases with IDCis missing for the S04 sample, which suggests that the threshold current is un- derestimated for this device. The comparable frequencies of all samples conrm that the saturation magnetization is constant throughout the alloy series. Furthermore, the quantitatively similar current tunability implies that the increased damping does not change the fundamental na- ture of the excited auto-oscillation mode. The linewidth of the SHNOs decreases rapidly after the auto-oscillation onset and then levels o for higher IDC values, as shown in Fig. 4(b). This behavior is consistent with previous studies on nanoconstriction-SHNOs made4 300 400 500 600 700 800 2.0 2.4 2.8 3.2 S02 S01 S03 S04 Ith DC(mA) Field (mT) FIG. 5. Threshold current ( Ith DC) as a function of external magnetic eld for the four devices of this study. of permalloy lms [11, 17]. The low-damping device S01 reaches its minimum level at
f11 MHz, while the SHNOs with higher damping materials all have a simi- lar minimum linewidth of
f5 MHz. The linewidth is inversely proportional to the mode volume [23], and the decrease in
fcan therefore be attributed to a spa- tial growth of the auto-oscillation region as the damping increases. Nevertheless, the active area of the device is conned to the nanoconstriction, which limits the reduc- tion in linewidth. The output power of the four nanoconstriction-SHNOs is shown as a function of IDCin Fig. 4(c). The power grows almost exponentially with increasing current for all samples. However, Pdrops dramatically as the Pt concentration increases. The AMR also decreases in the higher damping samples, but the reduction is too small to fully account for the drop in power. Together with the trend in linewidth, the evolution of the power contradicts the general assumption
f/=P [23{25]. This equa- tion is only valid in the vicinity of the threshold current and a direct comparison to the data is problematic, due to the experimental diculties of determining Ith DC. Still, the direct relation between the intrinsic oscillator power and the electrically measured power is put into question due to the remarkable decrease in the measured P. A number of factors could in uence the signal strength, e.g. rectication, spin-pumping, and the inverse spin-Hall ef- fect. The onset current for auto-oscillations was determined by current scans for external elds ranging between 0 :3 T and 0:8 T, and the results are shown in Fig. 5. The eld dependence of Ith DCis parabola-like for all samples. This kind of behavior has been predicted in a numerical study by Dvornik et al. [13]. The non-monotonic behav- ior of threshold current as a function of applied eld is a result of a re-localization of the auto-oscillation mode and a corresponding change in the spin-transfer-torque (STT) eciency. In weak oblique magnetic elds, the mode is of edge type and samples a signicant portion of the pure spin current, which is largest at the nanocon-striction edges due to the inhomogeneous current den- sity. When the eld strength increases, the mode shows an even stronger localization towards the region of the higher current density. Thereby, the STT eciency in- creases and the threshold current drops. When the eld strength increases further, the mode detaches from the edges and eventually transforms to the bulk type. As this transformation gradually reduces the spatial corre- lation between the spin current density and the location of the mode, the STT eciency drops and the threshold current increases. The lower eld tunability of Ith DCof the high damping samples imply an initially larger mode volume, which also was suggested by the evolution of the linewidth. The eld and current range with detectable auto- oscillations is strongly dependent on . The threshold current should increase linearly with damping [13] and the minimum Ith DCindeed scales with . The enhance- ment is smaller than predicted (a factor of three), which indicates that the increase in damping is accompanied with a higher STT eciency. A possible reason for the improved eciency is a larger SHE through a more trans- parent interface for alloyed lms. The origin of the ob- served damping dependence of the threshold eld is un- clear at this stage, calling for a closer inspection of the impact of the applied eld on the spectral characteristics. Thus, a further investigation of our devices is targeted towards the microwave emission as a function of eld with a constant IDC= 3.2 mA, i.e. above or at the pre- viously measured auto-oscillation threshold for all elds. While the peak frequencies are virtually identical for all the samples, see Fig. 6(a), the varied damping manifests in a clear pattern in Pand
f. The microwave power, shown in Fig. 6(c), rstly increases for all samples with increasing eld, peaks for an intermediate eld, and - nally drops relatively sharp until a point where no more oscillations are detectable. An opposite behavior can be seen for
f, which shows a minimum for intermediate elds. The eld at which the SHNOs emit their maxi- mum output power decreases monotonically from 0.64 T to 0.4 T with increasing damping. The same trend is visible for the point of minimum linewidth, which de- creases with increased damping from 0.71 T to 0.49 T, and is therefore at a typically 0.1 T larger eld than the respective maximum power. The lowest overall linewidth can be achieved for the lowest damping SHNO (S01) at high elds, where only this device still shows a detectable signal, i.e.,
f= 1:2 MHz at0Hext= 0:71 T. How- ever, at low applied elds 0Hext0:48 T a clear trend is noticeable towards smaller linewidths for the alloyed permalloy lms with larger damping. In light of this inverse trend, we can argue that auto- oscillations in nanoconstriction-SHNO should also be de- scribed in the framework of non-linear auto-oscillators, although the study in Ref. 13 has shown that oscilla- tions in nanoconstriction-SHNOs emerge from a local-5 Δf(MHz) (b) (c) 4 6 8 10 S01 S02S03 S04 1 10 100 300 400 500 600 700 800 0.1 1 Power (pW) Field (mT)(a)f(GHz) FIG. 6. (a) Frequency, (b) linewidth, and (c) integrated power of the auto-oscillations as a function of the applied external magnetic eld at a constant drive current IDC= 3:2 mA. ized linear mode. The generation linewidth of nanocon- tact spin torque oscillators, which are a prime example of non-linear auto-oscillators, has been studied analyti- cally [23, 26] and experimentally [27]. The linewidth as a function of current and magnetic eld angle was shown to follow the expression:
f=0 2kBT E0" 1 +N e2# ; (1) wherekB,T, andE0(IDC=Ith DC) are the Boltzmann con- stant, temperature and the average oscillator energy, re- spectively. Nis the nonlinear frequency shift, a material property that is determined by the internal magnetic eld and the magnetization [28]. eis the eective nonlinear damping rate and 0is the positive damping rate, and both have an explicit linear dependence on the Gilbert damping[23]. Assuming everything else equal amongst our devices, a decrease of the linewidth with can be thus expected, when the second term in the brackets in Eq. 1 dominates. This is likely for low to intermediateelds, since Ncan be calculated to take up the largest values under these conditions [28], which are thus in ac- cordance with the range of elds, where we observe the discussed linewidth vs. damping behavior in our devices. CONCLUSIONS In conclusion, we have fabricated a series of sam- ples where the magnetization is constant, while the spin wave damping is varied by a factor of three. We have shown that the damping of the magnetic layer in nanoconstriction-SHNOs has an important in uence on all the spectral characteristics of the devices. The re- sults of our study will encourage the application of tai- lored materials in SHNOs and can be used for a further understanding of the magnetodynamics in nanodevices, e.g. the coupling mechanisms in mutually synchronized SHNOs. ACKNOWLEDGMENTS We acknowledge nancial support from the China Scholarship Council (CSC), the G oran Gustafsson Foundation, the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation (KAW), and the Swedish Foundation for Strategic Research (SSF). 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2018-02-15

We study the impact of spin wave damping ($\alpha$) on the auto-oscillation properties of nano-constriction based spin Hall nano-oscillators (SHNOs). The SHNOs are based on a 5 nm Pt layer interfaced to a 5 nm Py$_{100-x-y}$Pt$_{x}$Ag$_{y}$ magnetic layer, where the Pt and Ag contents are co-varied to keep the saturation magnetization constant (within 10 %), while $\alpha$ varies close to a factor of three. We systematically investigate the influence of the Gilbert damping on the magnetodynamics of these SHNOs by means of electrical microwave measurements. Under the condition of a constant field, the threshold current scales with the damping in the magnetic layer. The threshold current as a function of field shows a parabolic-like behavior, which we attribute to the evolution of the spatial profile of the auto-oscillation mode. The signal linewidth is smaller for the high-damping materials in low magnetic fields, although the lowest observed linewidth was measured for the alloy with least damping.

Damping's effect on the magnetodynamics of spin Hall nano-oscillators

1802.05548v1

arXiv:1308.0450v2 [cond-mat.mes-hall] 5 Apr 2016Spin pumping damping and magnetic proximity effect in Pd and P t spin-sink layers M. Caminale,1,2,∗A. Ghosh,2,†S. Auffret,2U. Ebels,2K. Ollefs,3F. Wilhelm,4A. Rogalev,4and W.E. Bailey1,5,‡ 1Fondation Nanosciences, F-38000 Grenoble, France 2SPINTEC, Univ. Grenoble Alpes / CEA / CNRS, F-38000 Grenoble , France 3Fakult¨ at f¨ ur Physik and Center for Nanointegration (CENI DE), Universit¨ at Duisburg-Essen, 47057 Duisburg, Germany 4European Synchrotron Radiation Facility (ESRF), 38054 Gre noble Cedex, France 5Dept. of Applied Physics & Applied Mathematics, Columbia University, New York NY 10027, USA (Dated: April 6, 2016) We investigated the spin pumping damping contributed by par amagnetic layers (Pd, Pt) in both direct and indirect contact with ferromagnetic Ni 81Fe19films. We find a nearly linear dependence of the interface-related Gilbert damping enhancement ∆ αon the heavy-metal spin-sink layer thick- nesses t Nin direct-contact Ni 81Fe19/(Pd, Pt) junctions, whereas an exponential dependence is o b- servedwhenNi 81Fe19and(Pd, Pt)areseparated by3nmCu. Weattributethequasi-l inear thickness dependence to the presence of induced moments in Pt, Pd near t he interface with Ni 81Fe19, quan- tified using X-ray magnetic circular dichroism (XMCD) measu rements. Our results show that the scattering of pure spin current is configuration-dependent in these systems and cannot be described by a single characteristic length. I. INTRODUCTION As a novel means of conversion between charge- and spin-currents,spinHallphenomenahaverecentlyopened up new possibilities in magneto-electronics, with poten- tial applications in mesocale spin torques and electrical manipulation of domain walls1–9. However, several as- pects of the scattering mechanisms involved in spin cur- rent flow across thin films and interfaces are not entirely understood. Fundamental studies of spin current flow in ferromagnet/non-magnetic-meal (F/N) heterostructures in the form of continuous films have attempted to iso- late the contributions of interface roughness, microstruc- ture and impurities10–12. magnet/non-magnetic-meal (F/N) heterostructures in the form of continuous films have attempted to isolate the contributions of interface roughness, microstructure and impurities10–12. Proto- typical systems in this class of studies are Ni 81Fe19/Pt (Py/Pt)3,5–7,13–18andNi 81Fe19/Pd(Py/Pd)8,11,14,16,19,20 bilayers. Inthesesystems, PtandPdareemployedeither as efficient spin-sinks or spin/charge current transform- ers, in spin pumping and spin Hall experiments, respec- tively. Pd and Pt are metals with high paramagnetic susceptibility and when placed in contact with a ferro- magnetic layer (eg. Py, Ni, Co or Fe) a finite magnetic moment is induced at the interface by direct exchange coupling21–24. The role of the magnetic proximity effect on interface spin transport properties is still under debate. Zhang et al.25havereportedthat induced magnetic momentsin Pt and Pd films in direct contact with Py correlate strongly reduced spin Hall conductivities. This is ascribed to a spin splitting of the chemical potential and on the energy dependence of the intrinsic spin Hall effect. In standard spin pumping theory26, possible induced moments in N are supposed to be a priori included in calculations of the spin-mixing conductance g↑↓of a F|N interface27,28,which tends to be insensitive to their presence. Recent theoretical works, on the other hand, propose the need of a generalized spin pumping formalisms in- cluding spin flip and spin orbit interaction at the F |N interface, in order to justify discrepancies between exper- imental and calculated values of mixing conductance29,30. At present, it is still an open issue whether and how proximity-induced magnetic moments in F/N junctions arelinked to the varietyofthe spin-transportphenomena reported in literature10,17,31. Here, we present an experimental study of the pro- totypical systems: Py/Pd, Pt and Py/Cu/Pd, Pt het- erostructures. The objective of our study is to address the role of proximity induced magnetic moments in spin pumpingdamping. Tothisend, weemployedtwocomple- mentary experimental techniques. X-ray magnetic circu- lar dichroism (XMCD) is an element sensitive technique which allows us to quantify any static proximity-induced magnetic moments in Pt and Pd. Ferromagnetic reso- nance(FMR) measurementsprovideindirectinformation on the spin currents pumped out the Py layer by the pre- cessing magnetization, through the characterization of the Pd, Pt thickness dependence of the interface-related Gilbert damping α. In Fig. 3(Sec.IIIB), comparative measurements in Py/Cu/N and Py/N structures show a change of the N thickness dependence of ∆ α(tN) from an exponential to a linear-like behavior. A change in ∆α(tN) indicates a transformation in the spin scattering mechanism occurring at the interface, ascribed here to the presence of induced moments in directly exchange coupled F/N systems. Theoretical works predicted a deviation from a conventional N-thickness dependence when interface spin-flip scattering is considered in the pumping model29,30, howeverno functional form waspro- vided. For Py/N systems, we find that the experimental thickness dependence cannot be described by standard models16,26,32, but rather a linear function reproduces2 the data to a better degree of accuracy, by introducing a different characteristic length. We speculate that the spatial extent of spin current absorption in F/N systems shows an inverse proportionality to interfacial exchange coupling energy, obtained from XMCD, as proposed be- fore for spin polarized, decoupled interfaces in F 1/Cu/F 2 heterostructures14. II. EXPERIMENT The heterostructures were fabricated by DC mag- netron sputtering on ion-cleaned Si/SiO 2substrates in the form of substrate/seed/multilayer/cap stacks, where Ta(5nm)/Cu(5nm) bilayerwasemployedas seed. Ta/Cu is employed to promote <111>growth in Py and subse- quent fcc layers (Pd, Pt), and Ta is known to not affect the dampingstrongly17,32,33. Different stacksweregrown asmultilayer for each measurement. For FMR measurements, we have multilayer = Py(tF)/N(tN), Py(tF)/Cu(3nm)/N( tN) with N = Pd, Pt; an Al(3nm) film, oxidized in air, was used as cap. The smallest N layer thickness tNdeposited is 0.4nm, the maximum interdiffusion length observed for similar multilayers34. Samples with multilayer = Py(tN)/Cu(3nm) and no sink layer were also fabricated as reference for evaluation of the Gilbert damping en- hancement due tothe Pd orPtlayer. The tN-dependence measurementsofFMRweretakenforPythicknesses tF= 5and 10nm. Results from the tF= 10nm data set are shown in Appendix A. Measurements of the FMR were carried out at fixed frequency ωin the 4-24 Ghz range, by means of an in-house apparatus featuring an external magnetic field up to 0.9T parallel to a coplanar waveg- uide with a broad center conductor width of 350 µm. For XMCD measurements, given the low X-ray ab- sorption cross-sectionpresented by Pt and Pd absorption edges, a special set of samples was prepared, consisting of 20 repeats per structure in order to obtain sufficiently highsignal-to-noiseratio. Inthiscase,wehave multilayer = [Py(5nm)/N] 20, with N = Pd(2.5nm) and Pt(1nm); Cu(5nm)/Py(5nm)/Al(3nm) was deposited as cap. The Pt and Pd thicknesses were chosen to yield a damping enhancement equal about to half of the respective satu- ration value (as it will be shown later), i.e. a thicknesses for which the F/N interface is formed but the damp- ing enhancement is still increasing. XMCD experiments were carried out at the Circular Polarization Beamline ID-12 of the European Synchrotron Radiation Facility (ESRF)35. Measurementsweretakenintotalfluorescence yield detection mode, at grazing incidence of 10◦, with either left or right circular helicity of the photon beam, switching a 0.9T static magnetic field at each photon en- ergy value (further details on the method are in Ref.22). No correction for self-absorption effects is needed; how- ever XMCD spectra measured at the L 2,3edges of Pd have to be corrected for incomplete circular polarization rate 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 /s32/s80/s116/s32/s88/s65/s83 /s32/s65/s117/s32/s88/s65/s83 /s80/s100/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41 /s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121 /s32/s40/s107/s101/s86/s41/s80/s116 /s45/s48/s46/s48/s54/s45/s48/s46/s48/s51/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54 /s32 /s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41 /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 /s32/s80/s100/s32/s88/s65/s83 /s32/s65/s103/s32/s88/s65/s83/s88/s65/s83/s32/s40/s97/s46/s117/s46/s41 /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 /s88/s77/s67/s68/s32/s40/s97/s46/s117/s46/s41 FIG. 1. (Color online) X-ray absorption (XAS, left axis) and magnetic circular dichroism (XMCD, right axis) spectra at the L-edges of Pt (top panel) and Pd (bottom panel) for [Py(5nm)/Pt(1nm)] 20and [Py(5nm)/Pd(2.5nm)] 20multilay- ers. The dashed traces represent XAS spectra at L-edge of Ag and Au used as background of Pd and Pt, respectively, to extract the values of induced magnetic moment reported in Tab.I. L3and L2, respectively. The circular polarization rate is in excess of 95 % at the L 2,3edges of Pt. III. RESULTS AND ANALYSIS In order to study how the proximity-induced mag- netic moments may affect the absorption of spin-currents through interfaces, the static moment induced in Pt, Pd layers in direct contact with ferromagnetic Py in char- acterized first, by means of XMCD. The value of the induced moment extracted for the two Py/N systems is usedtoestimatetheinterfacialexchangeenergyactingon the two paramagnets. Afterwards, the dynamic response ofthe magnetization is addressedby FMR measurements in Py/N(direct contact) and Py/Cu/N(indirect contact) heterostructures. From FMR measurements carried out on both configurations as a function of N thickness, the damping enhancement due to the presence of the spin- sink layers Pt and Pd is obtained from the frequency- dependence of the FMR linewidth. The relation between the static induced moment and the spin pumping damp- ing is discussed by comparing the results of the direct with indirect contact systems.3 -0.06-0.04-0.020.00XMCD 11.5911.5811.5711.5611.55 Photon Energy (keV)1.0 0.5 0.0 3210tCu (nm) Area 1 nm Cu 0.5 nm Cu 0 nm Cu FIG. 2. (Color online) XMCD spectra at the L 3edge of Pt for [Py(5nm)/Cu(t Cu)/Pt(1nm)] 15, with t Cu= 0, 0.5 and 1nm. As inset the ares of the peak is plot as a function of Cu thickness. A. XMCD: Probing the induced magnetic moment In Fig.1we report X-ray absorption (XAS) and mag- netic circular dichroism (XMCD) spectra at the L2,3 edges of Pt (top panel) and Pd (bottom panel) taken on Py(5nm)/Pt(1nm) |20and Py(5nm)/Pd(2.5nm) |20, re- spectively. Rather intense XMCD signals have been de- tected at both Pt and Pd L 2,3edges, showing unambigu- ously that a strong magnetic moment is induced by di- rect exchange coupling at the Py |N interface. The static induced moment is expected to be ferromagnetically cou- pled with the magnetization in Py21. From the integrals of XMCD spectra, the induced magnetic moment on the Pt, Pd sitesis determined byapplying the sum rulesasin Ref.22(and references therein). In Py/Pd(2.5nm) |20, Pd atoms bear a moment of 0.12 µB/at, averaged over the whole volume of the volume, with an orbital-to-spinratio mL/mS= 0.05. In Py/Pt(1nm) |20, a magnetic moment 0.27µB/at is found on Pt, comparable to that reported for Ni/Pt epitaxial multilayers23, with a relatively high orbitalcharacter mL/mS= 0.18, ascomparedwithPdin- duced moment. The large difference in volume-averaged induced moment per atom comes from the different film thickness, hence volume, for Pt and Pd. Assuming that the induced magnetic moment is confined to the first atomic layers at the interface with Py23,24, one could estimate 0.32 µB/at for Pd and 0.30 µB/at for Pt36. When a3nm thick Cu interlayeris introducedbetween Py and N, a two ordersofmagnitude smaller induced mo- ment (0.0036 µB/at) was found for 2.5nm Pd22, while Pt showed an XMCD signal of the order of the experimen- tal sensitivity, ∼0.5·10−3µB/at. In Fig. 2XMCD spectra at the L 3edge of Pt are shown for Cu interlayer thicknesses 0, 0.5 and 1nm. For 0.5nm Cu the integral of XMCD signal at the L 3edge shrinks to 30%, while for 1nm it is reduced to zero within experimental error. This result could be explained either by a 3d growth of the Cu layer, allowing a fraction of the Pt layer to be in direct contact with Py for Cu coverages of 0.5nm, or byNχmol37S37N0abulktN/angbracketleftM/angbracketrightMiJex [cm3/mol][1/eV·at][nm][nm][µB/at][µB/at][meV] 10−4 Pd5.5±0.29.30.83±0.030.3892.50.1160.3242 Pt1.96±0.13.70.74±0.040.3921.00.270.30109 TABLE I. Spin-sink layer N properties in Py/N heterostruc- tures: experimental molar susceptibility χmolat 20◦C; den- sity of states N0calculated from tabulated χmol; Stoner pa- rameter S from Ref.37; bulk lattice parameter a; layer thick- nessestN; volume averaged induced magnetic moment /angbracketleftM/angbracketright from XMCD measurement in Fig. 1; interface magnetic mo- mentMi36; Py|N interfacial exchange energy per interface atomJex(Eq.1). diffusion of magnetic Ni atoms in Cu on a scale shorter than 1nm. The film then becomes continuous, and at 1nm coverage, no direct exchange coupling takes place between Py and Pt layers. For FMR measurements pre- sented in the followingsection, a 3nm thick Cu interlayer is employed, reducing also any possible indirect exchange coupling. From the values of induced moments in Pd and Pt, we can make a step forward and estimate the interfacial exchange coupling energies for the two cases. Equating interatomic exchange energy Jexand Zeeman energy for an interface paramagnetic atom, we have (see Appendix B1for the derivation) Jex=1 2/angbracketleftM/angbracketright µBN0StN ti(1) where/angbracketleftM/angbracketrightis the thickness-averaged paramagnetic moment, N0is the single-spin density of states (in eV−1at−1), S is the Stoner factor and ti= 2∗a/√ 3 is the polarized interface-layer thickness36. The 1/2fac- tor accounts for the fact that in XMCD measurements the N layer has both interfaces in contact with F. Un- der the simplifying assumption that all the magnetic mo- ment is confined to the interface N layer and assuming experimental bulk susceptibility parameters for χv, we obtainJPd ex= 42 meV for Pd and JPt ex= 109 meV for Pt (results and properties are summarized in Tab. I). Here the difference in estimated Jex, despite roughly equalMi, comes from the larger Stoner factor S for Pd. A stronger interfacial exchange energy in Pt de- notes a stronger orbital hybridization, yielding possibly a higher orbital character of the interfacial magnetic mo- ment in the ferromagnetic Py counterpart21. For com- parison, we consider the interatomic exchange param- etersJexin ferromagnetic Py and Co, investigated in Ref.14.Jexis estimated from the respective Curie tem- peratures TC, through Jex≃6kBTC/(m/µB)2, wherem is the atomic moment in µB/at (see Appendix B2). Ex- perimental Curie temperatures of 870K and 1388K give JCo ex=293meV for Co and JPy ex=393meV for Py, which are of the same order of the value calculated for Pt (de- tails about calculation in Appendix B2).4 1.0 0.5 0.0 14121086420 tPt (nm) Py/Pt Py/Cu/Pt12 10 8 6 4 2 0 468 12468 10 tPt (nm) Py/Pt tc = 2.4 nm Py/Cu/Pt λα = 1.8 nm 1.0 0.5 0.0∆α / ∆α0 14121086420 tPd (nm) Py/Pd Py/Cu/Pd6 5 4 3 2 1 0∆α (x103) 468 12468 10 tPd (nm) Py/Pd tc = 5.0 nm Py/Cu/Pd λα = 5.8 nma) b) c) d) FIG. 3. (Color online) Damping enhancement ∆ α, due to pumped spin current absorption, as a function of thickness tN for Py(5nm)/N and Py(5nm)/Cu(3nm)/N heterostructures, with N = Pd( tN) (panels a,c), Pt( tN) (panels b,d). Solid lines result from a fit with exponential function (Eq. 2) with decay length λα. Dashed lines represents instead a linear- cutoff behavior (Eq. 3) fortN< tc. Please notice in panels a, c the x-axis is in logarithmic scale. In panels b, c the dampin g enhancement is normalized to the respective saturation val ue ∆α0. In the following, the effect of these static induced mo- ments on the spin pumping damping of the heterostruc- tures characterized will be discussed. B. FMR: damping enhancement The main result of our work is now shown in Figure 3. In Fig. 3the damping enhancement ∆ αis plotted as a function of the spin-sink layer thickness tN, for Py/Pd, Py/Cu/Pd (panels a, c) and Py/Pt, Py/Cu/Pt (panels b, d). The enhancement ∆ αis compared with the damp- ingαof a reference structure Py(5nm)/Cu, excluding the sink layer N. Each value of αresults from established analysis of the linewidth of 11 FMR traces13,14, employ- ing ag-factor equal to 2.09 as a constant fit parameter for all samples. In Py/Cu/N systems (Fig. 3, green square markers), ∆αrises with increasing tNthickness to similar satura- tion values ∆ α0= 0.0027, 0.0031 for Pd and Pt, but reachedondifferentlengthscales,giventhedifferentchar-Ng↑↓ eff(Py|Cu/N) λαg↑↓ eff(Py|N)tc [nm−2][nm] [nm−2][nm] Pd 7.2 5.8±0.2145.0±0.3 Pt 8.3 2.4±0.1321.8±0.2 TABLE II. Mixing conductance values extracted from the damping enhancement ∆ αat saturation in Fig. 3, and re- spective length scales (see text for details). acteristic spin relaxation lengths of the two materials. From the saturation value, an effective mixing conduc- tanceg↑↓ eff(Py|Cu/N) = 7 .2−8.3nm−2is deduced in the frameworkof standardspin pumping picture13,17,19, with Py saturation magnetization µ0Ms= 1.04T. The fact that the spin-mixing conductance is not material depen- dent indicates that similar Cu |N interfaces are formed. The thickness dependence is well described by the expo- nentialfunction14,20 ∆α(tN) = ∆α0(1−exp(−2tN/λN α)) (2) as shown by the fit in Fig. 3a-b (continuous line). As a result, exponential decay lengths λPt α= 1.8nm and λPd α= 5.8nm are obtained for Pt and Pd, respectively. When the Pt, Pd spin-sink layers come into direct con- tact with the ferromagnetic Py, the damping enhance- ment ∆α(tN) changes dramatically. In Py/N systems (Fig.3a-b, trianglemarkers),the damping saturationval- ues become ∆ αPt 0= 0.0119 and ∆ αPd 0= 0.0054 for Pt and Pd, respectively a factor ∼2 and∼4 larger as com- pared to Py/Cu/N. Within the spin-pumping descrip- tion, a largerdamping enhancement implies a largerspin- currentdensitypumped outoftheferromagnetacrossthe interface and depolarized in the sink. In Py/N heterostructures, because of the magnetic proximity effect, few atomic layers in N are ferromagnet- ically polarized, with a magnetic moment decaying with distance from the Py |N interface. The higher value of damping at saturation might therefore be interpreted as the result of a magnetic bi-layer structure, with a thin ferromagnetic N characterized by high damping αN high coupled to a low damping αF lowferromagnetic Py38. To investigate whether damping is of bi-layer type, or truly interfacial, in Fig 4we show the tFthickness dependence of the damping enhancement ∆ α, for a Py( tF)/Pt(4 nm) series of samples. The power law thickness dependence adheres very closely to t−1 F, as shown in the logarithmic plot. The assumption of composite damping for syn- chronous precession, as ∆ α(t1) = (α1t1+α2t2)/(t1+t2), shown here for t2= 0.25nm and 1.0nm, cannot follow an inverse thickness dependence over the decade of ∆ α observed. Damping is therefore observed to have a pure interfacial character. In this case, the mixing conductances calculated from the saturation values are g↑↓ eff(Py|Pd) = 14nm−2and g↑↓ eff(Py|Pt) = 32nm−2. From ab initio calculations within a standard spin-pumping formalism in diffusive films10,29, it is found g↑↓ eff(Py|Pd) = 23nm−2for Pd and5 10 3 4 56789 20 30 40 tF (nm)110 2345678920∆α(10−3)Py(tF)/Pt(4) interfacial, K=0.055 bilayer, t2=0.25 nm t2=1.0 nm 26101520 30 tF (nm)051015202530α(10−3)Py(t)/Pt(4nm) Py(t) FIG. 4. Logarithmic plot of the damping enhancement ∆ α (triangle markers) as a function of the Py layer thickness tF, in Py(t F)/Pt(4nm). Solid and dashed lines represents, re- spectively, fits according to the spin pumping ( interfacial ) model ∆ α=Kt−1 Fand to a αlow(tF)/αhigh(t2)bilayermodel, witht2= 0.25,1.0nm.Inset: Gilbert damping αfor Py(t F) (square markers) and Py(t F)/Pt(4nm) (round markers). g↑↓ eff(Py|Pt) = 22nm−2for Pt. Theoretical spin mixing conductance from a standard picture does reproduce the experimental order of magnitude, but it misses the 2.3 factor between the Py |Pt and Py |Pd interfaces. Beyond a standard pumping picture, Liu and coworkers29intro- duce spin-flipping scattering at the interface and calcu- late from first principles, for ideal interfaces in finite dif- fusive films: g↑↓ eff(Py|Pd) = 15nm−2, in excellent agree- ment with the experimental value here reported for Pd (Tab.II), andg↑↓ eff(Py|Pt) = 25nm−2. Zhang et al.10 suggest an increase up to 25% of the mixing conductance can be obtained by introducing magnetic layers on the Pt side. The results here reported support the emerging idea that a generalized model of spin pumping including spin-orbit coupling and induced magnetic moments at F|N interfaces may be required to describe the response of heterostructures involving heavy elements. The saturation value of damping enhancement at ∆ α0 as a function of the Cu interlayer thickness is shown in Fig.5to follow the same trend of the XMCD signal (dashed line), reported from Fig. 2. Indeed, it is found that the augmented ∆ α0in Py/N junctions is drasti- cally reduced by the insertion of 0.5nm Cu at the Py |N interface17, and the saturation of the Py/Cu/N configu- ration is already reached for 1nm of Cu interlayer. As soonasacontinuousinterlayerisformedandnomagnetic moment is induced in N, ∆ α0is substantially constant with increasing Cu thickness. The N-thickness dependence of ∆ α(tN) in Py/N sys- tems before saturation is addressed in the following. At variance with the Py/Cu/N case, the thickness depen- dence of ∆ αis not anymore well described by an expo-1.0 0.5 0.0 3210 tCu (nm)1.0 0.5 0.0L3 XMCD area (norm.)Pt(3nm) XMCD Pt 3210 tCu (nm)1.0 0.5 0.0∆α0 (norm.)Pd(7nm) XMCD Pt FIG. 5. (Color online) Normalized damping enhancement ∆ α (left axis), due to spin pumping, as a function of interlayer thickness tCufor Py(5nm)/Cu( tCu)/N heterostructures, with N = Pd(7nm), N = Pt(3nm). The dashed line represents the XMCD signal (right axis) reported from inset in Fig. 2. nential behavior, as an exponential fit (with exponential decay length as only free fit parameter) fails to repro- duce the increase of ∆ αtowards saturation (solid lines in Fig.3a-b). More rigorous fitting functions employed in spin pumping experiments, within standard spin trans- port theory16,26,32, cannot as well reproduce the experi- mental data (see Appendix. A). It is worth mentioning that the same change of trend between the two configu- rations was observed for the same stacks with a 10nm thick Py layer (data shown in Appendix A, Fig.7). A change of the functional dependence of ∆ αontNre- flects a change in the spin-depolarization processes the pumped spin current undergoes, as for instance shown in Ref.30when interfacial spin-orbit coupling is introduced in the spin-pumping formalism. Experimentally, a linear thickness dependence with sharp cutoff has been shown to characterize spin-current absorption in spin-sink lay- ers exhibiting ferromagnetic order at the interface, as re- ported for F 1/Cu/F 2(tF2) junctions with F = Py, Co, CoFeB14. Given the presence of ferromagnetic order in N at the interface of F/N structures, the data are tenta- tively fit with a linear function ∆α= ∆α0tN/tN c (3) This linear function better reproduce the sharp rise of ∆α(dashed lines in Fig. 3a-b) and gives cutoff thick- nessestPt c= 2.4±0.2nm and tPd c= 5.0±0.3nm for Pt and Pd, respectively. The linearization is ascribed to the presence of ferromagnetic order in the paramagnetic Pd, Pt spin-sink layers at the interface with the ferromag- netic Py. The linear trend extends beyond the thickness for which a continuous layer is already formed (less than 1nm), and, especially for Pd, far beyond the distance within the non-uniform, induced moment is confined (up to0.9nm). InRef.14, thecutoff tcinF/Cu/Fheterostruc- turesis proposedto be on the orderofthe transversespin coherence length λJin ferromagnetically ordered layers. λJcan be expressed in terms of the exchange splitting6 6 5 4 3 2 1 0tc(nm) 2520151050 1/Jex(eV-1)PtPd Py Co FIG. 6. Effect of direct exchange strength on length scale of spin current absorption. Cutoff thickness tcextracted from the ∆α(tN) data in Fig. 3as a function of reciprocal interfa- cial exchange energy 1 /Jexextracted from XMCD in Fig. 1. Labels are given in terms of Jex. The Co and Py points are from Ref.14. energyJex, λJ=hvg 2Jex(4) wherevgis the electronic group velocity at the Fermi level. This form, found from hot-electron Mott polarimetry1, is expressed equivalently for free electrons asπ/|k↑−k↓|, which is a scaling length for geometrical dephasing in spin momentum transfer2. Electrons which enter the spin-sink at E Fdo so at a distribution of angles with respect to the interface normal, traverse a distribu- tionofpathlengths, andprecessbydifferentangles(from minority to majority or vice versa ) before being reflected back into the pumping ferromagnet. For a constant vg, it is therefore predicted that tcis inversely proportional to the exchange energy Jex. In Figure 6we plot the dependence of the cutoff thick- nesstN cupontheinverseoftheestimatedexchangeenergy Jex(Tab.I),asextractedfromtheXMCDmeasurements. A proportionality is roughly verified, as proposed for the transverse spin coherence length across spin polarized in- terfaces. Under the simplistic assumption that tc=λJ, from the slope of the line we extract a Fermi velocity of∼0.1·106m/s (Eq.4), of the order of magnitude ex- pectedforthematerialsconsidered39,40. Thesedatashow that, up to a certain extent, length scale for spin-current scattering shares common physical origin in ferromag- netic layers and paramagnetic heavy-metals, such as Pd and Pt, under the influence of magnetic proximity effect. This unexpected results is observed in spite of the fact that F 1/Cu/F 2and F/N systems present fundamental differences. In F/N structure, the induced moment in N is expected to be directly exchange coupled with the fer- romagneticcounterpart. Whereasin F 1/Cu/F 2, themag- netic moment in F 2(off-resonance) are only weakly cou- pled with the precession occurring in F 1(in-resonance), through spin-orbit torque and possible RKKY interac-tion. Magnetization dynamics in N might therefore be expected with its own pumped spin current, albeit, to the best of our knowledge, no experimental evidence of a dynamic response of proximity induced moments was reported so far. From these considerations and the experimental findings, counter-intuitively the proximity- induced magnetic moments appear not to be involved in the production of spin current, but rather to contribute exclusively with an additional spin-depolarization mech- anism at the interface. IV. CONCLUSIONS We have investigated the effect of induced magnetic moments in heavy metals at Py/Pt and Py/Pd inter- faces on the absorption of pumped spin currents, by analyzing ferromagnetic resonance spectra with varying Pt, Pd thicknesses. Static, proximity-induced magnetic moments amount to 0.32 and 0.3 µB/atom in Pd and Pt, respectively, at the interface with Py, as deduced from XMCD measurements taken at the L 2,3edges. We have shown that when the proximity induced moment in Pt and Pd is present, an onset of a linear-like thick- ness dependence of the damping is observed, in con- trast with an exponential trend shown by Py/Cu/Pd and Py/Cu/Pt systems, for which no induced moment is measured. These results point to the presence of an additional spin-flip process occurringat the interface and to a change of the character of spin current absorption in the ultrathin Pd and Pt paramagnets because of the interfacial spin polarization. The range of linear increase is proposed to be inversely proportional to the interfa- cial exchange energy in Py/Pt and Py/Pd, inferred from XMCD data. WEB acknowledges the Universit´ e Joseph Fourier and Fondation Nanosciences for his research stay at SPIN- TEC. This work was supported in part by the U.S. NSF- ECCS-0925829 and the EU EP7 NMP3-SL-2012-280879 CRONOS. MC is financed by Fondation Nanosciences. Appendix A: N-thickness dependence In order to confirm the results presented in the manuscript, additional sample series with thicker Py layer were fabricated and measured. The experimental results for 10nm thick Py layer are shown in Fig.s 7and 8forPdandPt, respectively. Wehavepresentedthedata here, rather than including them with the other plots in Figure3, to keep the figures from being overcrowded. As expected when doubling the ferromagnet thickness, the saturation values ∆ α0are about half of those measured for 5nm Py (Fig. 3). Confirming the data presented in the manuscript, it is observedagaina changeof thickness dependence of ∆ α(tN), fromexponential for Py/Cu/N (solid lines; Eq. 2,λα= 4.8nm and 1.4nm for Pd and Pt7 3 2 1 0∆α (x10-3) 345678 12345678 10 tPd (nm)Py/Pd Py/Cu/Pd Eq. 3 , Eq. 2 [16, 32] - ρPd=1.4E-7 [16, 32] - ρPd->ρ(tPd) FIG. 7. Damping enhancement ∆ α, due to pumped spin current absorption, as a function of thickness tPd for Py(10nm)/Pd and Py(10nm)/Cu(3nm)/Pd heterostruc- tures. Solid lines result from a fit with exponential functio n (Eq.2) with decay λα. Dashed lines represents instead a linear-cutoff behavior (Eq. 3) fortPd< tc. Short-dash and point-dash traces are fit to the data, employing equations from standard spin transport theory (see text for details)16,32. In bottom panel, ∆ αis normalized to the respective satura- tion value. respectively) to linear-like for Py/N (dashed lines; Eq. 3, tc= 5.3nm and 2nm for Pd and Pt respectively). The experimental data are also fitted with a set of equations derived from standard theory of diffusive spin transport16,26,32, describing the the dependence of ∆ α on the thickness of adjacent metallic layers (either N or Cu/N in our case) as follow ∆α=γ¯h 4πMstFMg↑↓ 1+g↑↓/gx ext(A1) with (Eq. 7 in Ref.16, and Eq. 6 in Ref.32) gN ext=gNtanhtN/λN sd gCu/N ext=gCugCucothtN/λN sd+gNcothtCu/λCu sd gCucothtN/λN sdcothtCu/λCu sd+gN(A2) wheregx=σx/λx sd,σxandλx sdare the electrical conduc- tivity and spin diffusion length of the non magnetic layer x. For the thin Cu layer, we used a resistivity ρCu= 1×107Ωm and a spin diffusion length λCu sd= 170nm32. For the Pt and Pd layers, two fitting models in which the conductivity of the films is either constant or thickness dependent areconsidered, as recently proposed by Boone and coworkers16. The values of conductivity, as taken di- rectly from Ref.16, will influence the spin diffusion length λN sdand spin mixing conductance g↑↓resulting from the fit, but will not affect the conclusions drawn about the overall trend. When a constant resistivity is used (short- dash, blue lines), the model basically corresponds to the6 5 4 3 2 1 0∆α (x10-3) 345678 12345678 10 tPt (nm) Py/Pt Py/Cu/Pt Eq. 3 , Eq. 2 [16, 32] - ρPt=1.7E-7 [16, 32] - ρPt->ρ(tPt) FIG. 8. Damping enhancement ∆ α, due to pumped spin current absorption, as a function of thickness tPtfor Py(10nm)/Pt and Py(10nm)/Cu(3nm)/Pt heterostructures. Solid lines result from a fit with exponential function (Eq. 2) with decay λα. Dashed lines represents instead a linear-cutoff behavior (Eq. 3) fortPt< tc. Short-dash and point-dash traces are fit to the data, employing equations from standard spin transport theory (see text for details)16,32. In bottom panel, ∆ αis normalized to the respective saturation value. simple exponential function in Eq. 2. It nicely repro- duces the data in the indirect contact case (Py/Cu/N) for both Pd (Fig. 7) and Pt (Fig. 8), but it fails to fit the direct contact (Py/N) configuration. When a thick- ness dependent resistivity of the form ρN=ρb N+ρs N/tNis used (dash-point, cyan lines)16, in Py/Cu/N systems, no significant difference with the other functions is observed for Pt, while for Pd a deviation from experimental trend is observed below 1.5nm. In Py/Nsystems, the fit better describes the rise at thicknesses shorter than the charac- teristic relaxation length, while deviates from the data around the saturation range. Models from standard spin transport theory cannot satisfactorily describe the experimental data for the di- rect contact Py/N systems. For this reason a different mechanismforthe spin depolarizationprocesseshasbeen proposed, considering the presence of induced magnetic moments in N in contact with the ferromagnetic layer. Appendix B: Interfacial interatomic exchange 1. Paramagnets We will show estimates for exchange energy based on XMCD-measured moments in [Py/(Pt, Pd)] repeatsu- perlattices. Calculations of susceptibility are validated against experimental data for Pd and Pt. Bulk suscepti- bilities will be used to infer interfacial exchange parame-8 tersJi ex. a. Pauli susceptibility For an itinerant electron sys- tem characterized by a density of states at the Fermi energyN0, if an energy ∆ Esplits the spin-up and spin- down electrons, the magnetization resulting from the (single-spin) exchange energy ∆ Eis M=µB/parenleftbig N↑−N↓/parenrightbig = 2µBN0S∆E(B1) whereN0is the density of states in # /eV/at,Sis the Stoner parameter, and 2∆ Eis the exchange splitting in eV. Moments are then given in µB/at. Solving for ∆ E, ∆E=M 2µBN0S(B2) If the exchange splitting is generated through the ap- plication of a magnetic field, ∆ E=µBH, µBH=M 2µBN0S(B3) and the dimensionless volume magnetic susceptibility can be expressed χv≡M H= 2µ2 BN0S (B4) In this expression, the prefactor can be evaluated through µ2 B= 59.218 eV ˚A3(B5) so with N0[=]/eV/at, χvtakes units of volume per atom, and is then also called an atomic susceptibility, in cm3/at, as printed in Ref37. b. Molar susceptiblity Experimentalvaluesaretabu- lated as molar susceptibilities. The atomic susceptibility χvcanbe contrastedwith the masssusceptibility χmand molar susceptibility χmol χmass=χv ρχmol=ATWT ρχv(B6) where ATWT is the atomic weight (g/mol) and ρis the density (g/cm3). These have units of χmass[=]cm3/g andχmol[=]cm3/mol. The molar susceptibility χmolis then χmol= 2µ2 BN0NAS (B7) in cm3/mol, where µBis the Bohr magneton, and 2N0S=χmol NAµ2 B(B8)Eq.B8providesaconvenentmethodtoestimateexper- imental unknowns, the density of states N0and Stoner parameter S, from measurements of χmol. Example: for Pd, the low-temperature measurement (differentfromtheroom-temperaturemeasurementinTa- bleI) isχmol∼7.0×10−4cm3/mol. In the denomina- tor, (NAµ2 B) = 2.622 ×10−6Ry·cm3/mol, The value 2N0Sconsistent with the experiment is 266/(Ry-at) or 19.6/(eV-at). For the tabulated measurement of S= 9.3, the inferred density of states is then N0= 1.05/eV/at. c. Interfacial exchange We canassumethat the Zee- man energy per interface atom is equal to its exchange energy, through the Heisenberg form M2 p χvVat= 2Ji exsfsp (B9) whereMpis the magnetization of the paramagnet, with the atomic moment of the paramagnet mpin terms of its per-atom spin sp, Mp=mp Vatmp= 2µBsp (B10) Vatis the volume of the paramagnetic site, sf,pare the per-atom spin numbers for the ferromagnetic and para- magnetic sites, and Ji exis the (interatomic) exchange en- ergy acting on the paramagnetic site from the ferromag- neticlayersonthe othersideoftheinterface. Interatomic exchangeenergyhasbeen distinguished fromintraatomic (Stoner) exchange involved in flipping the spin of a single electron. Rewriting Eq B9, M2 p χvVat= 2Ji exsfMp 2µBVat (B11) ifsf= 1/2, appropriate for 4 πMs∼10 kG, Ji ex= 2µBMp χv(B12) and substituting for χvthrough Eq B4, Ji ex=Mp µBN0S(B13) In the XMCD experiment, we measure the thickness- averaged magnetization as < M > in a [F/N]nsuper- lattice. We make a simplifying assumption that the ex- change acts only on nearest-neighbors and so only the near-interface atomic layer has a substantial magnetiza- tion. We can then estimate Mpfrom< M >through < M > t p= 2Mpti (B14)9 wheretiis the polarized interface-layer thickness of N36. Since the interface exists on both sides of the N layer, 2tiis the thickness in contact with F. Finally, Jex=1 2< M > µBN0Stp ti(B15) The exchange energy acting on each interface atom, from all neighbors, is JPt ex= 109 meV for Pt and JPd ex= 42 meV for Pd. Per nearest neighbor for an ideal F/N(111) interface, it is JPy|Pt= 36 meV and JPy|Pd= 14 meV. Per nearest neighbor for an inter- mixed interface (6 nn), the values drop to 18 meV and 7 meV, respectively. Since explicit calculations for these systems are not in the literature, we can compare indirectly with theo- retical values. Dennler41showed that at a (3 d)F/(4d)N interface (e.g. Co/Rh), there is a geometrical enhance- ment in the moment induced in Nper nearest-neighbor ofF. The 4d Natoms near the Finterface have larger induced magnetic moments per nn of Fby a factor of four. Specific calculations exist of JF|N(per neighbor) for dilute Co impurities in Pt and dilute Fe impurties in Pd42.JFe−Pd∼3 meV is calculated, roughly indepen- dent of composition up to 20% Fe. If this value is scaled up by a factor of four, to be consistent with the inter- face geometry in the XMCD experiment, it is ∼12 meV, comparable with the value for Pd, assuming intermixing. Therefore the values calculated have the correct order of magnitude. 2. Ferromagnets The Weiss molecular field, HW=βMs (B16) whereβis a constant of order 103, can be used to give an estimate of the Curie temperature, as TC=µBgJJ(J+1) 3kBHW (B17) Density functional theory calculations have been used toestimatethemolecularfieldrecently42,43; forspintype, theJ(J+1) term is substutited with < s >2, giving an estimate of TC=2< s >2J0 3kB(B18)where< s >is the number of spins on the atom as in EqB10; see the text by St¨ ohr and Siegmann44.< s > can be estimated from m=1.07µBfor Py and 1.7 µBfor Co, respectively. Then J0≃6kBTC (m/µB)2(B19) with experimental Curie temperatures of 870 and 1388 K, respectively, gives estimates of J0= 293 meV for Co andJ0= 393 meV for Py. Note that there is also a much older, simpler method. Kikuchi45has related the exchange energies to the Curie temperature for FCC lattices through J= 0.247kBTC (B20) Taking 12 NN, 12 Jgives a total energy of 222 meV for Py (870 K) and 358 meV for FCC Co (1400K), not too far off from the DFT estimates. d. Other estimates TheJ0exchange parameter is interatomic, describing the interaction between spin- clusters located on atoms. Reversing the spin of one of these clusters would change the energy J0. The Stoner exchange ∆ is different, since it is the energy involved in reversing the spin of a single electron in the electron sea. Generally ∆ is understood to be greater than J0because it involvesmore coloumbrepulsion; interatomic exchange can be screened more easily by spelectrons. This exchange energy is that which is measured by photoemissionandinversephotoemission. Measurements are quite different for Py and Co. Himpsel40finds an exchange splitting of ∆ = 270 meV for Py, which is not too far away from the Weiss J0value. 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2013-08-02

We investigated the spin pumping damping contributed by paramagnetic layers (Pd, Pt) in both direct and indirect contact with ferromagnetic Ni$_{81}$Fe$_{19}$ films. We find a nearly linear dependence of the interface-related Gilbert damping enhancement $\Delta\alpha$ on the heavy-metal spin-sink layer thicknesses t$_\textrm{N}$ in direct-contact Ni$_{81}$Fe$_{19}$/(Pd, Pt) junctions, whereas an exponential dependence is observed when Ni$_{81}$Fe$_{19}$ and (Pd, Pt) are separated by \unit[3]{nm} Cu. We attribute the quasi-linear thickness dependence to the presence of induced moments in Pt, Pd near the interface with Ni$_{81}$Fe$_{19}$, quantified using X-ray magnetic circular dichroism (XMCD) measurements. Our results show that the scattering of pure spin current is configuration-dependent in these systems and cannot be described by a single characteristic length.

Spin pumping damping and magnetic proximity effect in Pd and Pt spin-sink layers

1308.0450v2

Damping enhancement in coherent ferrite/insulating-paramagnet bilayers Jacob J. Wisser,1Alexander J. Grutter,2Dustin A. Gilbert,3Alpha T. N'Diaye,4 Christoph Klewe,4Padraic Shafer,4Elke Arenholz,4, 5Yuri Suzuki,1and Satoru Emori6, 1Department of Applied Physics, Stanford University, Stanford, CA, USA 2NIST Center for Neutron Research, Gaithersburg, MD, USA 3Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA 4Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA, USA 5Cornell High Energy Synchrotron Source, Ithaca, NY, USA 6Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA (Dated: October 29, 2019) High-quality epitaxial ferrites, such as low-damping MgAl-ferrite (MAFO), are promising nanoscale building blocks for all-oxide heterostructures driven by pure spin current. However, the impact of oxide interfaces on spin dynamics in such heterostructures remains an open question. Here, we investigate the spin dynamics and chemical and magnetic depth proles of 15-nm-thick MAFO coherently interfaced with an isostructural 1-8-nm-thick overlayer of paramagnetic CoCr 2O4 (CCO) as an all-oxide model system. Compared to MAFO without an overlayer, eective Gilbert damping in MAFO/CCO is enhanced by a factor of >3, irrespective of the CCO overlayer thickness. We attribute this damping enhancement to spin scattering at the 1-nm-thick chemically disordered layer at the MAFO/CCO interface, rather than spin pumping or proximity-induced magnetism. Our results indicate that damping in ferrite-based heterostructures is strongly in uenced by interfacial chemical disorder, even if the thickness of the disordered layer is a small fraction of the ferrite thickness. I. INTRODUCTION Emerging spintronic device schemes leverage magnon spin currents in electrically insulating magnetic oxides (e.g., ferrites), unaccompanied by dissipative motion of electrons, for computing and communications applications1,2. Low-dissipation spintronic devices become particularly attractive if insulating ferrite thin lms with low magnetic damping can serve as sources of magnon spin currents. Such low-damping ferrites include not only epitaxial garnet ferrites (e.g., YIG)3{11 that have been widely used in studies of insulating spintronics2{4,12{15, but also coherently strained epitaxial spinel ferrites16{18with crucial technical advantages over garnets, such as lower thermal budget for crystallization, higher magnon resonance frequencies, and potential to be integrated coherently with other spinels and perovskites with various functionalities19{22. In general, low-damping ferrite thin lms must be interfaced with other materials to realize spintronic devices. It is therefore essential to understand whether and how damping in the ferrite is impacted by the proximity to another material. For instance, to convert between electronic and magnonic signals through direct and inverse spin Hall or Rashba-Edelstein eects23, the low-damping ferrite needs to be interfaced with a nonmagnetic metal with strong spin-orbit coupling. Spin transport and enhanced damping through spin pumping24in ferrite/spin-orbit-metal structures has already been extensively studied3,4,12{15,25. Moreover, the low-damping ferrite can be interfaced with an insulating antiferromagnetic or paramagnetic oxide, in which signals can be transmitted as a pure magnon spin current26{40. While interfacing low-damping ferriteswith insulating anti/paramagnetic oxides has enabled prototypes of magnon spin valves37{39, the fundamental impact of insulating oxide interfaces on spin dynamics has remained mostly unexplored. In particular, it is an open question whether or how damping of the ferrite is enhanced from spin dissipation within the bulk of the adjacent anti/paramagnetic oxide or from spin scattering at the oxide interface. Here, we investigate how room-temperature magnetic damping in epitaxial ferrimagnetic spinel MgAl-ferrite (MgAl 1=2Fe3=2O4, MAFO) is impacted when interfaced with an overlayer of insulating paramagnetic spinel CoCr 2O4(CCO)41,42. This epitaxial MAFO/CCO bilayer is an isostructural model system, possessing a coherent interface with continuous crystal lattices between the spinel ferrite and paramagnet. We nd that the presence of MAFO/CCO interface increases damping by more than a factor of >3 compared to MAFO without an overlayer. We attribute this damping enhancement { which is comparable to or greater than spin pumping eects reported for ferrite/spin-orbit-metal bilayers { to spin scattering by the ultrathin ( 1 nm) chemically disordered layer at the MAFO/CCO interface. Our ndings show that spin scattering at oxide interfaces has a profound in uence on damping, even when the chemically disordered layer is a small fraction of the total magnetic layer thickness. II. FILM GROWTH AND STRUCTURAL CHARACTERIZATION Epitaxial thin lms of 15-nm-thick MAFO interfaced with 1.3-8 nm of CCO overlayer were grown on as-arXiv:1908.08629v2 [cond-mat.mtrl-sci] 26 Oct 20192 40 42 44 46MAO (004) MAFO/CCO (004) MAFO (004)log10(Intensity) (arb. units) 2q (deg)CCO (004) -0.180 -0.175 -0.1700.570.580.590.600.610.620.63- (115)-(a) (c) qip(Å-1)qop(Å-1) CCO (25 nm)MAO(b) -0.2-0.1 0.00.10.2MAFOCCOIntensity (arb. units) Dw004 (deg)MAFO/CCO Figure 1. (a) 2 -!scans of epitaxial MAFO(15 nm), CCO(25 nm), and MAFO(15 nm)/CCO(8 nm). The data are oset for clarity. (b) Rocking curve scans about the (004) lm peak for the lms shown in (a). (c) Reciprocal space map of epitaxial CCO(25 nm) coherently strained to the MAO substrate. received single-crystal MgAl 2O4(MAO) substrates via pulsed laser deposition. A KrF 248 nm laser was incident on stoichiometric targets of MAFO and CCO with uences of 1.5 J/cm2and1.3 J/cm2, respectively. Both lms were grown in 10 mTorr (1.3 Pa) O 2and were cooled in 100 Torr (13 kPa) O 2. MAFO lms were grown at 450C, whereas CCO lms were deposited at 300C in an attempt to minimize intermixing between the MAFO and CCO layers. These growth temperatures, much lower than >700C typically required for epitaxial garnets3{11, are sucient to fully crystallize MAFO and CCO. The low crystallization temperatures of the spinels oer an advantage over the oft-studied garnets, with more opportunities for isostructural integration with coherent interfaces. The MAFO lms exhibit a room-temperature saturation magnetization of100 kA/m and a Curie temperature of 400 K18. To obtain consistent ferromagnetic resonance results, MAFO lms were grown and subsequently characterized by ferromagnetic resonance (FMR) ex-situ; after surface cleaning with ultrasonication in isopropanol, CCO overlayers were then deposited as described above. Growth rates were calibrated via X-ray re ectivity. Our structural characterization of MAFO and CCO shows high-quality, coherently strained lms. In symmetric 2 -!X-ray diraction scans, only peaks corresponding to the (00 `) re ections are observed, indicating that the lms are highly epitaxial. Additionally, as seen in Fig. 1(a), Laue oscillations around the (004) Bragg re ections in both single-layer MAFO and CCO layers as well as MAFO/CCO bilayers denote smooth interfaces. Furthermore, MAFO, CCO, and MAFO/CCO samples all exhibit essentially the same lm-peak rocking curve widths (FWHM) of 0.06 (Fig. 1(b)). Reciprocal space mapping of the ( 115) re ection in 25-nm-thick single-layer CCO on MAO (Fig. 1(c)) reveals that the in-plane lattice parameter of the lm coincides with that of the substrate, indicating CCO is coherently strained to MAO. We note thatdespite the relatively large lattice mismatch between CCO and MAO of 3 %, coherently strained growth of CCO of up to 40 nm has been previously reported on MAO substrates41. For our CCO lm, we calculate an out-of-plane lattice constant c8:534A from the 2 -! scan; taking the in-plane lattice parameter a= 8:083A of the MAO substrate, the resulting tetragonal distortion of coherently strained CCO is c=a1:055, similar to that for coherently strained MAFO18. Structural characterization results underscore the quality of these epitaxial lms grown as single layers and bilayers. Considering the comparable high crystalline quality for MAFO, CCO, and MAFO/CCO { as evidenced by the presence of Laue oscillations and narrow lm-peak rocking curves { we conclude that MAFO/CCO bilayers (with the total thickness limited to 23 nm) are coherently strained to the substrate. In these samples where the substrate and lm layers are isostructural, we also do not expect antiphase boundaries43{46. Indeed, we nd no evidence for frustrated magnetism, i.e., high saturation eld and coercivity, that would arise from antiphase boundaries in spinel ferrites43{46; MAFO/CCO bilayers studied here instead exhibit soft magnetism, i.e., square hysteresis loops with low coercivity <0.5 mT, similar to our previous report on epitaxial MAFO thin lms18. Thus, MAFO/CCO is a high-quality all-oxide model system, which permits the evaluation of how spin dynamics are impacted by a structurally clean, coherent interface. III. FERROMAGNETIC RESONANCE CHARACTERIZATION OF DAMPING To quantify eective damping in coherently strained MAFO(/CCO) thin lms, we performed broadband FMR measurements at room temperature in a coplanar waveguide setup using the same procedure as our prior work16,18. We show FMR results with external bias3 magnetic eld applied in the lm plane along the [100] direction of MAFO(/CCO); essentially identical damping results were obtained with in-plane eld applied along [110]47. Figure 2(a) shows the frequency fdependence of half-width-at-half-maximum (HWHM) linewidth
Hfor a single-layer MAFO sample and a MAFO/CCO bilayer with a CCO overlayer thickness of just 1.3 nm, i.e., less than 2 unit cells. The linewidth is related to the eective Gilbert damping parameter effvia the linear equation:
H=
H0+heff g0Bf (1) where
H0is the zero-frequency linewidth, his Planck's constant,g2:05 is the Land e g-factor derived from the frequency dependence of resonance eld HFMR ,0is the permeability of free space, and Bis the Bohr magneton. It is easily seen from Fig. 2(a) that with the addition of ultrathin CCO, the damping parameter is drastically increased, i.e., >3 times its value in bare MAFO. Figure 2(b) shows that the damping enhancement seen in MAFO/CCO is essentially independent of the CCO thickness. This trend suggests that the damping enhancement is purely due to the MAFO/CCO interface, rather than spin dissipation in the bulk of CCO akin to the absorption of diusive spin current reported in antiferromagnetic NiO26,35,48. We note that other bulk magnetic properties of MAFO (e.g., eective magnetization, Land e g-factor, magnetocrystalline anisotropy) are not modied by the CCO overlayer in a detectable way. We also rule out eects from solvent cleaning prior to CCO growth or thermal cycling in the deposition chamber up to 300C, as subjecting bare MAFO to the same ex- situ cleaning and in-situ heating/cooling processes as described in Section II, but without CCO deposition, results in no measurable change in damping. The damping enhancement therefore evidently arises from the proximity of MAFO to the CCO overlayer. We consider two possible mechanisms at the MAFO/CCO interface for the observed damping enhancement: (1) Spin current excited by FMR in MAFO may be absorbed via spin transfer in an interfacial proximity-magnetized layer49of CCO, whose magnetic moments may not be completely aligned with those of MAFO. While CCO by itself is paramagnetic at room temperature, prior studies have shown that Co2+and Cr3+cations in epitaxial CCO interfaced with a spinel ferrite (e.g., Fe 3O4) can develop measurable magnetic order50. Such damping enhancement due to interfacial magnetic layer is analogous to spin dephasing reported for ferromagnets interfaced directly with proximity- magnetized paramagnetic metal (e.g., Pt, Pd)49. (2) Even if CCO does not develop proximity-induced magnetism, chemical disorder at the MAFO/CCO interface may enhance spin scattering. For instance, chemical disorder may lead to an increase of Fe2+ 0 10 20 300246810HWHM Linewidth (mT) Frequency (GHz)(a) (b)MAFO/CCO eff≈ 0.007 MAFO eff≈ 0.002 0 2 4 6 80.0000.0020.0040.0060.0080.0100.012eff CCO thickness (nm)Figure 2. (a) HWHM FMR linewidth versus frequency for MAFO(15 nm) and MAFO(15 nm)/CCO(1.3 nm). The eective Gilbert damping parameter effis derived from the linear t. (b) effplotted against the CCO overlayer thickness. The dashed horizontal line indicates the average of efffor MAFO without an overlayer.) cations at the MAFO surface, thereby increasing the spin-orbit spin scattering contribution to Gilbert damping in MAFO compared to its intrinsic composition dominated by Fe3+with weak spin-orbit coupling18,51. Another possibility is that chemical disorder at the MAFO/CCO interface introduces magnetic roughness that gives rise to additional spin scattering, perhaps similar to two-magnon scattering recently reported for ferromagnet/spin-orbit-metal systems52. In the following section, we directly examine interfacial proximity magnetism and chemical disorder to gain insight into the physical origin of the observed damping enhancement in MAFO/CCO. IV. CHARACTERIZATION OF INTERFACE CHEMISTRY AND MAGNETISM To evaluate the potential formation of a magnetized layer in the interfacial CCO through the magnetic proximity eect, we performed depth-resolved and element-specic magnetic characterization of MAFO/CCO bilayers using polarized neutron re ectometry (PNR) and soft magnetic X-ray spectroscopy. PNR measurements were performed using the PBR instrument at the NIST Center for Neutron Research on nominally 15-nm-thick MAFO layers capped with either thick (5 nm) or thin (3 nm)4 CCO overlayers. PNR measurements were performed in an in-plane applied eld of 3 T at temperatures of 300 K and 115 K, the latter case being slightly above the nominal 97 K Curie temperature of CCO41,42. Incident neutrons were spin-polarized parallel or anti-parallel to the applied eld both before and after scattering from the sample, and the re ected intensity was measured as a function of the perpendicular momentum transfer vector Q. The incident spin state of measured neutrons were retained after scattering, corresponding to the two non-spin- ip re ectivity cross sections ( ""and##). Since all layers of the lm are expected to saturate well below the applied eld of 3 T, no spin- ip re ectivity is expected and these cross sections were not measured. Since PNR is sensitive to the depth proles of the nuclear and magnetic scattering length density (SLD), the data can be tted to extract the chemical and magnetic depth proles of the heterostructure. In this case, we used the Re 1D software package for this purpose53. Figure 3(a,b) shows the 300 K re ectivities and spin asymmetry curves of a nominal MAFO (15 nm)/CCO (5 nm) sample alongside the depth prole (Fig. 3(c)) used to generate the ts shown. The best t prole (Fig. 3(c)) provides no evidence of a layer with proximity-induced magnetization in the CCO. Rather, we note that there appears to be a layer of magnetization suppression near both the MAO/MAFO and MAFO/CCO interfaces. Further, the interfacial roughnesses of both the MAO/MAFO and MAFO/CCO, 0.9(1) nm and 1.35(5) nm respectively, are signicantly larger than the CCO surface roughness of 0.27(3) nm and the bare MAFO surface roughness of <0.5 nm54. The interfacial roughnesses are signatures of chemical intermixing at the spinel-spinel interface leading to interfacial suppression of the magnetization and/or Curie temperature. Thus, we nd that the MAFO/CCO interface, although structurally coherent, exhibits a chemically intermixed region on the order of one spinel unit cell thick on either side. To obtain an upper limit of the proximity-induced interfacial magnetization in CCO, we performed Markov- chain Monte-carlo simulations as implemented in the DREAM algorithm of the BUMPS python package. These simulations suggest an upper limit (95% condence interval of) 7 emu/cc in the 1.5 nm of the CCO closest to the interface. In this case, the model evaluated the MAFO as a uniform structural slab but allowed for total or partial magnetization suppression at both interfaces, while the CCO layer was treated as a uniform slab with an allowed magnetization layer of variable thickness at the interface. However, we note that equivalently good ts are obtained using simpler models that t a single MAFO layer with magnetically dead layers at the interfaces and a completely nonmagnetic CCO layer. Equivalent results were obtained for the thick CCO sample at 115 K and for the thin CCO sample. We therefore conclude that the PNR results strongly favor a physical picture in which the Figure 3. (a) Spin-polarized neutron re ectivity and (b) spin asymmetry of a MAFO (15 nm)/CCO (5 nm) bilayer alongside theoretical ts. (c) Nuclear and magnetic scattering (scaled 10) length density prole used to generate the ts shown. Error bars represent 1 standard deviation. CCO is notmagnetized through the magnetic proximity eect. To conrm the PNR results and examine the eect of a CCO overlayer on the local environment of Fe cations in MAFO, we performed temperature-dependent X-ray absorption (XA) spectroscopy and X-ray magnetic circular dichroism (XMCD) measurements at Beamline 4.0.2 of the Advanced Light Source at Lawrence Berkeley National Laboratory. We note that the detection mode (total electron yield) used here for XA/XMCD is sensitive to the top 5 nm of the sample, such that Fe L edge signals from CCO-capped MAFO primarily capture the cation chemistry near the MAFO/CCO interface. Measurements were performed in an applied eld of 400 mT along the circularly polarized X-ray beam, incident at 30grazing from the lm plane. To minimize drift eects during the measurement, multiple successive energy scans were taken and averaged, switching both applied eld direction and photon helicity so that all four possible combinations of eld direction and helicity were captured at least once. XA and XMCD intensities were normalized such that the pre-edge is zero and the maximum value of the average of the (+) and () intensities is unity. In the case of the Co L- edge, measurements were taken with energy sweeps covering both Fe and Co edges, and for consistency both edges were normalized to the highest XAS signal, corresponding to the Fe L 3-edge. Figure 4(a) compares the XA of a bare MAFO lm5 Figure 4. (a) 300 K X-ray absorption spectra of MAFO and MAFO/CCO (3 nm) grown on MAO. (b) Photon helicity- dependent XA spectra and XMCD of the Fe L-edge for a MAFO/CCO (3 nm) bilayer at 300 K. (c) Co and (d) Cr L-edge XA and XMCD of the same bilayer. with one capped by 3 nm of CCO. The two XA lineshapes are nearly identical, indicating the same average Fe oxidation state and site-distribution in CCO-capped and uncapped MAFO lms. It is therefore likely that the reduced interfacial magnetization observed through PNR is a result of a defect-induced Curie temperature reduction, rather than preferential site-occupation of Co and Cr that might increase the Fe2+content in the intermixed interfacial region. We further note that although a large XMCD signal is observed on the Fe-edge at 300 K (Fig. 4(b)), neither the Co nor Cr L edges exhibit any signicant magnetic dichroism, as shown in Figs. 4(c)-(d). Similar results are obtained on the Cr L edge at 120 K. Consistent with the PNR results, we thus nd no evidence for a net magnetization induced in the CCO through the interfacial magnetic proximity eect. Our nding of suppressed interfacial magnetism in MAFO/CCO is reminiscent of earlier reports of magnetic dead layers in epitaxially-grown ferrite- based heterostructures55{57. For example, prior PNR experiments have revealed magnetic dead layers at the interfaces of ferrimagnetic spinel Fe 3O4and antiferromagnetic rock-salt NiO or CoO, even when the interfacial roughness is small (e.g., only 0.3 nm)55,56. A magnetic dead layer of 1 spinel unit cell has also been reported at the interface of Fe 3O4and diamagnetic rock-salt MgO grown by molecular beam epitaxy57. We note that in these prior studies, the spinel ferritelms interfaced with the rock salts (NiO, CoO, MgO) possess antiphase boundaries. Suppressed magnetism is known to result from antiphase boundaries, as they frustrate the long-range magnetic order and reduce the net magnetization of the ferrite44. By contrast, there is no evidence for antiphase boundaries in all- spinel MAFO/CCO grown on spinel MAO; therefore, the suppressed magnetism at the MAFO/CCO interface cannot be attributed to antiphase-boundary-induced magnetic frustration. Another possible scenario is that magnetic dead layer formation is a fundamental consequence of the charge imbalance between dierent lattice planes, as recently shown in a recent report of (polar) Fe 3O4undergoing atomic reconstruction to avoid \polar catastrophe" when grown on (nonpolar) MgO58. In our study on all- spinel heterostructures, there may also be some degree of charge mismatch depending on the relative populations of cations on the tetrahedrally- and octahedrally- coordinated sites at the MAFO/CCO interface, although the charge mismatch is expected to be only 1, i.e., a factor of5-6 smaller than that in MgO/Fe 3O458. Thus, atomic reconstruction driven by charge imbalance appears unlikely as a dominant source of the magnetic dead layer in MAFO/CCO. We instead tentatively attribute the dead layer to atomic intermixing driven by diusion across the MAFO/CCO interface during CCO overlayer deposition. V. DISCUSSION Our PNR and XA/XMCD results (Section IV) indicate that the damping enhancement observed in Section III arises from chemical disorder, rather than proximity- induced magnetism, at the MAFO/CCO interface. We emphasize that this interfacial disordered layer is conned to within 2 spinel unit cells. We also note that this interfacial disorder is due to atomic intermixing, but not structural defects (e.g., dislocations, antiphase boundaries), in this coherent bilayer system of MAFO/CCO. Nevertheless, this ultrathin chemically disordered layer alone is evidently sucient to signicantly increase spin scattering. Considering that the cation chemistry of Fe in MAFO does not change substantially (Fig. 4(a)), the interfacial spin scattering is likely driven by magnetic roughness, leading to a mechanism similar to two-magnon scattering that accounts for a large fraction of eective damping in metallic ferromagnet/Pt bilayers52. We now put in context the magnitude of the damping enhancement
eff, i.e., the dierence in the eective Gilbert damping parameter between CCO-capped and bare MAFO,
eff=bilayer effferrite eff; (2) by comparing it with ferrite/spin-orbit-metal systems where spin pumping is often considered as the source6 0.0000.0020.0040.0060.008 MAFO/CCO [this study] MAFO/W [Riddiford] MAFO/Pt [Riddiford]YIG/Pt [Wang]Daeff YIG/Pt [Sun] Figure 5. Comparison of the enhancement of the eective Gilbert damping parameter
efffor MAFO/CCO and ferrite/spin-orbit-metal bilayers. YIG/Pt [Sun], YIG/Pt [Wang], and MAFO/Pt(W) [Riddiford] are adapted from Refs.59,60, and61respectively. The values of
efffrom the literature are normalized for the saturation magnetization of 100 kA/m and magnetic thickness of 15 nm for direct comparison with our MAFO/CCO result. of damping enhancement. Since damping enhancement from spin pumping or interfacial scattering scales inversely with the product of the saturation of magnetization Msand the magnetic layer thickness tm, the values of
efftaken from the literature59{61are normalized for direct comparison with the MAFO lms studied here with Ms= 100 kA/m and tm= 15 nm. As summarized in Fig. 5,
efffor MAFO/CCO is comparable to { or even greater than {
eff for ferrite/metal bilayers. This nding highlights that the strength of increased spin scattering in a ferrite due to interfacial chemical disorder can be on par with spin dissipation due to spin pumping in metallic spin sinks. More generally, this nding suggests that special care may be required in directly relating
eff to spin pumping across bilayer interfaces (i.e., spin- mixing conductance52), particularly when the FMR- driven magnetic layer is directly interfaced with a spin scatterer. Furthermore, the strong interfacial spin scattering { even when the oxide interface is structurally coherent and the chemically disordered layer is kept to just <2 unit cells { poses a signicant challenge for maintaining low damping in ferrite/insulator heterostructures. This challenge is partially analogous to the problem of reduced spin polarization in tunnel junctions consisting of spinelFe3O4and oxide barriers (e.g., MgO)62{65, which is also likely due to interfacial chemical disorder and magnetic dead layers. However, we emphasize that the problems of antiphase boundaries43{46and charge-imbalance-driven atomic reconstruction58, which have posed intrinsic challenges for devices with MgO/Fe 3O4interfaces, are likely not applicable to all-spinel MAFO/CCO. It is therefore possible that deposition schemes that yield sharper interfaces, e.g., molecular beam epitaxy, can be employed to reduce interfacial imperfections and hence spin scattering at MAFO/CCO for low-loss all-oxide device structures. VI. CONCLUSIONS We have shown that eective damping in epitaxial spinel MgAl-ferrite (MAFO) increases more than threefold when interfaced coherently with an insulating paramagnetic spinel of CoCr 2O4(CCO). This damping enhancement is not due to spin pumping into the bulk of CCO. Our depth-resolved characterization of MAFO/CCO bilayers also reveals no proximity-induced magnetization in CCO or signicant change in the cation chemistry of MAFO. We attribute the giant damping enhancement to spin scattering in an ultrathin chemically disordered layer, conned to within 2 spinel unit cells across the MAFO/CCO interface. Our results demonstrate that spin dynamics in ferrite thin lms are strongly impacted by interfacial disorder. Acknowledgements - This work was supported in part by the Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Oce of the Assistant Secretary of Defense for Research and Engineering and funded by the Oce of Naval Research through grant no. N00014-15-1-0045. J.J.W. was supported by the U.S. Department of Energy, Director, Oce of Science, Oce of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract No. DESC0008505. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-1542152. This research used resources of the Advanced Light Source, which is a DOE Oce of Science User Facility under contract no. DE-AC02-05CH11231. We thank Brian J. Kirby for technical assistance on PNR analysis. semori@vt.edu 1A. Homann and S. D. Bader, Opportunities at the Frontiers of Spintronics, Phys. Rev. Appl. 4, 047001 (2015). 2A. V. Chumak, V. I. Vasyuchka, A. A. 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2019-08-23

High-quality epitaxial ferrites, such as low-damping MgAl-ferrite (MAFO), are promising nanoscale building blocks for all-oxide heterostructures driven by pure spin current. However, the impact of oxide interfaces on spin dynamics in such heterostructures remains an open question. Here, we investigate the spin dynamics and chemical and magnetic depth profiles of 15-nm-thick MAFO coherently interfaced with an isostructural $\approx$1-8-nm-thick overlayer of paramagnetic CoCr$_2$O$_4$ (CCO) as an all-oxide model system. Compared to MAFO without an overlayer, effective Gilbert damping in MAFO/CCO is enhanced by a factor of $>$3, irrespective of the CCO overlayer thickness. We attribute this damping enhancement to spin scattering at the $\sim$1-nm-thick chemically disordered layer at the MAFO/CCO interface, rather than spin pumping or proximity-induced magnetism. Our results indicate that damping in ferrite-based heterostructures is strongly influenced by interfacial chemical disorder, even if the thickness of the disordered layer is a small fraction of the ferrite thickness.

Damping enhancement in coherent ferrite/insulating-paramagnet bilayers

1908.08629v2

arXiv:1606.06610v1 [hep-th] 21 Jun 2016Torsion Effects and LLG Equation. Cristine N. Ferreira†a, Cresus F. L. Godinho‡b, J. A. Helay¨ el Neto∗c †N´ ucleo de Estudos em F´ ısica, Instituto Federal de Educa¸ c ˜ ao, Ciˆ encia e Tecnologia Fluminense, Rua Dr. Siqueira 273, Campos dos Goytacazes, 28 030-130 RJ, Brazil ‡Grupo de F´ ısica Te´ orica, Departamento de F´ ısica, Univer sidade Federal Rural do Rio de Janeiro, BR 465-07, 23890-971, Serop´ edica, RJ, Brazil ∗Centro Brasileiro de Pesquisas F´ ısicas (CBPF), Rua Dr. Xav ier Sigaud 150, Urca, 22290-180, Rio de Janeiro, Brazil Abstract Based on the non-relativistic regime of the Dirac equation coupled to a torsion pseudo-vector, we study the dynamics of magnetization and how it is affected by the presence of torsion. We consider that torsion interacting terms in Dirac equation appear in two ways one of these is thhrough the covariant derivativ e considering the spin connection and gauge magnetic field and the other is through a n on-minimal spin torsion coupling. We show within this framework, that it is possible to obtain the most general Landau, Lifshitz and Gilbert (LLG) equation includ ing the torsion effects, where we refer to torsion as a geometric field playing an impo rtant role in the spin coupling process. We show that the torsion terms can give us tw o important landscapes in the magnetization dynamics: one of them related with d amping and the other related with the screw dislocation that give us a global effect lik e a helix damping sharped. These terms are responsible for changes in the magnetiz ation precession dynamics. acrisnfer@iff.edu.br ,bcrgodinho@ufrrj.br ,chelayel@cbpf.br 11 Introduction The discovery of the graphene-like systems and topological insulators systems introduced a new dynamic in the applications of the framework of the high e nergy physics in low energy systems in special condensed matter systems. The fact that t hese systems can be described by Dirac equations give us new possibilities for theoretica l and experimental applications. In this direction there are some effects in condensed matter sy stems still without a full description as the magnetizable systems that we described i n this work. In this sense the constructionofthetheoreticalframeworksthatcan,insom elimit, beobtainedinlowenergy systems is the crucial importance to understand the invaria nces and interactions in certain limits. One of the important effects that we can study is relate d with the spin systems. So, in this work we deal with the new framework to study the spi n systems considering the Dirac equation in non-relativistic limit with torsion i nteraction[1]. Spin systems are generally connected with magnetic systems. It is well known that spin angular momentum isanintrinsicpropertyofquantumsystems. Whenamagnetic fieldisapplied, eachmaterial presents some level of magnetization, and Quantum Mechanic s says that magnetization is related to the expectation value of the spin angular momentu m operator. In the case of ferromagnetic materials, they can have a large magnetiza tion even under the action of a small magnetic field and the magnetization process is alw ays followed by hysteresis, and the magnetization is uniform and lined up with the magnet ic field, usually these materials exhibit a strong ordering process that results in a parallel line up the spins[2]. In materials graphene type we also can generate magnetic momen t. In this form it is possible to study the transport phenomena [3]. Overlapping between e lectronic wave functions are interactions well understood, again thanks to Quantum Mech anics, however thereare other kinds of interactions occurring such as magnetocrystallin e anisotropy, connected with the temperature dependence[4] and demagnetization fields [5], acting in low range. In such systems if we only consider the precession we will not reach t he right limit. Certainly, the precession equation has to include a damping term providing the magnetization alignment with the magnetic field after a finite time [6]. In order to simu late these phenomena, several physical models have been presented. However, the L andau-Lifshitz model is still the one widely used in the description of the dynamics of ferr omagnetic media. In their pioneering work [7] in 1935, Landau and Lifshitz proposed a n ew theory based on the following dynamical equation: ∂tM=/vectorHeff×/vectorM+α M2s/vectorM×(/vectorM×/vectorHeff), (1) where/vectorHeffdenote an effective magnetic field, with the gyromagnetic rati o absorbed, inter- acting with the magnetization M=|/vectorM|. The first term is the precession of the magnetiza- tion vector around the direction of the effective magnetic fiel d and the second one describes a damping of the dynamics. With this theory we are able to comp ute the thickness of walls between magnetic domains, and also understand the domain fo rmation in ferromagnetic 2materials. This theory, which now goes under the name of micr omagnetics, has been in- strumental in the understanding and development of magneti c memories. Landau and Lifshitz considered the Gibbs energy G of a magnetic materia l to be composed of three terms: exchange, anisotropy and Zeeman energies (due to the external magnetic field), and postulated that the observed magnetization per unit volume M field would correspond to a local minimum of the Gibbs energy. Later researchers added other terms to G such as magnetoelastic energy and demagnetization energy. They al so derived the Landau Lifshitz (LL) equation using only physical arguments and not using th e calculus of variations. In subsequent work, Gilbert [8] realized a more convincing for m for the damping term, based on a variational approach, and the new combined form was then called Landau Lifshitz Gilbert (LLG) equation, today it is a fundamental dynamic sy stem in applied magnetism. Nowadays, the scientific and technological advances provid e a wide spectrum of ma- nipulations to the spin degrees of freedom. The complete for mulation for magnetization dynamics also include the excitation of magnons and their in teraction with other degrees of freedom, that remains as a challenge for modern theory of m agnetism [9]. These amaz- ing and reliable kinds of procedures are propelling spintro nics as a consolidated sub-area of Condensed Matter Physics [10]. Since the experimental ad vances are increasingly pro- viding high-precision data, many theoretical works are bei ng presented [11, 14, 12, 13] and including strange materials, as the topological insula tors, connected with the mag- netocondutivity [15] and graphene like structures [16] for a deeper understanding of the phenomenon including the spin polarization super currents for spintronics [17], holographic understanding of spin transport phenomena [18] and non-rel ativistic background [19]. The torsion field appears as one of the most natural extension s of General Relativity along with the metric tensor, which couples to the energy-mo mentum distribution, inspects the details of the spin density tensor. Actually, in General Relativity, fermions naturally couple to torsion by means of their spin. In this work we consider that the torsion interacts with the m atter in two types one of these is present in covariant derivative that contains the s pin connection related with the Christoffell symbol given by the metric of the curve space time and the contorsion given by the torsion that have two antisymmetric index. The other c ontribution is given by the non minimal spin torsion coupling that is important to the co nsistence of the theory. It is possible to study the non relativistic approach to the torsi on in connection with the spin particles [20], in this work we only consider the torsion con tribution considering the plane space-time. Our work is organized as follows, in Section 2, we analyse the torsion coupling in relativistic limit, in Section 3 the modified version of Paul i equation is presented. Our approach starts with a field theoretical action where a Dirac fermion is non-minimally coupled in the presence of a torsion term, a low relativistic approximation is considered and the equivalent Pauli equation is then obtained. We deriv e a very similar expression to the Landau-Lifshitz-Gilbert (LLG) equation from our Pauli equation with torsion. Under specific conditions for the magnetic moment, we show that the LLG equation can be 3established with damping and dislocations terms. 2 TheRelativistic andNon- Relativistic Discussions for Sp in Coupling with Torsion In this Section, let us understand the way to describe the spi n interaction by taking into account the torsion coupling. In our framework there are two terms in Dirac action, one of these is connected with spin current effect from the spin con nection, the other is a non- minimal spin torsion coupling whose effects are the subject of this work. Dirac’s equation is relativistic and we should justify why we use it in our mode l. Dispersion relations in Condensed Matter Physics (CMP) linear in the velocity appea r in a wide class of models and one adopts the framework of Dirac’s equation to approach them. However, the speed of light, c, is suitably replaced by the Fermi velocity, vF. Here, this is not what we are doing. We actually start off from the Dirac’s equation and we t ake, to match with effects of CMP, the non-relativistic regime, for the electron moves with velocities v≤c 300. So, contrary to an analogue model where we describe the phenomen a by a sort of relativity with c replaced by vF, we here consider that the non-relativistic electrons of ou r system is a remnant of a more fundamental relativistic world. The non- relativistic limit is also more complete because it brings effects that do not directly appear in Galilean Physics. This is why we have taken the viewpoint of associating our physics to the Dirac’s equation. 2.1 The Dirac Model for Torsion and the Spin Current Interpre tation In this sub-section, we consider the microscopic discussio n that gives the explicit form of the spin current in function of the gauge potential and torsi on coupling. The scenario we are setting up is justified by the following chain of argument s: (i) We are interested in spin effects. We assume that there is a space-time structures ( torsion) whose coupling with the matter spin becomes relevant. But, we are actually inter ested in the possible non- relativistic effects stemming from this coupling, which is mi nimal and taken into account in the covariant derivative though the spin connection. (ii ) The other point we consider is that, amongst the three irreducible torsion components, its pseudo-vector piece is the only one that couples to the charged leptons. Then, with this results in mind, we realize that the electron spin density may non-minimally couple, in a Pauli- like interaction, to the field-strength of the torsion pseudo-vector degree of fr eedom. So, our scenario is based on the relevant role space-time tor sion, here modeled by a pseudo-vector, may place in the non-relativistic electron s of spin systems in CMP. The spin current that we talk about is the spin magnetic moment and in g eneral is not conserved alone. Thequantity that is conserved is the total magnetic m oment that is the composition between both /vectorJ=/vectorJS+/vectorJL. In form that ∂µJµ= 0 where the spin current can be defined in related to the three component spin current as Jµ S=ǫµab ρJρ aband/vectorJLis the spin orbit 4coupling . In analogy with the charge current, defined by the d erivative of the action in relation to the gauge field Aµwe used the definition where the spin current is the derivativ e of the action in relation with the spin connection. We consid er here the spin current as the derivative of the action in relation to the spin connecti onωab µthen the spin current is given by Jµ ab=δS δωabµ, (2) whereωab µis ωab µ=ea ν∇µeνb+ea λΓλ µνeνb+ea λKλ µνeνb, (3) andKλ µνis the contorsion given by Kλ µν=−1 2(Tλ µ ν+Tλ ν µ−Tλ µν), (4) whereTλ µνis the torsion. In this work we consider two terms for torsion , on of these is the totally anti symmetric tensor, that respecting the duality relation given by Tµνλ=ǫµνλρSρ whereSρis the pseudo-vector part of torsion.1The other term that we consider is the 2- form tensor TµνwhereTµν=∂µSν−∂νSµthis term is analog to the field strength of the electromagnetic gauge potential Aµthat in our case is changed to pseudo-vector Sµ. The invariant fermionic action that contained these contribut ions for torsion is given by S=/integraldisplay d4xi¯ψ(γµDµ+λTµνΣµν+m)ψ, (5) where the covariant derivative is Dµ=Dµ−iηωab µΣabthat contain the covariant gauge derivativeDµ=∂µ−ieAµand the spin connection covariant derivative. We consider the flat space-time where the only contribution f or the spin connection is the contortion. In this form we have a spin current given by Jµ ab=1 2¯ψγµΣabψ. (6) We consider the ansatz where ωab µcontain the total antisymmetric part of the contorsion given by ωκλ µ=µǫκλρ µSρ. (7) We used the splitting γµΣκλ=ǫµ κλργργ5+δµ κγλ−δµ λγκThat give us the current in the form Jµ αλ=δS δΓabµ=1 2ǫµ αλρ¯ψγργ5ψ=Ji jk+J0 ij (8) 1Considering space times with torsion Tα βγ, the afine connection is not symmetric, Tα βγ= Γα βγ−Γα γβ, and we can split it into three irreducible components, where one of them is the pseudo-trace Sκ=1 6ǫαβγκTαβγ. 5where the currents are Ji jk=1 2ǫi jk¯ψγ0γ5ψ; (9) J0 ij=1 2ǫijk¯ψγkγ5ψ, (10) then the current part of the action coming from the covariant derivative is given by Scurr=−η/integraldisplay d4xγµγ5Sµ=η/integraldisplay d4x/vectorS·/vectorJ. (11) The action (5) considers the temporal component of the torsi on pseudo-vector S0= 0, we have can be written as S=/integraldisplay d4xi¯ψ/parenleftBig γµ∂µ+igγµAµ+iηγµγ5Sµ+λTµνΣµν+m/parenrightBig ψ, (12) Our sort of gravity background does not exhibit metric fluctu ations. The space-time is taken to be flat, and we propose a scenario such that the type of gravitational background is parametrized by the torsion pseudo-trace Sµ, whose origin may be traced back to one geometrical defect. 2.2 Dirac equation in presence of torsion Now, we discuss the Dirac equation given by eq. (12). From the action above, taking the variation with respect to ( δS/δ¯ψ), a modified Dirac’s equation reads as below: [iγµ∂µ−ηγµγ5Sµ−eγµAµ+λΣµν∂µSν+m]ψ= 0. (13) For a vanishing λ−parameter, the equation (13) has been carefully studied in [ 21, 22, 23, 24, 26]. The generation, manipulation, and detection of a spin curre nt, as well as the flow of electron spins, are the main challenges in the field of spintr onics, which involves the study of active control and manipulation of the spin degree of free dom in solid-state systems, [27, 10, 28]. A spin current interacts with magnetization by exchanging the spin-angular momentum, enabling the direct manipulation of magnetizati on without using magnetic fields [29, 30]. The interaction between spin currents and ma gnetization provides also a method for spin current generation from magnetization pre cession, which is the spin pumping [31, 32]. We showed in last Sections that there are tw o type of deformations induced by the torsion. Both of these can be generating a spin current. After a suitable separation of components ( µ= 0,1,2,3), the equation of motion can be written as, i∂tψ=iαi∂iψ+ηγ5S0ψ−ηαiγ5Siψ+eA0ψ+ −eαiAiψ−iλ 4ǫijkβγ5αk∂iSjψ+βmψ. (14) 6Definingagauge-invariant momentum, πj=i∂j−eAj, andusingthatΣ ij=−i 4ǫijkγ5αk withγ0=β, the effective Hamiltonian takes the form, H=αkπk+ηγ5S0−ηαkγ5Sk+eA0+ −iλ 4ǫijkβγ5αk∂iSj+βm. (15) where we have the matricial definitions: αi=/parenleftbigg0σi σi0/parenrightbigg ,γ5=/parenleftbigg0 1 1 0/parenrightbigg ,β=/parenleftbigg1 0 0−1/parenrightbigg , (16) In the Heisenberg picture, the position, /vector x, and momentum, /vector π, operators obey two different kinds of relations; we consider the torsion as a func tion of position only, S=S(/vector x), so that ˙/vector x=/vector α ˙/vector π=e(/vector α×/vectorB)+e/vectorE+ηγ5∂ ∂x(/vector α·/vectorS)ˆx. (17) One reproduces the usual relation for ˙/vector x, while the equation for ˙/vector πpresents a new term apparently giving some tiny correction to the Lorentz force . However, if we consider the torsion in a broader context, now as a momentum- and position-dependent background field, S=S(/vector x,/vectork), we have to deal with the following picture, ˙/vector x=/vector α−ηγ5∂ ∂k(/vector α·/vectorS)ˆk ˙/vector π=e(/vector α×/vectorB)+e/vectorE+ηγ5∂ ∂x(/vector α·/vectorS)ˆx+ +eηγ5∂A ∂x∂ ∂k(/vector α·/vectorS)ˆk. (18) The two sets of dynamical equations above are clearly showin g us the small corrections inducedbythetorsionterm. Nevertheless, wearestill here intherelativistic domain, andit is necessary change this framework for a better understandi ng of the SHE phenomenology. For this reason, in next Section we are going to approach the s ystem by going over into its non-relativistic regimen 2.3 Non-Relativistic Approach with torsion In this sub-section, we consider the Dirac equation in its no n-relativistic limit. One im- portant requirement for the Dirac equation is that it reprod uces what we know from non- relativistic quantum mechanics. We can show that, in the non -relativistic limit, two com- ponents of the Dirac spinor are large and two are quite small. To make contact with the 7non-relativistic description , we go back to the equations w ritten in terms of ϕandχof the four component spinor ψ=eimt√ 2m/parenleftbiggϕ χ/parenrightbigg , just prior to the introduction of the γma- trices. we obtain two equation on of these for ϕand the other for χ. We can solved in χand substiuted in the Dirac equation given by (12) and take th e non-relativistic regi- men (|/vector p|<< m). So, in this physical landscape, from now and hereafter, ou r goal is to consider a low-relativistic approximation based on an exte nded Pauli equation version by including torsion as presented before. Employing the Hamil tonian (15), we could carry out our calculations in the framework of the Fouldy-Wouthuy sen transformations; however for the sake of our approximation at lowest order in v/c, we take that SHE is adequately well described by the low-relativistic Pauli equation. We a re considering that the electron velocities are in the range of Fermi’s velocity. In this case , we arrive at the version given below for the Pauli’s equation: i∂ϕ ∂t=/bracketleftBig(/vector p−e/vectorA)2 2m−e 2m(/vector σeff·/vectorB)+eA0−(σ·/vectorSeff)+ −λ 8m(/vector∇×/vectorS)·(/vector σ×/vector p)+iλ 8m(/vector∇×/vectorS)·/vector p+ −eλ 8m(/vector∇×/vectorS)·(/vector σ×/vectorA)+ieλ 8m(/vector∇×/vectorS)·/vectorA/bracketrightBig ϕ. (19) The equation above displays the usual Pauli terms, but corre cted by new terms due to the torsion coupling. The second and the fourth contributio ns in the RHS of eq.(19) can be thought of as effective terms for /vector σand/vectorS, respectively given by σeff=/vector σ+η 2m/vectorS+iη 2m(/vector σ×/vectorS) (20) /vectorSeff=η/vectorS+iλ 4(/vector∇×/vectorS). (21) The fifth contribution in the RHS is proportional to the Rashb a SO coupling term; this term yields an important effect on the behavior of spin. 3 From the Modified Pauli Equation to Unfold in LLG In this Section, we consider the magnetization equation der ivation given by Dirac non- relativistic limit take into account the presence of torsio n. Let us start by considering the modified Pauli equation eq.(19) and find the magnetization eq uation. By using the Landau gauge/vectorA=H/vector xand taking that /vector x·/vector σ= 0 (the spins are aligned orthogonally to the plane of motion), give us the Hamiltonian of the full system in the n on-relativistic limit as: 8H=(/vector p−e/vectorA)2 2m−e 2m(/vector σeff·/vectorB)+eA0−(σ·/vectorSeff)−λ 8m(/vector∇×/vectorS)·(/vector σ×/vector p)+iλ 8m(/vector∇×/vectorS)·/vector p, (22) with/vectorB(t) =µ0/vectorH(t), whereµ0is the gyromagnetic ratio. The Pauli equation associated with (22) reads as follows below i∂ϕ ∂t=/bracketleftBig(/vector p−eA)2 2m−e 2m(/vector σeff·/vectorH)+eA0−(/vector σ·/vectorSeff)+ −λ 8m(/vector∇×/vectorS)·(/vector σ×/vector p))+iλ 8m(/vector∇×/vectorS)·p/bracketrightBig ϕ. (23) Let us consider the magnetization vector equation related w in the spin magnetic mo- ment/vector µ=e 2m/vector σ. In our approach we consider the magnetization is defined by /vectorM= (/vector µϕ)†ϕ−ϕ†(/vector µϕ) whereϕgiven by Pauli equation (23) and ϕ†ϕ= 1 and/vectorˆSϕ=/vectorSϕ, with the notation/vectorˆSis a torsion operator and /vectorSis the torsion autovalue. We have, by the manipulation of Pauli equation the magnetization equat ion associated with a fermionic state when we applied a external magnetic field /vectorHconsidering the Pauli product algebra as1 2(σiσj−σjσi) =iǫijkσkand1 2(σiσj+σjσi) =δij. The magnetization equation that arrive that is ∂/vectorM ∂t=/vectorM×/vectorH+η/vectorM×/vectorS+β(/vectorM×/vectorL) + +ηe 2m2(/vectorS×/vectorH) +eλ 2m(/vector∇×/vectorS), (24) with the magnetic moment given by /vectorL=/vector r×/vector pand/vectorSas the torsion pseudo-vector. We can observed that there are two terms that arrived by the covaria nt derivative Dµdefined by the coupling constant ηand the other is the parameter that arrived by non-minimal sp in torsion coupling with the coupling constant λ. Where the effect of the new terms given whenη/ne}ationslash= 0 andλ/ne}ationslash= 0. We consider the scalar product of the magnetization /vectorM, the magnetic field /vectorHand the torsion pseudo-vector /vectorSwith the equation (28) and we obtain2 ∂t/bracketleftBig1 2(/vectorM·/vectorM)/bracketrightBig =ηe 2m2/bracketleftBig /vectorM·(/vectorS×/vectorH)+λm η/vectorM·(/vector∇×/vectorS)/bracketrightBig ; (25) 2We used the vectorial relating given by A·(B×C) =B·(C×A) =C·(A×B) the other is A×(B×C) = (A·C)B−(A·B)Cand∇·(A×B) =B·(∇×A)−A·(∇×B). 9∂t/bracketleftBig1 2(/vectorH·/vectorM)/bracketrightBig =/bracketleftBig η/vectorM·(/vectorS×/vectorH)+β/vectorM·(/vectorL×/vectorH) + +e 2mλ/vectorH·(/vector∇×/vectorS)/bracketrightBig ; (26) ∂t/bracketleftBig1 2(/vectorS·/vectorM)/bracketrightBig =/vectorM·(/vectorS×/vectorH)+β/vectorM·(/vectorL×/vectorH). (27) ∂t/bracketleftBig1 2(/vectorL·/vectorM)/bracketrightBig =/vectorM·(/vectorL×/vectorH)+η/vectorM·(/vectorL×/vectorS) + +ηe 2m2/vectorS·(/vectorL×/vectorH)+eλ 2m/vectorL·(/vector∇×/vectorS),. (28) With the equations dysplayed in (25)-(27), it is possible to inspect the general behavior of the magnitude of the magnetization, /vectorM, that precesses around the magnetic field, /vectorH. In our framework, the magnetization also precesses around the torsion vector /vectorM·/vectorS. Without torsion, we have∂/vectorM ∂t=/vectorM×/vectorH, so that/vectorM·/vectorM= constant and /vectorM·/vectorH= constant as in the usual case of the electron under the action of a time-depende nt external magnetic field, with the Zeeman term given by the Hamiltonian HM=/vectorM·/vectorH. 3.1 Planar torsion analysis with damping Here, we intend to analyze some possibilities of solutions t o the magnetization that respect the conditions given by (25)-( 27). The magnitude of the magn etization is not constant in general, as we can see in equation (25), but, if this quantity is constant, there comes out a constraint given by /vectorM·(/vectorS×/vectorH) =−λm η/vectorM·(/vector∇×/vectorS). (29) If we considerd(/vectorL·/vectorM) dt= 0, we have /vectorS·(/vectorL×/vectorH) =−mλ η/vectorL·(/vector∇×/vectorS). (30) This expression describes us the case where /vectorM·/vectorH/ne}ationslash= 0 and/vectorS·/vectorM/ne}ationslash= 0; then, there is a the damping angle in both directions given by the precession aro und the magnetic field /vectorHand around the torsion pseudo-vector /vectorS: ∂t/bracketleftBig1 2(/vectorH·/vectorM)/bracketrightBig =λm(e 2m2/vectorH−/vectorM)·(/vector∇×/vectorS); (31) ∂t/bracketleftBig1 2(/vectorS·/vectorM)/bracketrightBig =−λm η/vectorM·(/vector∇×/vectorS). (32) 10Let us consider the first proposal in a very particular and ver y simple case for a planar torsion field, /vectorS=1 2χ(xˆy−yˆx); this choice allows us to realize the curl of torsion as an effective magnetic field, /vector∇×/vectorS=/vectorBeff=χˆz. If we pick up the configuration of Fig. 1, we find the relation of the angle between the magnetic field /vectorHand the magnetization /vectorM. Figure 1: Magnetization vector rotating around the magneti c field/vectorHwith damping given by the dynamics of the angle ζ. The system {M,H}rotates around the vector /vectorSin the xy- plane also with damping given by the angle φ. We consider φ=ωφtwithωφ=ωθ=2λmχ ηS; with this configuration ζ=ωζt=λχm ηHM/parenleftBig H+ηM/parenrightBig t. We started off by discussing the case where the torsion is plan ar with the magnitude of the magnetization being constant, /vectorM·/vectorM= 0, and ∂t/bracketleftBig1 2(/vectorS·/vectorM)/bracketrightBig =−λχm η/vectorM·/vector z,. (33) this gives us the magnetic momentum precession around the to rsion. For consistency, we show that this result is compatible with the equation ∂t/bracketleftBig1 2(/vectorH·/vectorM)/bracketrightBig =λmχ(e 2m2/vectorH−/vectorM)·/vector z. (34) In the case of the Fig. 1, the magnetization precesses around the magnetic field and around the planar torsion vector both with damping. 113.2 Helix-Damping Sharped Effect in a Planar Torsion Configur ation Now, let us consider the most general case, where the magnitu de of the magnetization is not constant, but with ( /vectorL·/vectorM) = 0. The configuration is considered in Fig. 2. With the Figure 2: In this picture we show the effect of the torsion in mag netization dynamics. The green vector is the magnetization vector and the blue vector is the external magnetic field. expressions (25)-(27), we can readily write the magnitude o f the magnetization ∂t/bracketleftBig1 2(/vectorM·/vectorM)/bracketrightBig =eη 2m2/bracketleftBig ∂t/bracketleftBig1 2(/vectorS·/vectorM)/bracketrightBig +λm η/vectorM·(/vector∇×/vectorS)/bracketrightBig . (35) By using of the equation (35) and considering ∂t/bracketleftBig 1 2(/vectorS·/vectorM)/bracketrightBig /ne}ationslash= 0, we can see, the example of the Fig. 3, that ∂t/bracketleftBig 1 2(/vectorM·/vectorM)/bracketrightBig /ne}ationslash= 0. This possibility gives us that the magnitude of magnetization is not constant, as in the usual LLG. This effect is the effect of torsion that gives us that the rotational lines do not return around thems elves. Equation (35 ) does not involve the explicit dependence of th e magnetic field. We choose to work out the equation ∂t/bracketleftBig1 2(/vectorHeff·/vectorM)/bracketrightBig =e 2mλχ/vectorH·/vector z, (36) where/vectorHeff=/vectorH−η/vectorSgives us the explicit form of the magnetic field interaction w ith the 12Figure 3: In this draw we show the effect of the torsion in magnet ization dynamics. The green vector is the magnetization vector and the blue vector is the external magnetic field. In this representation we used |/vectorM|=M(t),|/vectorS|= constant and |/vectorH|= constant with ωθ=2mλχ ηSthenM(t) =λeχ ωθmsinωθt. magnetization. We notice that this quantity is different from zero, then the an gle between the magnetic field and the magnetization is not constant; this yields us th e damping precessing effect of the magnetization vector around the magnetic field. The comp osition between these two effects, dislocation and damping, is what we refer to as the hel ix-sharped with damping, effect where the damping effect can be see in Fig. 4. We can show tha t there are two magnetization effects: the damping given by the longitudinal magnetization function m(t)l and dislocations given by the longitudinal magnetization f unctionm(t)tas we can see in Fig. 2. The trajectory of the magnetization is the conical in creasing spiral, where the modulus of magnetization increases with the time. In the tor sion plan the behavior is given by Fig3. 13Figure 4: Damping behavior in torsion plane. Show the behavi or of theφ=ωφtdynamic. 4 Concluding Remarks In this work, we have considered that the magnetization equa tion is a non-relativistic remnant of the non-relativistic limit of the Dirac equation with torsion couplings. We have considered two types of couplings: one of these related with the spin current in Dirac equation, defined by the spin connection. When we derived the action in relation with the spin connection we obtain the spin current, this descrip tion is analog to the charged current when we have the derivation of the action in relation with the gauge field. Werefertotheothertermasthenon-minimaltorsiontermand Itgives ustherotational of the torsion. We have analyzed this term in the general cont ext and observed that it is possible to recover the Landau Lifshitz in the case were the t orsion is zero. Then, we can point out that the non-relativistic limit of the Dirac equat ion reproduces the usual case where the magnetization vector precesses around the magnet ic field. When we introduce the torsion terms we analyze, in the general regime the magni tude of magnetization /vectorM·/vectorM, the precession of the magnetization around the magnetic fiel d/vectorH·/vectorM, and the precession of the magnetization around the torsion pseudo-vector /vectorS·/vectorMis not constant. When the magnitude of the magnetization is constant, in the c ase where the torsion is planar, there are two possible magnetization precessions o ne around the magnetic field and other around the planar torsion pseudo-vector. In both dyna mics, there occurs damping. An interesting example has been analyzed in Fig. 1, where we s how that it is possible to realized an apparatus in some experimental device. In thi s sense, our framework can reproduce the LLG equation. The most general approach shoul d consider that the mag- nitude of the magnetization is not constant. In this case, as we can see from Fig. 2, the loop drawn by the magnetization damping but the is not remain in the same plane. This 14effect is typically a torsion effect, were the lines are not close d. This effect seems to be like a dislocation in the material that presents topologica l defects like solitons and vortices. Both dislocation and damping give us what we refer to as the he lix-damping sharp effect, wich is a new feature of the models with torsion[33]. We can observethat this resultis thenewfeatureintroduced bytheplanartorsion, ifwe consider the comparation with dampingand dislocations ter ms presented in LLG equation. This may help in the task of setting up new apparatuses and may be experimental purposes to explore such characteristics in this phenomenon. We have found that /vectorM·/vectorM= constant, its consequence is the dislocation effect. The damping effect is the usual one, where the angle dynamic can crease and decrease with the time. In this w ork we does not study the polarization of the spins that is subject of next work when we will consider these systems in terms or the spinup and spin down dynamic. In the literature, this effect is named pumped spin current [34], and we shall study the possibility of this current when the system is in a helix-sharped configuration[35]. References [1] A. Dyrdal, J. Barnas, Phys. Rev. B 92, 165404 (2015) [2] Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi, Phys. Rev . Lett.114, 016603 (2015). [3] K. M. McCreary, A. G. Swartz, W. Han, J. Fabian, R.K. Kawak ami Phys. Rev. Lett. 109, 186604 (2012) [4] I. A. Zhuravlev, V. P. Antropov andK. D. Belashchenko, Ph ys.Rev. Lett. 115, 217201 (2015). [5] V. Flovik, F. Maci, J. M. Hernndez, R. Brucas, M. Hanson an d E. Wahlstrm, Phys. Rev. B92, 104406 (2015). [6] I. Turek, J. Kudrnovsky and V. Drchal, Physical Review B 92, 214407 (2015). [7] L.D. Landau, E.M. Lifshitz, ”On the theory of the dispers ion of magnetic permeability in ferromagnetic bodies”, Phys. Z. Soviet Union 8, 153 (1935); M. Lakshmanan, ”The fascinating world of the Landau Lifshitz Gilbert equation: an overview”, Phil. Trans. R. Soc. A 369, 1280 (2011). [8] Gilbert, T. L. ”A phenomenological theory of damping in f erromagnetic materials”, IEEE Trans. Magn. 40, 34433449 (2004). [9] S. Iihama, S. Mizukami, H. Naganuma, M. Oogane, Y. Ando, a nd T. Miyazaki, Phys. Rev. B89, 174416 (2014). 15[10] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton ,S. von Moln´ ar,M. L. Roukes, A. Y. Chtchelkanova,D. M. Treger , Science 294,1488 (2001). [11] D. Culcer et al., Phys. Rev. Lett. 93, 046602 (2004). [12] J. Yao and Z. Q. Yang, Phys. Rev. B 73, 033314 (2006). [13] E. G. Mishchenko, A.V. Shytov and B. I. Halperin Phys. Re v. Lett.93, 226602 (2004). [14] Sh. Murakami, N. Nagaosa, S.-C. Zhang, Science 301, 1348 (2003). [15] P. Adroguer, E. L. Weizhe, D. Culcer and E. M. Hankiewicz , Phys. Rev. B 92, 241402 (2015). [16] Xin-Zhong Yan and C. S. Ting, Physical Review B 92, 165404 (2015) . [17] M. Eschrig, Rept. Prog. Phys. 78, no. 10, 104501 (2015). [18] K. Hashimoto, N. Iizuka and T. Kimura, Phys. Rev. D 91, no. 8, 086003 (2015). [19] K. B. Fadafan and F. Saiedi, Eur. Phys. J. C 75612 (2015). [20] K. Bakke, C. Furtado and J. R. Nascimento, Eur. Phys. J. C 60, 501 (2009) : erratum Eur. Phys. J. C 64169 (2009) [21] I.L. Buchbinder and I.L. Shapiro, Phys. Lett. B 151263 (1985) [22] V.G. Bagrov, I.L. Buchbinder and I.L. Shapiro, Sov. J. P hys.353 (1992), hep-th/9406122 [23] R. T. Hammond, Phys. Rev. D 5212 (1995). [24] I.L. Shapiro, Phys. Rep. 357113 (2002) [25] R. T. Hammond Rep. Prog. Phys. 65599 (2002). [26] L.H. Ryder and I.L. Shapiro, Phys. Lett. A 24721 (1998) [27] Zutic I, Fabian J and Sarma S D, Rev. Mod. Phys. 76, 323 (2004). [28] Maekawa S., Concepts in Spin Electronics (Oxford: Oxfo rd University Press), (2006). [29] Grollier J, Cros V, Hamzic A, George J M, Jaffr‘es H, Fert A, Faini G, Youssef J B and Legall H., Appl.Phys. Lett. 78, 3663 (2001). [30] Ando K, Takahashi S, Harii K, Sasage K, Ieda J, Maekawa S. and Saitoh E., Phys. Rev. Lett. 101, 036601 (2008). 16[31] Tserkovnyak Y, Brataas A. and Bauer G. E. W. , Phys. Rev. L ett.88, 117601 (2002). [32] Mizukami S, Ando Y and Miyazaki T., Phys. Rev. B 66104413 (2002). [33] S. Azevedo, J. Phys. A 34, 6081 (2001). [34] P. A. Andreev, Phys. Rev. E 91, 033111 (2015) [35] TaikiYoda, TakehitoYokoyama, ShuichiMurakami, Scie ntificReports5, 12024(2015). 17

2016-06-21

Based on the non-relativistic regime of the Dirac equation coupled to a torsion pseudo-vector, we study the dynamics of magnetization and how it is affected by the presence of torsion. We consider that torsion interacting terms in Dirac equation appear in two ways one of these is thhrough the covariant derivative considering the spin connection and gauge magnetic field and the other is through a non-minimal spin torsion coupling. We show within this framework, that it is possible to obtain the most general Landau, Lifshitz and Gilbert (LLG) equation including the torsion effects, where we refer to torsion as a geometric field playing an important role in the spin coupling process. We show that the torsion terms can give us two important landscapes in the magnetization dynamics: one of them related with damping and the other related with the screw dislocation that give us a global effect like a helix damping sharped. These terms are responsible for changes in the magnetization precession dynamics.

Torsion Effects and LLG Equation

1606.06610v1

arXiv:1001.2845v1 [cond-mat.other] 16 Jan 2010Resonance Damping in Ferromagnets and Ferroelectrics A. Widom Physics Department, Northeastern University, Boston, MA U SA S. Sivasubramanian NSF Nanoscale Science & Engineering Center for High-rate Na nomanufacturing, Northeastern University, Boston MA USA C. Vittoria and S. Yoon Department of Electrical and Computer Engineering, Northe astern University, Boston, MA USA Y.N. Srivastava Physics Department and & INFN, University of Perugia, Perug ia IT The phenomenological equations of motion for the relaxatio n of ordered phases of magnetized and polarized crystal phases can be developed in close analo gy with one another. For the case of magnetized systems, thedrivingmagnetic fieldintensityto ward relaxation was developedbyGilbert. For the case of polarized systems, the driving electric field intensity toward relaxation was developed by Khalatnikov. The transport times for relaxation into the rmal equilibrium can be attributed to viscous sound wave damping via magnetostriction for the mag netic case and electrostriction for the polarization case. PACS numbers: 76.50.+g, 75.30.Sg I. INTRODUCTION It has long been of interest to understand the close analogiesbetween orderedelectric polarized systems, e.g. ferroelectricity , and ordered magnetic systems, e.g. fer- romagnetism . Atthe microscopiclevel, the sourceofsuch ordering must depend on the nature of the electronic en- ergy spectra. The relaxation mechanism into thermal equilibrium state must be described by local electric field fluctuationsforthe electricpolarizationcaseandbymag- netic intensity fluctuations for the magnetization case; Specifically, the field fluctuations for each case Gpol ij(r,r′,t) =1 ¯h/integraldisplayβ 0/an}bracketle{t∆Ej(r′,−iλ)∆Ei(r,t)/an}bracketri}htdλ, Gmag ij(r,r′,t) =1 ¯h/integraldisplayβ 0/an}bracketle{t∆Hj(r′,−iλ)∆Hi(r,t)/an}bracketri}htdλ, wherein β=¯h kBT,(1) determine the relaxation time tensor for both cases via the fluctuation-dissipation formula1–4 τij=/integraldisplay∞ 0lim V→∞/bracketleftbigg1 V/integraldisplay V/integraldisplay VGij(r,r′,t)d3rd3r′/bracketrightbigg dt.(2) We have unified the theories of relaxation in ordered po- larized systems and ordered magnetized systems via the Kubo transport time tensor in Eqs.(1) and (2). The transport describing the relaxation of or- dered magnetization is the Landau-Lifshitz-Gilbert equation5–7. This equation has been of considerable recent interest8–10in describing ordered magnetic reso- nancephenomena11–14. The equationdescribingthe elec-tric relaxation of an ordered polarization is the Landau- Khalatinikov-Tani equation15–17. This equation can be simplymodeled18–21witheffectiveelectricalcircuits22–25. Information memory applications26–29of such polarized system are of considerable recent interest30–32. The unification of the magnetic Gilbert-Landau- Lifshitz equations and the electric Landau-Khalatnikov- Tani equations via the relaxation time tensor depends on the notion of a nonequilibrium driving field . For the magnetic case, the driving magnetic intensity Hddeter- mines the relaxation of the magnetization via the torque equation ˙M=γM×Hd, (3) whereinγisthegyromagneticratio. Fortheelectriccase, the driving electric field Eddetermines the relaxation of the polarization via the equation of motion for an ion of chargeze m¨r=zeEd. (4) The unification of both forms of relaxation lies in the close analogy between the magnetic driving intensity Hd and the electric driving field Ed. In Sec.II the thermodynamics of ordered magnetized and polarized systems is reviewed. The notions of mag- netostrictionand electrostrictionaregiven aprecise ther- modynamic definition. In Sec.III, the phenomenology of the relaxation equations are presented. The magnetic driving intensity Hdand the electric driving field Edare defined in terms of the relaxation time tensor Eq.(2). In Sec.IV, we introduce the crystal viscosity tensor. From a Kubo formula viewpoint, the stress fluctuation correlax-2 ation Fijkl(r,r′,t) =1 ¯h/integraldisplayβ 0/an}bracketle{t∆σkl(r′,−iλ)∆σij(r,t)/an}bracketri}htdλ,(5) determines the crystal viscosity ηijkl=/integraldisplay∞ 0lim V→∞/bracketleftbigg1 V/integraldisplay V/integraldisplay VFijkl(r,r′,t)d3rd3r′/bracketrightbigg dt.(6) For models of magnetic relaxation wherein acoustic heat- ing dominates via magnetostriction33and for models of electric relaxation wherein acoustic heating dominates via electrostriction, the relaxation time tensor in Eq.(2) can be related to the viscosity tensor Eq.(6). An inde- pendent microscopic derivation of viscosity induced re- laxationisgiveninAppendixA. IntheconcludingSec.V, the sound wave absorption physics of the viscous damp- ing mechanism will be noted. II. THERMODYNAMICS Our purpose is to review the thermodynamic proper- ties of both magnetically ordered crystals and polariza- tion ordered crystals. The former is characterized by a remnant magnetization Mfor vanishing applied mag- netic intensity H→0 while the latter is characterized by a remnant polarization Pfor vanishing applied electric fieldE→0. A. Magnetically Ordered Crystals Letwbe the enthalpy per unit volume. The funda- mental thermodynamic law determining the equations of state for magnetically ordered crystals is given by dw=Tds+H·dM−e:dσ, (7) whereinsistheentropyperunitvolume, Tisthetemper- ature,eis the crystal strain and σis the crystal stress. The magnetic adiabatic susceptibility is defined by χ=/parenleftbigg∂M ∂H/parenrightbigg s,σ. (8) If N=M M⇒N·N= 1 (9) denotes a unit vector in the direction of the magnetiza- tion, then the tensor Λ ijkldescribing adiabatic magne- tostriction coefficients may be defined as34 2ΛijklNl=M/parenleftbigg∂eij ∂Mk/parenrightbigg s,σ=−M/parenleftbigg∂Hk ∂σij/parenrightbigg s,M.(10)When the system is out of thermal equilibrium, the driv- ing magnetic intensity is Hd=H−/parenleftbigg∂w ∂M/parenrightbigg s,σ−τ·/parenleftbigg∂M ∂t/parenrightbigg ,(11) wherein τare the relaxation time tensor transport co- efficients which determine the relaxation of the ordered magnetic system into a state of thermal equilibrium. B. Ordered Polarized Crystals The fundamental thermodynamic law determining the equations of state for ordered polarized crystals is given by dw=Tds+E·dP−e:dσ, (12) wherein wis the enthalpy per unit volume, sis the en- tropy per unit volume, Tis the temperature, eis the crystal strain and σis the crystal stress. The electric adiabatic susceptibility is defined by χ=/parenleftbigg∂P ∂E/parenrightbigg s,σ. (13) The tensor βijkdescribing adiabatic electrostriction co- efficients may be defined as34 βijk=/parenleftbigg∂eij ∂Pk/parenrightbigg s,σ=−/parenleftbigg∂Ek ∂σij/parenrightbigg s,P.(14) The piezoelectric tensor is closely related to the elec- trostriction tensor via γijk=/parenleftbigg∂eij ∂Ek/parenrightbigg s,σ=/parenleftbigg∂Pk ∂σij/parenrightbigg s,E=βijmχmk.(15) When the system is out of thermal equilibrium, the driv- ing electric field is Ed=E−/parenleftbigg∂w ∂P/parenrightbigg s,σ−τ·/parenleftbigg∂P ∂t/parenrightbigg ,(16) wherein τis the relaxation time tensor transport coef- ficients which determine the relaxation of the ordered polarized system into a state of thermal equilibrium. III. RESONANCE DYNAMICS Here we shall show how the magnetic intensity Hd drives the magnetic resonance equations of motion in magnetically ordered systems. Similarly, we shall show howtheelectricfield Eddrivesthe polarizationresonance equations of motion for polarized ordered systems.3 A. Gilbert-Landau-Lifshitz Equations The driving magnetic intensity determines the torque on the magnetic moments according to ∂M ∂t=γM×Hd. (17) Employing Eqs.(11) and (17), one finds the equations for magnetic resonance in the Gilbert form ∂M ∂t=γM×/bracketleftBigg H−/parenleftbigg∂w ∂M/parenrightbigg s,σ−/parenleftbiggα γM/parenrightbigg ·∂M ∂t/bracketrightBigg ,(18) wherein the Gilbert dimensionless damping tensor αis defined as α= (γM)τ. (19) OnemaydirectlysolvetheGilbert equationsforthe driv- ing magnetic intensity according to Hd+α·/parenleftbig N×Hd/parenrightbig =H−/parenleftbigg∂w ∂M/parenrightbigg s,σ.(20) Eqs.(17) and (20) expressthe magneticresonancemotion in the Landau-Lifshitz form. B. Landau-Khalatnikov-Tani Equations The driving electric field gives rise to a polarization response according to ∂2P ∂t2=/parenleftBigg ω2 p 4π/parenrightBigg Ed, (21) whereinωpis the plasma frequency. A simple derivation of Eq.(21) may be formulated as follows. In a large vol- umeV, the polarization due to charges {zje}is given by P=/parenleftbigg/summationtext jzjerj V/parenrightbigg . (22) If the drivingelectric field acceleratesthe chargesaccord- ing to mj¨rj=zjeEd, (23) then Eq.(21) holds true with the plasma frequency ω2 p= 4πe2lim V→∞/bracketleftBigg/summationtext j(z2 j/mj) V/bracketrightBigg = 4πe2/summationdisplay anaz2 a ma,(24) whereinnais the density of charged particles of type a. The polarization resonance equation of motion follows from Eqs.(16) and (21) as17 /parenleftbigg4π ω2p/parenrightbigg∂2P ∂t2+τ·∂P ∂t+∂w(P,s,σ) ∂P=E.(25)The electric field Einduces the polarization Pat reso- nant frequencies which are eigenvalues of the tensor Ω for which Ω2=ω2 pχ−1 4π≡ω2 p(ǫ−1)−1. (26) Thedecayratesforthepolarizationoscillationsareeigen- values of the tensor Γfor which Γ=ω2 pτ 4π. (27) Ifthedecayratesarelargeonthescaleofthetheresonant frequencies, then the equation of motion is over damped so that min jΓj≫max iΩiimplies τ·∂P ∂t+∂w(P,s,σ) ∂P=E. (28) Eq.(28) represents the Landau-Khalatnikov equation for polarized systems. IV. HEATING RATE PER UNIT VOLUME Let us here consider the heating rate implicit in relax- ation processes. Independently of the details of the mi- croscopic mechanism for generating such heat, the rates of energy dissipation are entirely determined byτ. Ex- plicitly, the heating rates per unit volume for magnetiza- tion and polarization are given, respectively, by ˙qM=∂M ∂t·τ·∂M ∂t, (29) and ˙qP=∂P ∂t·τ·∂P ∂t. (30) Finally, the notion of crystal viscosity ηijklis introduced into elasticity theory35via the heating rate per unit vol- ume from rates of change in the strain ∂e/∂t; It is ˙qe=∂eij ∂tηijkl∂ekl ∂t. (31) Crystal viscosity is employed to describe, among other things, sound wave attenuation. Our purpose is to de- scribe how heating rates in Eqs.(29) and (30) can be re- lated to the heating rate in Eq.((31)). This allows us to expressthetransportcoefficients τintermsofthecrystal viscosity. A. Relaxation via Magnetostriction From the magnetostriction Eq.(10), it follows that magnetic relaxation gives rise to a strain ∂eij ∂t=2 MΛijklNk∂Ml ∂t, (32)4 and thereby to the heating rate, ˙q=4 M2∂Mi ∂t(ΛmnqiNq)ηmnrs(ΛrskjNk)∂Mj ∂t,(33) in virtue of Eq.(31). Employing Eqs.(29) and (33), we find that the magnetic relaxation transport coefficient in the magnetostriction model τij=4 M2(ΛmnqiNq)ηmnrs(ΛrskjNk).(34) The Gilbert damping tensor follows from Eqs.(19) and (34) as αij=4γ M(ΛmnqiNq)ηmnrs(ΛrskjNk).(35) The central relaxation tensor Eq.(35) describes the mag- netic relaxation in terms of the magnetostriction coeffi- cients and the crystal viscosity. B. Relaxation via Electrostriction FromtheelectrostrictionEq.(14), it followsthatatime varying polarization gives rise to a time varying strain ∂eij ∂t=βijk∂Pk ∂t, (36) and thereby to the heating rate, ˙q=∂Pi ∂tβkliηklmnβmnj∂Pj ∂t, (37) in virtue of Eq.(31). Employing Eqs.(30) and (37), we find that the electric relaxation transport coefficient in the electroostriction model τij=βkliηklmnβmnj. (38) ThecentralrelaxationtensorEq.(38) describesthe polar- ization relaxation time tensor coefficients in terms of the electrostriction coefficients and the crystal viscosity. The implications ofthe electrostrictionmodel forthe Landau- Khalatnikov equation is to the authors knowledge a new result. V. CONCLUSIONS For ordered polarized and magnetized systems, we have developed phenomenological equations of motion inclose analogywith one another. For the magnetized case, the relaxation is driven by the magnetic intensity Hd yielding the Gilbert equation of motion7. For the polar- ized case, the relaxation is driven by the electric field Ed yielding the Tani equation of motion17. In both cases, the relaxation time tensor τis determined by the crystal viscosity as derived in the Appendix A; i.e. in Eqs.(A3) and (A6). The viscosity can be measured independently from the magnetic or electrical relaxation by employing sound absorption techniques36. Appendix A: Kubo formulae From the thermodynamic Eq.(10), the fluctuations in the magnetic intensity are given by magnetostriction, i.e. ∆Hk(r,t) =−/parenleftbigg2ΛijklNl M/parenrightbigg ∆σij(r,t).(A1) Eqs.(A1), (1) and (5) imply Gmag ij(r,r′,t) = 4 M2(ΛmnqiNq)Fmnrs(r,r′,t)(ΛrskjNk).(A2) Employing Eqs.(A2), (2) and (6), one finds the central result for the magnetic relaxation time tensor; It is τmag ij=4 M2(ΛmnqiNq)ηmnrs(ΛrskjNk) =αij γM.(A3) From the thermodynamic Eq.(14), the fluctuations in the electric intensity are given by electrostriction, i.e. ∆Ek(r,t) =−βijk∆σij(r,t). (A4) Eqs.(A4), (1) and (5) imply Gpol ij(r,r′,t) =βkliFklmn(r,r′,t)βmnj.(A5) Employing Eqs.(A5), (2) and (6), one finds the central result for the electric relaxation time tensor; It is τpol ij=βkliηklmnβmnj. (A6) 1S. Machlup and L. Onsager, Phys. Rev. 91, 1505 (1953). 2S. Machlup and L. Onsager, Phys. Rev. 91, 1512 (1953). 3H.B. Callen, Fluctuation, Relaxation, and Resonance inMagnetic Systems , Editor D. ter Haar, p 176, Oliver & Boyd Ltd., London (1962). 4R. Kubo, M. Toda and N. Hashitsume, Statistical Physics5 II, Nonequilibrium Statistical Mechanics Springer, Berlin (1998). 5L. Landau and L. Lifshitz, Phys. Zeit. Sowjetunion 8,153 (1935). 6T. L. Gilbert, Armor Research Foundation Rep. No. 11 Chicago, IL. (1955). 7T. L. Gilbert, IEEE Trans. Mag. 40, 3443 (2004). 8M. Fahnle, D. Steiauf, and J. Seib, J. Phys. D: Appl. Phys. 41, 164014 (2008). 9E. Rossi, O.G. Heinonen, andA.H. MacDonald, Phys. Rev. B 72, 174412 (2005). 10V. Kambersky, Phys. Rev. B 76, 134416 (2007). 11I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064403 (2009). 12A. Brataas, Y. Tserkovnyak, and G.E.W. Bauer, Phys. Rev. Lett. 101, 037207 (2008). 13K. Gilmore, Y.U. Idzerda, and M.D. Stiles, Phys. Rev. Lett.99, 027204 (2007). 14R.D. McMichael and A. Kunz, J. Appl. Phys. 91, 8650 (2002). 15L. D. Landau and I. M. Khalatnikov, Dok. Akad. Navk SSSR96469, (1954). 16Y. Makita, I. Seo, and M. Sumita, J. Phys. Soc. Japan. 28, 5268 (1970). 17K.Tani,J. Phys. Soc. Japan. 26, 93 (1969). 18S. Sivasubramanian, A. Widom, and Y. N. Srivastava, Fer- roelectrics 300, 43 (2004). 19H. Li, G. Subramanyam and J. Wang, Integrated Ferro- electrics 97, 69 (2009). 20V. In, A. Palacions, A.R. Bulsara, P. Longhini, A. Kho, J. Neff, S. Baflio and B. Ando, Phys. Rev. E 73, 066121 (2008). 21A. Gordon, Solid State Communications 147, 201 (2008). 22S. Sivasubramanian, A. Widom, and Y. N. Srivastava, IEEE (UFFC) 50, 950 (2003). 23H. Li and G. Subramanyam, IEEE Trans. on Ultrasonics,Ferroelectrics and Frequency Control 56, 1861 (2009). 24D. Guyomar, B. Ducharorne and G. S´ ebald, J. Phys. Appl. Phys.D 40, 6048 (2007). 25B. And´ o, S. Baglio, A.R. Bulsara and V. Marletta, IEEE Trans. on Instrumentation and Measurements , (2009). DOI 10.1109/TIM.2009.2025081 26K. Yoshihisa, Y. Kaneko, H. Tanaka, K. Kaibara, S. Koyama, K. Isogai, T. Yamada and Y. Shimada, Jap. J. Appl. Phys. 46, 2157 (2007). 27H. Kohlstedt, Y. Mustafa, A. Gerber, A. Petraru, M. Fit- silis, R. Meyer, U. Bttger and R. Waser, Microelectronic Engineering 80, 296 (2005). 28L. Eun Sun, D. Jung, K. Young Min, K. Hyun Ho, H. Young Ki, J. Park, K. Seung Kuk, K. Jae Hyun, W. Hee San KIM and A. Woo Song, Jap. J. Appl. Phys. 47, 2725 (2008). 29R. E. Jones, Jr. in the Custom Integrated Circuits Confer- ence, 1998, Proc. of the IEEE , 431 (1998). 30I. Vrejoiu, M. Alexe, D. Hesse and U. Gsele, J. Vac. Sci. & Tech. B: Microelectronics and Nanometer Structures 27, 498 (2009). 31N. Inoue and Y. Hayashi, IEEE Trans. on Elect. Dev. 48, 2266 (2001). 32T. Mikolajick, C. Dehm, W. Hartner, I. Kasko, M. Kast- ner, N. Nagel, M. Moert and C. Mazure, Microelectronics Reliability 41, 947 (2001). 33C. Vittoria, S. D. Yoon and A. Widom, Phys. Rev. B 81, 014412 (2010). 34L.D. Landau and E.M. Lifshitz, “ Electrodynamics of Con- tinuos Media ”, Pergamon Press, Oxford (1960). 35L.D. Landau and E.M. Lifshitz, “ Theory of Elasticity ”, Secs.10, 23 and 34, Pergamon Press, Oxford (1986). 36A.B. Bhatia, Ultrasonic Absorption , Oxford University Press, Oxford (1967).

2010-01-16

The phenomenological equations of motion for the relaxation of ordered phases of magnetized and polarized crystal phases can be developed in close analogy with one another. For the case of magnetized systems, the driving magnetic field intensity toward relaxation was developed by Gilbert. For the case of polarized systems, the driving electric field intensity toward relaxation was developed by Khalatnikov. The transport times for relaxation into thermal equilibrium can be attributed to viscous sound wave damping via magnetostriction for the magnetic case and electrostriction for the polarization case.

Resonance Damping in Ferromagnets and Ferroelectrics

1001.2845v1

A two-stage approach to relaxation in billiard systems of locally conned hard spheres Pierre Gaspard1,a)and Thomas Gilbert1,b) Center for Nonlinear Phenomena and Complex Systems, Universit e Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium We consider the three-dimensional dynamics of systems of many interacting hard spheres, each individually conned to a dispersive environment, and show that the macroscopic limit of such systems is characterized by a coecient of heat conduction whose value reduces to a dimensional formula in the limit of vanishingly small rate of interaction. It is argued that this limit arises from an eective loss of memory. Similarities with the diusion of a tagged particle in binary mixtures are emphasized. PACS numbers: 05.20.Dd,05.45.-a,05.60.-k,05.70.Ln The derivation of the macroscopic transport equations of hydrodynamics and the computa- tion of the associated coecients for systems de- scribed at the microscopic level by Hamilton's equations of classical mechanics is a central prob- lem of non-equilibrium statistical physics. The periodic Lorentz gas provides an example where this program can be achieved and Fick's law of diusion established1{3. Furthermore, by tweak- ing the system's geometry so that tracer particles hop from cell to cell at nearly vanishing rates, memory eects disappear and the dynamics of tracers is well approximated by a continuous time random walk4. In this regime, the diusion coef- cient takes on a simple limiting value, given by a dimensional formula5, by which we mean that its expression reduces to the square of the length scale of the cell separation multiplied by the hop- ping rate. Similarly, it was found that heat trans- port in systems of conned hard disks with rare interactions reduces to a stochastic process of energy exchanges which obeys Fourier's law of heat conduction, with the coecient of heat con- ductivity also given by a dimensional formula6{8, where the timescale is that of energy exchanges among particles in neighboring cells. Here we ex- tend these ndings to mechanical systems of con- ned hard spheres, whose stochastic limit was al- ready studied elsewhere9, and review in some de- tails the analogy with the problem of mass trans- port in the periodic Lorentz gas. I. INTRODUCTION A long-standing problem in non-equilibrium statisti- cal mechanics has been to provide a derivation from a)Electronic mail: gaspard@ulb.ac.be b)Electronic mail: thomas.gilbert@ulb.ac.berst principles of Fourier's law of heat conduction in insulating materials in the framework of Hamiltonian mechanics10. A centerpiece of the puzzle is to identify the conditions under which scale separation occurs so the time evolution at the microscopic level can somehow be reduced to a hydrodynamic equation at the macroscopic level, which is an especially challenging problem for in- teracting particle systems. As emphasized in earlier papers6{8, the following two- step program, which consists of (i) identifying an in- termediate level of description|a mesoscopic scale| where the Newtonian dynamics can be consistently ap- proximated by a set of stochastic equations, and (ii) subsequently analyzing the statistical properties of this stochastic system and computing its transport properties in the hydrodynamic scaling limit, can be successfully achieved in a class of chaotic billiard systems composed of many hard disks trapped in a semi-porous material which prevents mass transfer and yet allows energy trans- fer through elastic collisions among neighboring disks. Under the assumption that collisions among moving disks are rare compared to wall collision events, the global multi-particle probability distribution of the system typ- ically reaches local equilibrium at the kinetic energy of each individual particle before energy exchanges proceed. This mechanism naturally yields a stochastic description for the process of energy exchanges in the system. In a subsequent paper9, it was shown that the same reduction applies to systems of conned hard spheres. Our purpose in this paper is twofold. Our main objec- tive is to establish a comparison between the process of energy transfer in models of interacting hard spheres with local connement rules, on the one hand, and the diu- sive motion of tracer particles in low-dimensional billiard tables such as the nite-horizon periodic Lorentz gas, on the other. Specically, we show that the two systems, un- der equivalent assumptions of local equilibration, whose accuracy can be precisely controlled by tuning the sys- tems' parameters down to a critical geometry, are both amenable to stochastic descriptions in the form of mas- ter equations, whether for the distribution of energies in the case of the former systems, or that of mass in thearXiv:1111.6272v1 [cond-mat.stat-mech] 27 Nov 20112 latter. Taking the hydrodynamic limit of these stochas- tic systems, we obtain explicit values of the transport coecients of heat conduction and diusion respectively, which turn out to share the remarkable property that they are given by simple dimensional formulae, i. e. by the square of the mesoscopic length scale multiplied re- spectively by the rates of energy or mass transfer. Turning back to the billiard dynamics, we address our second objective, which is to show that, under the as- sumption of separation of two characteristic timescales, one associated with the local dynamics, the other with energy transfers, the process of heat transport in three- dimensional billiard systems of conned hard spheres is well-approximated by the corresponding stochastic pro- cess of energy exchanges. We do so by considering the Helfand moment of thermal conductivity11and compute the linear divergence in time of its mean squared change. Plotting our results as functions of the system's sizes for dierent parameter values, we compute the innite-size extrapolations to obtain the heat conductivity and ob- serve a very good convergence to the value obtained for the stochastic systems for parameter values where an ef- fective separation of timescales is observed. The paper is organized as follows. The problem of mass transport in a spatially periodic billiard table is consid- ered in Sec. II. Starting from the pseudo-Liouville equa- tion for the billiard dynamics, we derive a continuous- time random walk which describes the stochastic jumps of particles across a lattice, and analyze its transport properties, comparing it to that of the billiard. The same procedure is applied in Sec. III to two-dimensional billiards of conned hard disks. In Sec. IV, we turn to a three-dimensional billiard system and discuss our nu- merical results. Conclusions are drawn in Sec. V. II. A CONTINUOUS-TIME RANDOM WALK APPROACH TO MASS TRANSPORT We consider for the sake of the example a periodic ar- ray of two-dimensional semi-dispersing Sinai billiard ta- bles in the form of square billiard cells bounded by at walls of sizes l, and connected to each other by small openings of relative widths . All the cells are identical and contain circular obstacles arranged so that there are no periodic trajectory that avoid these obstacles. Inde- pendent pointwise tracer particles move across this array, performing elastic collisions on the obstacles and walls. An example is shown in gure 1. The diusive properties of this model were studied in some details in Ref. [12]. There the attention was on memory eects and their im- pact on the value of the diusion coecient. Here we will focus on the derivation of the kinetic prediction of this coecient, which yields the aforementioned dimen- sional formula, disregarding the memory eects. The conclusions are similar to those obtained by Machta and Zwanzig5in the framework of the periodic Lorentz gas. The continuous-time approach we present below is some- FIG. 1. Periodic billiard table on a square lattice with a typical trajectory. Here the central disks in the initial and nal cells are color-lled. One considers the process of mass transport across the periodic cells. The model has several parameters in terms of which a dimensional formula of the diusion coecient is computed. what similar to that of Zwanzig4. The phase-space conguration of this system refers to a single tracer particle and is specied by the triplet fn;r;vg, where n= (nx;ny)2Z2denotes the lattice index of the cell where the tracer is located, r, its posi- tion within the cell and vits velocity. Letp(n;r;v;t) denote the probability distribution of the tracer. Its time evolution is determined by the pseudo-Liouville operator13, which comprises four dier- ent types of contributions, corresponding to as many dif- ferent types of events: 1. Free advection of the tracer inside the cell is ac- counted for by the term v
@r; 2. Collisions of the tracer with any of the circular ob- stacles inside the cell are determined by the opera- torsK(d), wheredrefers to a specic disk of radius dat position qd: K(d)p(n;r;v;t) =dZ ^e
v>0d^e(^e
v) h (rqd^e)p(n;r;v2^e(^e
v);t) (rqd+^e)p(n;r;v;t)i ; (1) 3. Collisions of the tracer with any of the at walls of cell n, with operator W(j), wherejtakes on the valuesj= 1;:::; 4, corresponding to the right, bottom, left and top walls: letting rxandvxdenote the position and velocity components of the tracer along the horizontal axis, wall #1 is the right wall3 at positionl=2 so that, if the wall is continuous with respect to the vertical axis, i. e. 0, the collision operator acts according to W(1)p(n;r;v;t) =jvxj(rxl=2) (2) h (vx)p(n;r;vx;vy;t)(vx)p(n;r;v;t)i ; with similar expressions for W(2),W(3), andW(4); 4. Jump events between neighboring cells take place when a wall collision event occurs at a position where the wall is actually open, which amounts to subtracting the action of the wall operator (2) above, keeping track of the cell indices: using the same set of four indices as above for every wall and assuming that wall #1 has a slit of width centered abouty= 0, the jump operator J(1) is J(1) p(n;r;v;t) =jvxj(rxl=2)(ry)(vx) (3) h p(n+ (1;0);r;v;t)p(n;r;vx;vy;t)i ; where (x) = 1 ifjxj=2 and 0 otherwise. Collecting these terms together, we obtain the time evo- lution ofp(n;r;v;t), given by the pseudo-Liouville equa- tion: @tp(n;r;v;t) = (4)n v
@r+P dK(d)+P jW(j)o p(n;r;v;t) +P jJ(j) p(n;r;v;t): The terms on the right-hand side of this equation are grouped so as to distinguish the local terms, which do not act on the cell index in the distribution, from the non-local ones, i. e. the jump terms, which act on the cell index. Also note that the velocity amplitude is in- variant under every term of this equation and is therefore a conserved quantity which plays no role once it has been xed by the initial condition. It it an immediate consequence of the ergodicity of the local dynamics that the following distribution, P(leq)(n;t) =Z drZ dv(v2v2 0)p(n;r;v;t);(5) is invariant under the local terms of the pseudo-Liouville equation (4). We thus refer to P(leq)(n;t) as the lo- cal equilibrium distribution . Its time evolution occurs through jump events only : @tP(leq)(n;t) = P jR drR dv(v2v2 0)J(j) p(n;r;v;t): (6) In order to close this equation for P(leq)(n;t), we make thelocal equilibrium approximation , p(n;r;v;t) =1 A(v2v2 0)P(leq)(n;t); (7)whereAis the area of the billiard, e. g. A=l2(2 1+2 2) in the case of gure 1 (where 1denotes the radius of the disk at the center of the cell and 2that of the disks at the cell corners). Plugging Eq. (7) into (6), the pseudo-Liouville equa- tion (4) reduces to the following continuous-time random walk @tP(leq)(n;t) =X jv0 A[P(leq)(n+ej;t)P(leq)(n;t)]; (8) where ej2f(1;0);(0;1)g. In the continuum scaling limit, ~ r=ln,l!0,v0l2, this reduces to a diusion equation, @tP(~ r;t) =Dr2P(~ r;t); (9) with diusion coecient D=l2(J); (10) here written in terms of the frequency (J)of a jump event in a specic direction, identical for all four direc- tions, (J)=v0 A: (11) Equation (10) is what we refer to as a dimensional formula : the diusion coecient associated with the continuous-time random walk process (8) is the product between the scale of displacements squared and hopping rate. The fact that it is an exact property of the stochas- tic process is remarkable. As far as the transport properties of the billiard table go, Eq. (10) is a mere approximation: its validity relies on the local equilibrium approximation (7). In contrast, Eq. (11) is an exact result for the billiard dynamics as well|valid for all parameter values|as it relies solely on the ergodicity of the billiard table14. The closure approximation (7) is expected to be exact in the limit of vanishing window widths, !0, whereby the hopping rate (11) diverges and the diusion coe- cient (10) vanishes. In practice however, we may expect p(n;r;v;t) to converge to the local equilibrium distribu- tionP(leq)(n;t) between successive jumps so long as the typical number of disk collision events within a cell is large. This can be veried numerically, as shown on the top panel of gure 2. These numerical results were al- ready reported in Ref. [12]. Similar results are equally obtainable for three dimensional billiard systems such as the three-dimensional periodic Lorentz gas15. III. HEAT TRANSPORT IN TWO-DIMENSIONAL CONFINING BILLIARDS Similar considerations apply to models of heat trans- port with mass connement6. Classes of such mod- els were initially introduced by Bunimovich et. al in4 ææææææææææææææææææææææææææææææææææ 0.00.10.20.30.40.50.60.70.30.40.50.60.70.80.91.0 dDl-2nHJL-1 0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6 dnHJL-1 FIG. 2. (Top) Diusion coecient as measured from the mean-squared displacement divided by the number of jumps for the Sinai billiard table shown in gure 1. (Bottom) Corre- sponding decay time. The dots show numerical measurements and the red line is the formula (11). Ref. [16]. There, the authors proved the ergodicity of two-dimensional billiard tables consisting of an arbitrary large number of unit cells placed side by side, each con- taining a single disk trapped within the cell's boundaries, but in such a way that collisions may still take place be- tween disks belonging to neighboring cells. Figure 3 shows an example of such a system17. In Ref. [7], we introduce the following parametrization of the model in terms of two parameters: , which char- acterizes the timescale of wall collision events, and m, which characterizes that of binary collisions. On the one hand, the dynamics within an isolated cell of widthlboils down to the motion of a point particle in an area bounded by the exterior intersection of four disks of radius ,l=2 < l=p 2. In the absence of interaction with neighboring particles, the mean free path of a particle is given by `=jBj=j@Bj, whereBand @Bdenote respectively the area and perimeter of the billiard cell. The corresponding wall collision timescale is obtained by multiplying the mean free path by the speed of the particle. Interactions among neighboring particles on the other hand take place provided the radius of the moving par- ticlesmis larger than a critical value, m>  c= p 2l2=4. The corresponding timescale can be com- FIG. 3. Example of a two-dimensional billiard table consist- ing of many moving hard disks (colored), each trapped within conning walls (black disks), here arranged in a square tiling. The centers of the moving disks are bound to the areas delim- ited by the exterior intersections of the black circles around the xed disks (the solid broken lines sample their trajecto- ries). The parameters (radii of the black circles and moving disksm) are so chosen that (i) black circles overlap, so the moving disks are conned, and (ii) collisions are allowed (such as in the upper right corner), whereby energy exchanges take place. The mobile disks are color coded from blue to red with growing kinetic energies. puted and shown to diverge with ( mc)3. Letting (W)(T) and(B)(T) respectively denote the frequencies of wall collision and binary collision events measured at equilibrium temperature T, the following separation of timescales is assumed, (B)(T)(W)(T); (12) which occurs when m!c. Under this assumption, we showed6,8that the heat conductivity of spatially extended billiard systems, which are innite size limits of systems such as depicted in Fig. 3, reduces, up to a dimensional factor l2, to the frequency of binary collisions, (B)(T) =l2(B)(T): (13) Furthermore the heat conductivity scales with the ther- mal speed, (B)(T)p T. Although Eqs. (10) and (13) are very much alike, the derivation of the latter is much more involved than that of the former. Proceeding as in Sec. II, the rst part of the pro- gram, which is to reduce the pseudo-Liouville equation describing the time evolution of the billiard system to a5 continuous-time stochastic process of energy exchanges, follows closely along lines which led from Eqs. (4) to (8). Indeed, considering the time evolution acting on the N-particle phase-space distribution pN(fri;vig;t), one readily notices that the local equilibrium distribution P(leq) N(1;:::;N;t) (14) ZNY a=1dradva(amv2 a=2)pN(fri;vig;t) is left unchanged by the advection and wall collisionterms. Therefore, and provided binary collision events are rare with respect to wall collision events, we can make a closure approximation similar to Eq. (7) to obtain the time evolution of the local equilibrium distribution in the form of a stochastic process. The result is a master equation which describes the time evolution of the Ncell system with energy variables f1;:::;Ngin terms of energy exchanges between neigh- boring cells at respective energies aandbof amount, specied by a stochastic kernel W: @tP(leq) N(1;:::;N;t) =1 2NX a;b=1Z dh W(a+;bja;b)P(leq) N(:::;a+;:::;b;:::;t ) W(a;bja;b+)P(leq) N(:::;a;:::;b;:::;t )i ; (15) where the expression of Wcan be obtained by direct com- putation of the collision integrals, which, after rescaling the time variable to the units of the frequency of binary collision events and thus absorbing all the parameters into the time scale, yields the universal function8: W(a;bja;b+) =r 2 38 >>>>>>< >>>>>>:q 1 aK b+ a q 1 b+K a b+ q 1 aK b a q 1 bK a b; (16) whose detion intervals correspond respectively to b< <max(ba;0),max(ba;0)<< 0, 0<< max(ab;0), and max( ab;0)<<a. HereK(m) denotes the complete elliptic intergal of the rst kind18. A system of Nisolated cells whose time evolution is specied by the master equation (15) reaches a micro- canonical equilibrium state whose total energy can be parametrized in terms of the temperature according to 1+:::+N=NT. The corresponding energy exchange frequency is N(T) =p Th 1 +O(1=N)i ; (17) whose innite-size limit is simply (T) = limN!1N(T) =p T(in the chosen units of time). An expression of the heat conductivity of the system described by Eq. (15) is obtained by considering the Einstein-type relation satised by the variance of the as- sociated Helfand moment, which measures the spread of energy as a function of time. For the system of Nenergy cells aligned along a one- dimensional ring, the Helfand moment is dened accord-ing to HN(t) =NX i=1ii(t); (18) wherei(t) is the state of the energy at site iat timet. This quantity evolves in time by discrete steps, when en- ergy exchanges occur. Let fngn2Ndenote the sequence of times at which successive energy exchanges take place. Assuming cells iandi+ 1 exchange some amount of en- ergy at time n, we can write the corresponding change in the Helfand moment as i(n0)i(n+ 0). Computing the mean squared change in the Helfand moment as a function of time, we obtain an expression of the thermal conductivity according to (T) = lim N!1N(T); (19) where we dened the nite Nconductivity to be N(T) =1 N(kBT)2lim n!11 2nD [HN(n)HN(0)]2E : (20) In Ref. [8], it was argued that the innite Nlimit of this quantity is determined by the static correlations only, yielding the result N(T) =l2p Th 1 +O(1=N)i : (21) Comparing Eqs. (17) and (21), we obtain the an- nounced result (T) =l2(T); (22) which is an exact result for the stochastic system evolved by the master equation (15) and does not make explicit6 use of the form (16), except for some symmetries9. The corresponding result (13) for the billiard dynamics is ob- tained by plugging back the proper timescale of binary collisions and letting m!cso that the separation of timescales (12) is eective. Unlike mass transport in Sinai billiard tables for which the transport equation (9) follows directly from the continuous-time random walk (8), so that there only re- mains the problem of comparing the diusion coecient of the billiard to the dimension formula (10) in the ap- propriate parameter regime, the problem of computing the heat conductivity associated with heat transport in billiard systems of many conned particles proceeds in two separate steps. Having carried out the reduction of the billiard's pseudo-Liouville equation to a master equation for a stochastic energy exchange system, the rst step is to es- tablish the cancellation of dynamical correlations in such a system, which yields the identity lim N!1N=N=l2. This is however a delicate result19and requires in-depth knowledge of the spectral properties of the master equa- tion (15). The approach we took in Ref. [8], though equivalent, uses dierent techniques, based on analyz- ing the rst few orders of the gradient expansion of the kinetic equation to obtain the expression of the heat cur- rent in terms of the local temperature gradient in a non- equilibrium stationary state, i. e. Fourier's law. The second step of the program is to go back to the billiard dynamics and investigate the convergence of the heat conductivity to the binary collision frequency in the limitm!c. For two-dimensional billiard systems such as the ones shown in Fig. 3, the agreement was found to be rather satisfactory7. For the three-dimensional billiard systems we turn to now, this agreement is even better. IV. THREE-DIMENSIONAL BILLIARD FOR HEAT TRANSPORT Consider a three-dimensional billiard of conned hard spheres such as shown in Fig. 4. The reduction of the pseudo-Liouville equation governing the phase-space evo- lution of probability densities pN(fri;vig;t) to a master equation similar to Eq. (15) was already carried out in Ref. [9]. The corresponding kernel, in the appropriate time units, is found to have the universal form W(a;bja;b+) =r 88 >>>< >>>:q b+ ab 1p max(a;b)q a ab;(23) whose denition intervals correspond respectively to b<  <max(ba;0),max(ba;0)<  < max(ab;0), and max( ab;0)<<a. FIG. 4. (Top) Hard-sphere particle trapped in a cuboid cell with cylindrical edges. (Bottom) A system made out of many copies of such cells which form a spatially periodic structure. The cells are semi-porous in the sense that particles are pre- vented from escaping, and can yet partially penetrate into the neighboring cells, thus allowing energy transfer through collisions among neighboring particles. As with the two- dimensional case, the likelihood of binary collision events can be controlled by tuning the geometry of the cell. Extensive numerical investigations of the master equa- tion (15) with the stochastic kernel (23) were presented in Ref. [9], supporting with high precision the validity of Eq. (22). Here we focus on the billiard dynamics and consider the mean squared change in time of the Helfand moment associated with the distribution of energy in bil- liard systems formed by one-dimensional lattices of bil- liard cells such as shown in Fig. 4. Letfngn2Zdenote the times at successive binary col-7 lision events. As we now need to account for the motion of particles within their respective cells, there are two types of contributions to changes in the Helfand moment between two successive binary collision events. The lead- ing contribution arises from the energy exchanges which take place at binary collision events. Thus, when a binary collision occurs between particles jandk, the Helfand moment changes by the amount [ xj(n)xk(n)][j(n+ 0)j(n0)], wherexj(n) andxk(n) denote the po- sitions of particles jandkalong the direction of the lattice spatial extension at time n. The other contri- bution to the Helfand moment arises from the advec- tion of particles within their respective cells according toP a[xa(n)xa(n1)]a(n1). The range of allowed parameter values is identical to the two-dimensional case: l=2 < l=p 2, andc< m<, wherecp 2l2=4. Takingl= 1, we x = 0:50 soc= 0 and use the six parameter values m= 0:20;0:25;:::; 0:45. For each one of them, we x the total energy to be E= 3N=2 (T= 1) and vary the system size from N= 3 and up to N= 100 cells. We typically run 103trajectories, each for a dura- tion of 103Nunits of the average time between collision events. We compute: (i) the Helfand moment versus time and take the mean and standard deviation of the trans- position of Eq. (20) to obtain the conductivities (B) N; (ii) the binary and wall collision frequencies (B) Nand(W) N by directly averaging their numbers with respect to time. We then use linear ts in 1 =NforN10 of(B) Nand (B) Nto obtain the extrapolation (B) 1= limN!1(B) Nand (B) 1= limN!1(B) N. Here and below, explicit tempera- ture dependencies are dropped. N=25 0.200.250.300.350.400.450.0010.010.11 rm-rcnNHBLnNHWL FIG. 5. Ratio between the binary and wall collision frequen- cies as a function of the parameter mforN= 25 (= 0:50, c= 0). Measurements of the ratio between (B) Nand(W) N, by which we can assess the eectiveness of the separation of timescales (12), are displayed in Fig. 5 for N= 25 (other system sizes yield similar values). The results of the computations of (B) 1=(B) 1for the dierent parame- ter valuesmare displayed in Fig. 6 and are found to be in very good agreement (within two digits) of the dimen- 0.200.250.300.350.400.450.00.51.01.52.0 rm-rck¥HBLn¥HBL-1FIG. 6. Ratio between the heat conductivity of the innite length system and the corresponding collision frequency be- tween neighboring particles as a function of the parameter m (= 0:50 andc= 0). The error bars show the widths of the computed 0.95 condence intervals. sional formula (13) for values of mas large as 0 :30. In Fig. 7, the details of the tting procedure used to obtain the values of (B) 1and(B) 1are shown for the dierent parameter values as functions of N. V. CONCLUDING REMARKS By controlling the rate at which a tracer hops from cell to cell in a Sinai billiard table or that of interaction among neighboring disks or spheres in a high-dimensional billiard with local connement rules, one identies a limit of vanishing rate where the complicated phase-space dy- namics are replaced by stochastic processes which ac- count for the transport of mass in the rst case, or (ki- netic) energy in the second. Furthermore, the trans- port coecients of these stochastic processes have the same simple dimensional expression, given by the length scale of transfers squared multiplied by the corresponding rates. As emphasized in this paper, the dimensional formulae we obtained for the transport coecients of the models of mass and heat transport we have considered are but two faces of the same coin. Indeed, in both cases the ac- curacy of our approximation of the transport coecients of the billiards in terms of hopping or collision rates relies on the ecient separation of two timescales, namely the timescale of local collision events must be much shorter than that characterizing the transfer of mass or energy. In other words, the relaxation to local equilibrium pre- cedes mass or energy transfers. We can in fact view the notion of relaxation to local equilibrium as a low-dimensional transposition of that of local thermal equilibrium, which is at the heart of many theories involving hydrodynamic scaling limits and typi- cally assumes a large number of degrees of freedom10. In our billiards, the relaxation to local equilibrium occurs8 rm=0.45 k¥HBLn¥HBL=1.957 0204060801001.01.21.41.61.82.0 NkNHBLn¥HBL,nNHBLn¥HBL rm=0.4 k¥HBLn¥HBL=1.335 0204060801000.91.01.11.21.3 NkNHBLn¥HBL,nNHBLn¥HBL rm=0.35 k¥HBLn¥HBL=1.066 0204060801000.800.850.900.951.001.051.10 NkNHBLn¥HBL,nNHBLn¥HBL rm=0.3 k¥HBLn¥HBL=0.986 0204060801000.750.800.850.900.951.00 NkNHBLn¥HBL,nNHBLn¥HBL rm=0.25 k¥HBLn¥HBL=0.987 0204060801000.700.750.800.850.900.951.00 NkNHBLn¥HBL,nNHBLn¥HBL rm=0.2 k¥HBLn¥HBL=0.992 0204060801000.750.800.850.900.951.00 NkNHBLn¥HBL,nNHBLn¥HBL FIG. 7. Heat conductivities (B) Nand binary collision frequencies (B) Nmeasured as functions of the system size Nfor dierent values of the parameter m(= 0:50). The solid curves correspond to the linear ts of (B) Nand(B) Nas functions of 1 =N. The intercepts yield the innite size estimates (B) 1and(B) 1. on the constant energy surface of a single particle. But one could instead consider systems of trapped gases, in which case the relaxation to local equilibrium would in- volve transfers of energy among the particles in the same trap. In such a case, provided relaxation to local equi- librium takes place on timescales much shorter than that of energy transfer between neighboring traps, a similar dimensional expression of the heat conductivity of the corresponding stochastic process in terms of the prod- uct of length scale squared and rate of energy transfer would yield an accurate approximation of the transport coecient of the billiard system. We end with a remark concerning the closure (7) which relies on the ergodicity of the local dynamics on the sur-face of constant energy. As noted already by Zwanzig4, Sinai billiards cannot be replaced by polygonal ones, for which the notion of ergodicity is weaker since the ve- locity directions take on values in a discrete set. In energy exchange processes, however, ergodicity of the many-particle billiard may be restored through the sole interaction among neighboring particles. As shown in Ref. [20], an elastic string-type interaction between par- ticles trapped in polygonal boxes provides a simple model of a system which, on the one hand, cannot be accurately described by a master equation similar to Eq. (15), even when interactions are rare, but on the other hand, has a well-dened heat conductivity which is well approxi- mated by a dimensional formula. An understanding of9 the transport properties of this system beyond the Boltz- mann hypothesis remains to be elucidated. ACKNOWLEDGMENTS We dedicate this paper to the memory of Sasha Losku- tov in appreciation for his co-organizing the conference Billiards'2011 in Ubatuba, SP, Brazil. TG also wishes to thank E. Leonel for his warm hospitality. The au- thors would further like to thank F. Barra, L. Buni- movich, R. Lefevere, M. Lenci, C. Liverani, V. Rom- Kedar, D. P. Sanders and D. Sz asz for stimulating discus- sions which took place at dierent stages of this project. They acknowledge nancial support by the Belgian Fed- eral Government under the Interuniversity Attraction Pole project NOSY P06/02 and FRS-FNRS under con- tract C-Net NR/FVH 972. TG is nancially supported by the Fonds de la Recherche Scientique FRS-FNRS and receives additional support through FRFC conven- tion 2,4592.11. 1L. A. Bunimovich and Y. G. Sinai, \Markov partitions for dis- persed billiards," Commun. Math. Phys. 78, 247{280(1980). 2L. A. Bunimovich and Y. G. Sinai, \Statistical properties of Lorentz gas with periodic conguration of scatterers," Commun. Math. Phys. 78, 479{497 (1981). 3L. A. Bunimovich, Y. G. Sinai, and N. Chernov, \Statistical properties of two-dimensional hyperbolic billiards," Russ. Math. Surv. 46, 47{106 (1991). 4R. Zwanzig, \From classical dynamics to continuous time random walks," J. Stat. Phys. 30, 255{262 (1983). 5J. Machta and R. Zwanzig, \Diusion in a periodic lorentz gas," Phys. Rev. Lett. 50, 1959 (1983). 6P. Gaspard and T. Gilbert, \Heat conduction and Fourier's law by consecutive local mixing and thermalization," Phys. Rev. Lett. 101, 020601 (2008).7P. Gaspard and T. Gilbert, \Heat conduction and Fourier's law in a class of many particle dispersing billiards," New J. Phys. 10, 3004 (2008). 8P. Gaspard and T. Gilbert, \On the derivation of Fourier's law in stochastic energy exchange systems," J. Stat. Mech. P11021 (2008). 9P. Gaspard and T. Gilbert, \Heat transport in stochastic energy exchange models of locally conned hard spheres," J. Stat. Mech. , P08020 (2009). 10F. Bonetto, J. L. Lebowitz, and L. Rey-Bellet, \Fourier's law: a challenge for theorists," Mathematical Physics 2000, 2052, A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski eds. (Imperial College, London, 2000).. 11E. Helfand, \Transport coecients from dissipation in a canoni- cal ensemble," Phys. Rev. 119, 1{9 (1960). 12T. Gilbert and D. P. Sanders, \Persistence eects in deterministic diusion," Phys. Rev. E 80, 41121 (2009). 13J. R. Dorfman and M. H. Ernst, \Hard-sphere binary-collision operators," J. Stat. Phys. 57, 581{593 (1989). 14N. Chernov and R. Markarian, \Introduction to the ergodic the- ory of chaotic billiards, 2nd edition" Instituto de Matem atica Pura e Aplicada, Rio de Janeiro (2003). 15T. Gilbert, H. C. Nguyen, and D. P. Sanders, \Diusive prop- erties of persistent walks on cubic lattices with application to periodic lorentz gases," J. Phys. A 44, 065001 (2011). 16L. A. Bunimovich, C. Liverani, A. Pellegrinotti, and Y. M. Suhov, \Ergodic systems of n balls in a billiard table," Commun. Math. Phys. 146, 357 (1992). 17For technical reasons pertaining to the probability of direct re- collisions between two particles, this example actually falls out of the class of systems considered in Ref. [16]. This is however unimportant for our own considerations. 18M. Abramowitz and I. A. Stegun, \Handbook of mathematical functions : with formulas, graphs, and mathematical tables," (Dover, New York, 1972). 19A. Grigo, K. Khanin, and D. Sz asz, \Mixing rates of par- ticle systems with energy exchange," arXiv math-ph (2011), 1109.2356v1. 20T. Gilbert and R. Lefevere, \Heat conductivity from molecular chaos hypothesis in locally conned billiard systems," Phys. Rev. Lett. 101, 200601 (2008).

2011-11-27

We consider the three-dimensional dynamics of systems of many interacting hard spheres, each individually confined to a dispersive environment, and show that the macroscopic limit of such systems is characterized by a coefficient of heat conduction whose value reduces to a dimensional formula in the limit of vanishingly small rate of interaction. It is argued that this limit arises from an effective loss of memory. Similarities with the diffusion of a tagged particle in binary mixtures are emphasized.

A two-stage approach to relaxation in billiard systems of locally confined hard spheres

1111.6272v1

arXiv:0805.1320v2 [cond-mat.str-el] 25 Aug 2008Spin dynamics in (III,Mn)V ferromagnetic semiconductors: the role of correlations M. D. Kapetanakis and I. E. Perakis Department of Physics, University of Crete, and Institute o f Electronic Structure & Laser, Foundation for Research and Technology-Hellas, Heraklion , Crete, Greece (Dated: November 6, 2018) We address the role of correlations between spin and charge d egrees of freedom on the dynamical properties of ferromagnetic systems governed by the magnet ic exchange interaction between itiner- ant and localized spins. For this we introduce a general theo ry that treats quantum fluctuations beyond the Random Phase Approximation based on a correlatio n expansion of the Green’s function equations of motion. We calculate the spin susceptibility, spin–wave excitation spectrum, and mag- netization precession damping. We find that correlations st rongly affect the magnitude and carrier concentration dependence of the spin stiffness and magnetiz ation Gilbert damping. PACS numbers: 75.30.Ds, 75.50.Pp, 78.47.J- Introduction— Semiconductors displaying carrier– induced ferromagnetic order, such as Mn–doped III-V semiconductors, manganites, chalcogenides, etc, have re- ceived a lot of attention due to their combined magnetic and semiconducting properties [1, 2]. A strong response of their magnetic properties to carrier density tuning via light, electrical gates, or current[3, 4, 5] canlead to novel spintronics applications [6] and multifunctional magnetic devices combining information processing and storage on a single chip. One of the challenges facing such magnetic devices concerns the speed of the basic processing unit, determined by the dynamics of the collective spin. Two key parameters characterize the spin dynam- ics in ferromagnets: the spin stiffness, D, and the Gilbert damping coefficient, α.Ddetermines the long– wavelength spin–wave excitation energies, ωQ∼DQ2, whereQis the momentum, and other magnetic prop- erties.Dalso sets an upper limit to the ferromagnetic transition temperature: Tc∝D[1]. So far, the Tcof (Ga,Mn)As has increased from ∼110 K [2] to ∼173 K [1, 7]. It is important for potential room temperature ferromagnetism to consider the theoretical limits of Tc. TheGilbertcoefficient, α, characterizesthedampingof the magnetization precession described by the Landau– Lifshitz–Gilbert (LLG) equation [1, 8]. A microscopic expression can be obtained by relating the spin suscepti- bility of the LLG equation to the Green’s function [9] ≪A≫=−iθ(t)<[A(t),S− Q(0)]> (1) withA=S+ −Q,S+=Sx+iSy.∝angbracketleft···∝angbracketrightdenotes the average over a grand canonical ensemble and SQ= 1/√ N/summationtext jSje−iQRj, whereSjare spins localized at N randomly distributed positions Rj. The microscopic ori- gin ofαisstill notfully understood[9]. Amean–fieldcal- culation of the magnetization damping due to the inter- play between spin–spin interactions and carrier spin de- phasingwasdevelopedin Refs.[9, 10]. Themagnetization dynamics can be probed with, e.g., ferromagnetic res- onance [11] and ultrafast magneto–optical pump–probe spectroscopy experiments [5, 12, 13, 14]. The interpre-tation of such experiments requires a better theoretical understanding of dynamical magnetic properties. In this Letter we discuss the effects of spin–charge cor- relations, due to the p–d exchange coupling of local and itinerant spins, on the spin stiffness and Gilbert damp- ingcoefficient. Wedescribequantumfluctuationsbeyond the Random Phase Approximation (RPA) [15, 16] with a correlationexpansion[17]ofhigherGreen’sfunctionsand a 1/S expansion of the spin self–energy. To O(1/S2), we obtain a strong enhancement, as compared to the RPA, of the spin stiffness and the magnetization damping and a different dependence on carrier concentration. Equations of motion— The magnetic propertiescan be describedby the Hamiltonian [1] H=HMF+Hcorr, where the mean field Hamiltonian HMF=/summationtext knεkna† knaknde- scribes valence holes created by a† kn, wherekis the mo- mentum, nis the band index, and εknthe band disper- sion in the presenceof the mean field created by the mag- netic exchangeinteraction[16]. The Mn impurities act as acceptors, creating a hole Fermi sea with concentration ch, and also provide S= 5/2 local spins. Hcorr=βc/summationdisplay q∆Sz q∆sz −q+βc 2/summationdisplay q(∆S+ q∆s− −q+h.c.),(2) whereβ∼50–150meV nm3in (III,Mn)V semiconductors [1] is the magnetic exchane interaction. cis the Mn spin concentration and sq= 1/√ N/summationtext nn′kσnn′a† k+qnakn′the hole spin operator. ∆ A=A− ∝angbracketleftA∝angbracketrightdescribes the quan- tum fluctuations of A. The ground state and thermo- dynamic properties of (III,Mn)V semiconductors in the metallic regime ( ch∼1020cm−3) are described to first approximation by the mean field virtual crystal approxi- mation,HMF, justified for S→ ∞[1]. Most sensitive to the quantum fluctuations induced by Hcorrare the dy- namical properties. Refs.[9, 15] treated quantum effects toO(1/S) (RPA). Here we study correlations that first arise atO(1/S2). By choosing the z–axis parallel to the ground state local spin S, we have S±= 0 and Sz=S. The mean hole spin, s, is antiparallel to S,s±= 0 [1].2 The spin Green’s function is given by the equation ∂t≪S+ −Q≫=−2iSδ(t)+βc≪(s×S−Q)+≫ −i∆≪s+ −Q≫+βc N× /summationdisplay kpnn′≪(σnn′×∆Sp−k−Q)+∆[a† knapn′]≫,(3) where ∆ = βcSis the mean field spin–flip energy gap ands= 1/N/summationtext knσnnfknis the ground state hole spin. fkn=∝angbracketlefta† knakn∝angbracketrightis the hole population. The first line on the right hand side (rhs) describes the mean field pre- cession of the Mn spin around the mean hole spin. The second line on the rhs describes the RPA coupling to the itinerant hole spin [10], while the last line is due to the correlations. The hole spin dynamics is described by (i∂t−εkn′+εk−Qn)≪a† k−Q↑ak↓≫ =βc 2√ N/bracketleftbigg (fk−Qn−fkn′)≪S+ −Q≫ +/summationdisplay qm≪(σn′m·∆Sq)∆[a† k−Qnak+qm]≫ −/summationdisplay qm≪(σmn·∆Sq)∆[a† k−Q−qmakn′]≫/bracketrightbigg .(4) The firstterm on the rhsgivesthe RPAcontribution[10], while the last two terms describe correlations. The correlation contributions to Eqs.(3) and (4) are determined by the dynamics of the interactions be- tween a carrier excitation and a local spin fluctuation. This dynamics is described by the Green’s functions ≪∆Sp−k−Q∆[a† knapn′]≫, whose equations of motion couple to higher Green’s functions, ≪Sa†aa†a≫and ≪SSa†a≫, describingdynamicsof threeelementaryex- citations. To truncate the infinite hierarchy, we apply a correlation expansion [17] and decompose ≪Sa†aa†a≫ into all possible products of the form ∝angbracketlefta†aa†a∝angbracketright ≪S≫, ∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪a†a≫,∝angbracketlefta†a∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketleftS∝angbracketright ≪ a†aa†a≫c, where≪a†aa†a≫cis obtained after sub- tracting all uncorrelated contributions, ∝angbracketlefta†a∝angbracketright ≪a†a≫, from≪a†aa†a≫(we include all permutations of mo- mentum and band indices) [18]. Similarly, we decompose ≪SSa†a≫into products of the form ∝angbracketleftSS∝angbracketright ≪a†a≫, ∝angbracketleftS∝angbracketright∝angbracketlefta†a∝angbracketright ≪S≫,∝angbracketleftS∝angbracketright ≪∆S∆[a†a]≫, and∝angbracketlefta†a∝angbracketright ≪ ∆S∆S≫. This corresponds to decomposing all opera- torsAinto average and quantum fluctuation parts and neglecting products of three fluctuations. We thus de- scribe all correlations between any twospin and charge excitations and neglect correlations among threeor more elementary excitations (which contribute to O(1/S3)) [18]. In the case of ferromagnetic β, as in the mangan- ites, we recover the variational results of Ref.[19] and thus obtain very good agreement with exact diagonaliza- tionresultswhilereproducingexactlysolvablelimits (one electron, half filling, and atomic limits, see Refs.[18, 19]).When treating correlations in the realistic (III,Mn)V system, the numerical solution of the above closed sys- tem of equations of motion is complicated by the cou- pling of many momenta and bands and by unsettled is- sues regarding the role on the dynamical and magnetic anisotropy properties of impurity bands, strain, localized states, and sp–d hybridization [1, 20, 21, 22, 23]. In the simpler RPA case, which neglects inelastic effects, a six– band effective mass approximation [16] revealed an order of magnitude enhancement of D. The single–band RPA model [15] also predicts maximum Dat very small hole concentrations, while in the six–band model Dincreases and then saturates with hole doping. Here we illustrate the main qualitative features due to ubiquitous corre- lations important in different ferromagnets [19, 24] by adopting the single–band Hamiltonian [15]. We then dis- cuss the role of the multi–band structure of (III,Mn)V semiconductors by using a heavy and light hole band model. In the case of two bands of spin– ↑and spin– ↓states [15], we obtain by Fourier transformation ≪S+ −Q≫ω=−2S ω+δ+ΣRPA(Q,ω)+Σcorr(Q,ω),(5) whereδ=βcsgives the energy splitting of the local spin levels. Σ RPAis the RPA self energy [15, 16]. Σcorr=βc 2N/summationdisplay kp/bracketleftBigg (Gpk↑+Fpk)ω+εk−εk+Q ω+εk−εk+Q+∆+iΓ −(Gpk↓−Fpk)ω+εp−Q−εp ω+εp−Q−εp+∆+iΓ/bracketrightBigg (6) is the correlated contribution, where Gσ=≪S+∆[a† σaσ]≫ ≪S+≫, F=≪∆Sza† ↑a↓≫ ≪S+≫.(7) Γ∼10-100meV is the hole spin dephasing rate [25]. Sim- ilar to Ref.[10] and the Lindblad method calculation of Ref.[14], we describe such elastic effects by substituting the spin–flip excitation energy∆ by ∆+ iΓ. We obtained GandFbysolvingthecorrespondingequationstolowest order in 1/S, with βSkept constant, which gives Σcorrto O(1/S2). More details will be presented elsewhere [18]. Results— Firstwestudythe spinstiffness D=DRPA+ Dcorr ++Dcorr −. The RPA contribution DRPAreproduces Ref.[15]. The correlated cotributions Dcorr +>0 and3 0 0.1 0.2 0.3 0.4 0.5 p00.020.040.060.08D/D0 D DRPA DRPA+D(-) 0 0.2 0.4 p00.020.040.06 50 100 150 βc (meV)00.010.02D/D0 50 100 150 βc (meV)00.020.040.06a) βc =70meV b) βc =150meV c) p =0.1 d) p =0.5 FIG. 1: (Color online) Spin stiffness Das function of hole doping and interaction strength for the single–band model. c= 1nm−3, Γ=0,D0=/planckover2pi12/2mhh,mhh= 0.5me. Dcorr −<0 were obtained to O(1/S2) from Eq.(6) [18]: Dcorr −=−/planckover2pi12 2mhS2N2/summationdisplay kp/bracketleftBigg fk↓(1−fp↓)εp(ˆp·ˆQ)2 εp−εk +fk↑(1−fp↑)εk(ˆk·ˆQ)2 εp−εk/bracketrightBigg , (8) Dcorr +=/planckover2pi12 2mhS2N2/summationdisplay kpfk↓(1−fp↑)× /bracketleftBig εk(ˆk·ˆQ)2+εp(ˆp·ˆQ)2/bracketrightBig × /bracketleftbigg2 εp−εk+1 εp−εk+∆−∆ (εp−εk)2/bracketrightbigg ,(9) whereˆQ,ˆk, andˆ pdenote the unit vectors. For ferromagnetic interaction, as in the manganites [19, 24], the Mn and carrier spins align in parallel. The Hartree–Fock is then the state of maximum spin and an exact eigenstate of the many–body Hamiltonian (Na- gaoka state). For anti–ferromagnetic β, as in (III,Mn)V semiconductors, the ground state carrier spin is anti– parallel to the Mn spin and can increase via the scat- tering of a spin– ↓hole to an empty spin– ↑state (which decreases Szby 1). Such quantum fluctuations give rise toDcorr +, Eq.(9), which vanishes for fk↓= 0.Dcorr −comes from magnon scattering accompanied by the creation of aFermi seapair. In the caseofaspin– ↑Fermi sea, Eq.(8) recovers the results of Refs.[19, 24]. We evaluated Eqs.(8) and (9) for zero temperature after introducing an upper energy cutoff corresponding to the Debye momentum, k3 D= 6π2c, that ensures the correct number of magnetic ion degrees of freedom [15].0 0.1 0.2 0.3 0.4 0.5 p00.20.4D/D0 0 0.1 0.2 0.3 0.4 0.5 p00.20.4 0 0.1 0.2 0.3 εF (eV)00.010.02D/D0 0 0.1 0.2 0.3 0.4 0.5 εF (eV)00.020.04a) βc =70meV b) βc =150meV c) βc =70meV d) βc =150meV FIG. 2: (Color online) Spin stiffness Dfor the parameters of Fig. 1. (a)and(b): two–bandmodel, (c)and(d): dependence on the Fermi energy within the single–band model. Figs. 1(a) and (b) show the dependence of Don hole doping, characterized by p=ch/c, for two couplings β, while Figs. 1(c) and (d) show its dependence on βfor two dopings p. Figure 1 also compares our full result, D, withDRPAandDRPA+Dcorr −. It is clear that the cor- relations beyond RPA have a pronounced effect on the spin stiffness, and therefore on Tc∝D[1, 7] and other magnetic properties. Similar to the manganites [19, 24], Dcorr −<0 destabilizes the ferromagneticphase. However, Dcorr +stronglyenhances Das comparedto DRPA[15] and also changes its p–dependence. The ferromagnetic order and Tcvalues observed in (III,Mn)V semiconductors cannot be explained with the single–band RPA approximation [15], which predicts a smallDthat decreases with increasing p. Figure 1 shows that the correlations change these RPA results in a profound way. Even within the single–band model, the correlations strongly enhance Dand change its p– dependence: Dnow increases with p. Within the RPA, such behavior can be obtained only by including multiple valence bands [16]. As discussed e.g. in Refs.[1, 7], the main bandstructure effects can be understood by con- sidering two bands of heavy ( mhh=0.5me) and light ( mlh=0.086me) holes. Dis dominated and enhanced by the more dispersive light hole band. On the other hand, the heavily populated heavy hole states dominate the static properties and EF. By adopting such a two–band model, we obtain the results of Figs. 2(a) and (b). The main difference from Fig. 1 is the order of magnitude en- hancement of all contributions, due to mlh/mhh= 0.17. Importantly,thedifferencesbetween DandDRPAremain strong. Regarding the upper limit of Tcdue to collective effects, we note from Ref.[7] that is is proportional to D and the mean field Mn spin. We thus expect an enhance- ment, as compared to the RPA result, comparable to the4 0 0.5 100.020.04αα αRPA 0 0.5 100.020.04 0 0.5 1 p00.020.04α 0 0.5 1 p00.020.04a) βc =70meV b) βc =100meV c) βc =120meV d) βc =150meV FIG. 3: (Color online) Gilbert damping as function of hole doping for different interactions β.c= 1nm−3,Γ = 20meV. difference between DandDRPA. The dopingdependence of Dmainlycomesfromits de- pendence on EF, shown in Figs. 2(c) and (d), which dif- fers strongly from the RPA result. Even though the two band model captures these differences, it fails to describe accuratelythe dependence of EFonp, determined by the successive population of multiple anisotropicbands. Fur- thermore, thespin–orbitinteractionreducesthe holespin matrix elements [22]. For example, |σ+ nn′|2is maximum when the bandstates arealsospin eigenstates. The spin– orbit interaction mixes the spin– ↑and spin– ↓states and reduces|σ+ nn′|2. From Eq.(3) we see that the deviations fromthe meanfield resultaredetermined bythe coupling to the Green’s functions ≪σ+ nn′∆[a† nan′]≫(RPA),≪ ∆Szσ+ nn′∆[a† nan′]≫(correctiontoRPAdueto Szfluctu- ationsleadingto Dcorr +>0), and≪∆S+σz nn′∆[a† nan′]≫ (magnon–Fermi sea pair scattering leading to Dcorr −<0). Both the RPA and the correlation contribution arising from ∆Szare proportional to σ+ nn′. Our main result, i.e. therelativeimportance of the correlation as compared to the RPA contribution, should thus also hold in the real- istic system. The full solution will be pursued elsewhere. We now turn to the Gilbert damping coefficient, α= 2S/ω×Im≪S+ 0≫−1atω→0 [9]. We obtain to O(1/S2) thatα=αRPA+αcorr, where αRPArecovers the mean–field result of Refs [9, 10] and predicts a linear dependence on the hole doping p, while αcorr=∆2 2N2S2/summationdisplay kpIm/bracketleftBigg fk↓(1−fp↑) ∆+iΓ× /parenleftbigg1 εp−εk−δ+1 εp−εk+∆+iΓ/parenrightbigg/bracketrightBigg (10) arises from the carrier spin–flip quantum fluctuations.Fig.(3) compares αwith the RPA result as function of p. The correlations enhance αand lead to a nonlinear dependence on p, which suggests the possibility of con- trolling the magnetization relaxation by tuning the hole density. A nonlinear dependence of αon photoexcitation intensity was reported in Ref.[13] (see also Refs.[12, 21]). We conclude that spin–charge correlations play an im- portant role on the dynamical properties of ferromag- netic semiconductors. For quantitative statements, they must be addressed together with the bandstructure ef- fects particular to the individual systems. The correla- tions studied here should play an important role in the ultrafast magnetization dynamics observed with pump– probe magneto–optical spectroscopy [12, 13, 14, 21, 22]. This work was supported by the EU STREP program HYSWITCH. [1] T. Jungwirth et al., Rev. Mod. Phys. 78, 2006. [2] H. Ohno, Science 281, 951 (1998). [3] S. Koshihara et al., Phys. Rev. Lett. 78, 4617 (1997). [4] H. Ohno et al., Nature 408, 944 (2000). [5] J. Wang et al., Phys. Rev. Lett. 98, 217401 (2007). [6] S. A. Wolf et al., Science 294, 1488 (2001). [7] T. K. Jungwirth et al., Phys. Rev. B 72, 165204 (2005). [8] L. D. Landau, E. M. Lifshitz, and L. P. Pitaeviski, Sta- tistical Physics, Part 2 (Pergamon, Oxford, 1980). [9] J. Sinova et. al., Phys. Rev. B69, 085209 (2004); Y. Tserkovnyak, G.A.Fiete, andB. I.Halperin, Appl.Phys. Lett.84, 25 (2004). [10] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Sta t. Sol.23, 501 (1967). [11] S. T. B. Goennenwein et al., Appl. Phys. Lett. 82, 730 (2003). [12] J. Wang et al., J. Phys: Cond. Matt. 18, R501 (2006). [13] J. Qi et al., Appl. Phys. Lett. 91, 112506 (2007). [14] J. Chovan, E. G. Kavousanaki, and I. E. Perakis, Phys. Rev. Lett. 96, 057402 (2006); J. Chovan and I. E. Per- akis, Phys. Rev. B 77, 085321 (2008). [15] J. K¨ onig, H–H Lin and A. H. MacDonald, Phys. Rev. Lett.84, 5628, (2000); M. Berciu and R. N. Bhatt, Phys. Rev. B66, 085207 (2002). [16] J. K¨ onig, T. Jungwirth, and A. H. MacDonald, Phys. Rev. B64, 184423 (2001). [17] J. Fricke, Ann. Phys. 252, 479 (1996). [18] M. D. Kapetanakis and I. E. Perakis, arXiv:0806.0938v1 . [19] M. D. Kapetanakis, A. Manousaki, and I. E. Perakis, Phys. Rev. B 73, 174424 (2006); M. D. Kapetanakis and I. E. Perakis, Phys. Rev. B 75, 140401(R) (2007). [20] K. S. Burch et. al., Phys. Rev. Lett. 97, 087208 (2006). [21] J. Wang et. al., arXiv:0804.3456; K. S. Burch at. al., Phys. Rev. B 70, 205208 (2004). [22] L. Cywi´ nski and L. J. Sham, Phys. Rev. B 76, 045205 (2007). [23] X. Liu et. al., Phys. Rev. B 71, 035307 (2005); K. Hamaya et. al., Phys. Rev. B 74, 045201 (2006). [24] D. I. Golosov, Phys. Rev. Lett. 84, 3974 (2000); N. Shannon and A. V. Chubukov, Phys. Rev. B 65, 104418 (2002).5 [25] T. Jungwirth et. al., Appl. Phys. Lett. 81, 4029 (2002).

2008-05-09

We address the role of correlations between spin and charge degrees of freedom on the dynamical properties of ferromagnetic systems governed by the magnetic exchange interaction between itinerant and localized spins. For this we introduce a general theory that treats quantum fluctuations beyond the Random Phase Approximation based on a correlation expansion of the Green's function equations of motion. We calculate the spin susceptibility, spin--wave excitation spectrum, and magnetization precession damping. We find that correlations strongly affect the magnitude and carrier concentration dependence of the spin stiffness and magnetization Gilbert damping.

Spin dynamics in (III,Mn)V ferromagnetic semiconductors: the role of correlations

0805.1320v2

arXiv:1506.04851v1 [math.AP] 16 Jun 2015Fast Energy Decay for Wave Equations with Variable Damping Coefficients in the 1-D Half Line Ryo IKEHATA∗and Takeshi KOMATSU Department of Mathematics, Graduate School of Education, Hiro shima University Higashi-Hiroshima 739-8524, Japan April 15, 2019 Abstract We derive fast decay estimates of the total energy for wave equa tions with localized variable damping coefficients, which are dealt with in the one dimensiona l half line (0 ,∞). The variable damping coefficient vanishes near the boundary x= 0, and is effective critically near spatial infinity x=∞. 1 Introduction We consider the 1-dimensional initial-boundary value prob lem for linear dissipative wave equation in the half line (0 ,+∞): utt(t,x)−uxx(t,x)+V(x)ut(t,x) = 0,(t,x)∈(0,∞)×(0,∞), (1.1) u(0,x) =u0(x), ut(0,x) =u1(x), x∈(0,∞), (1.2) u(t,0) = 0, t∈(0,∞), (1.3) where the initial data ( u0,u1) are compactly supported with the size R >0: u0∈H1 0(0,∞), u1∈L2(0,∞),suppui⊂[0,R],(i= 0,1). Let us first talk about several related results of the Cauchy p roblem in RNof the equation: utt(t,x)−∆u(t,x)+V(x)ut(t,x) = 0. (1.4) In the case when V(x) = constant >0, Matsumura [9] derived a historical Lp-Lqdecay estimates of solutions of (1.4). Later on more detail investigations s uch as the asymptotic profiles of solutions were studied by Nishihara [14] and the references therein. In the case of mixed problems of (1.4) in the exterior domain Ω ⊂RNNakao [13] and Ikehata [4] have derived precise decay estimates of the tota l energy for the initial data belonging to the energy class and weighted L2-class, respectively. In these cases they have dealt with th e localized damping coefficient V(x), which is effective only near infinity such as V(x)> V0for |x| ≫1, and the star-shaped boundary ∂Ω. In particular, fast decay estimates of the total ∗Corresponding author: ikehatar@hiroshima-u.ac.jp Keywords and Phrases: Localized damping; Wave equation; Mu ltiplier method; Total energy decay; Weighted initial data. 2010 Mathematics Subject Classification. Primary 35L20; Se condary 35L05, 35B33, 35B40. 1energy like E(t) =O(t−2) was obtained in [4] by a special multiplier method develope d in [6], which modified the Morawetz one [12]. Racke [15] also derived theLp-Lqestimates for solutions to (1.4) with constant V(x) in the exterior domain case by relying on so called the gener alized Fourier transform (in fact, more general form of equations a nd systems are considered). For related results in the case when the damping is effective near i nfinity one can cite the papers written by Aloui-Ibrahim-Khenissi [1] and Daoulatli [2], w here they studied the fast decay of the total and local energy under the so called GCC assumption due to Lebeau [8]. Exponential decay of the total energy for wave equations with mass (i.e., Klein-Gordon type) and localized damping terms was investigated by Zuazua [19] in the exterio r domain case. Furthermore, by Watanabe [18] an effective application of the method develope d in [4] is recently introduced, and there a global existence and decay estimates of solution s to a nonlinear Cauchy problem of the quasilinear wave equation with a localized damping near infinity are studied. On the other hand, recently in [7] the critically decaying da mping coefficient V(x) near spatial infinity was considered for (1.4) in the whole space RN, in fact, they have studied so called the critical case with δ= 1 below: V0 (1+|x|)δ≤V(x)≤V1 (1+|x|)δx∈RN, (1.5) and they have announced the fact that E(t) =O(t−N) ifV0> N, while if V0≤N, then (roughly speaking) E(t) =O(t−V0). In this connection, we call it sub-critical damping in the case of (1.5) withδ∈[0,1), while in the case when δ >1, (1.5) is called as super-critical damping. The precise decay order of several norms of solutions for the Cau chy problem in RNof (1.4) with (1.5) were investigated by Todorova-Yordanov [16] in the su b-critical case. Furthermore, in the super critical case it is well-known by Mochizuki [10] that t he solution of the Cauchy problem of (1.4) is asymptotically free. While, in the constant damping case with V(x) = constant of (1.1)-(1.3) (i.e., problem in the half line) was studied in [3], and he derived E(t) =O(t−2) (t→+∞). In the localized damping case, it seems that there are no any previous researc h papers that are dealing with the half space problem and critical damping near infinity. In thi s connection, one notes that in [5] Ikehata-Inoue have derived E(t) =O(t−1) ast→+∞forV0>1 (see also Mochizuki-Nakazawa [11] and Uesaka [17] for the topics on the energy decay proper ty). In fact, they treated a more general nonlinear equation like utt(t,x)−∆u(t,x)+V(x)ut(t,x)+|u(t,x)|p−1u(t,x) = 0, in the critical damping case of (1.5). So, a natural question arises whether one can derive fast dec ay result like E(t) =O(t−2) in the case when V(x) vanishes near the boundary and is effective critically (i.e. ,δ= 1 in (1.5)) only near spatial infinity x=∞. The main purpose of this paper is to answer this natural question by modifying a technical method originally develo ped in [5]. Our assumptions on V(x) are as follows: (A)V∈L∞(0,∞)∩C([0,∞)), and there exist three constants V0>2, V1∈[V0,+∞), L2>0 such thatV0 1+x≤V(x)≤V1 1+x, x∈[L2,+∞), 0≤V(x) (x∈[0,∞)). Our new results read as follows. 2Theorem 1.1 LetV(x)satisfy the assumption (A). If the initial data (u0,u1)∈H1 0(0,∞)× L2(0,∞)further satisfies suppu 0∪suppu 1⊂[0,R] for some R >0, then for the solution u∈C([0,∞);H1 0(0,∞))∩C1([0,∞);L2(0,∞))to problem (1.1)−(1.3)one has E(t) =O(t−2),(t→ ∞). Remark 1.1 In the 1-D whole space case, if we consider the corresponding Cauchy problem, we can have E(t) =O(t−1),(t→ ∞) provided that V0>1 (cf., [7], [17]). So, the results seem to reflect the half spa ce property itself. This discovery of the new number V0>2in the half space case is one of our maximal contribution. Even if one takes L2= 0 formally, the obtained result in Theorem 1.1 is new. Remark 1.2 In the case when V0≤2, we still do not know the exact rate of decay for E(t). However, from [7, Theorem 4.1] one can state a conjecture tha tE(t) =O(t−V0) (t→+∞) if V0≤2. Furthermore, the result will be generalized to the N-dimensional half space problems of (1.1)-(1.3). We can present a conjecture that if V0> N+1, then E(t) =O(t−(N+1)) (t→+∞) in the half space case. Remark 1.3 The assumption ( A) implies that the damping coefficient V(x) can vanish near the boundary x= 0, so the damping is effective only near infinity. But, the effect iveness of the damping V(x) near infinity corresponds to so called the critical case of ( 1.5) with δ= 1 (see [7]). In this sense, the result obtained in Theorem 1.1 is der ived under two types of difficult situations, that is, one is vanishing damping, and the other is critical one. This paper is organized as follows. In section 2 we shall prov e Theorem 1.1 by relying on a multiplier method which was originally introduced by the fir st author’s collaborative work [5] and [6]. Notation. Throughout this paper, /bardbl·/bardblmeans the usual L2(Ω)-norm. The total energy E(t) to the solution u(t,x) of (1.1) is defined by E(t) :=1 2(/bardblut(t,·)/bardbl2+/bardblux(t,·)/bardbl2). Furthermore, one sets as a L2(0,∞) inner product: (u,v) :=/integraldisplay+∞ 0u(x)v(x)dx. A weighted function space is defined as follows: u∈L1,1/2(0,∞) iffu∈L1(0,∞) and /bardblu/bardbl1,1/2:=/integraldisplay∞ 0√ 1+x|u(x)|dx <+∞. Furthermore, one sets X1(0,∞) :=C([0,∞);H1 0(0,∞))∩C1([0,∞);L2(0,∞)). 32 Proof of Results We first prepare an important lemma, which is borrowed from [3 , Lemma 2.1]. Lemma 2.1 It holds that sup 0≤x<+∞(|u(x)|√1+x)≤ /bardblux/bardbl for allu∈H1 0(0,∞). Lemma 2.2 Letu∈X1(0,∞)be the solution to problem (1.1)−(1.3). Then it is true that d dtE(t)+F(t) = 0,(t≥0) where E(t) =1 2d dt/integraldisplay∞ 0[f(u2 t+u2 x)+2guut+(gV−gt)u2+2huxut]dx, F(t) =1 2/integraldisplay∞ 0(2fV−ft−2g+hx)u2 tdx+1 2/integraldisplay∞ 0(2g−ft+hx)u2 xdx +1 2/integraldisplay∞ 0(gtt−gtV)u2dx+/integraldisplay∞ 0(hV−ht)uxutdx−1 2/integraldisplay∞ 0d dx(hux)dx. Moreover, f=f(t), g=g(t)and h=h(t,x)are all smooth functions specified later on. Proof.Since one considers the weak solutions u(t,x) of the problem (1.1)-(1.3), we can assume to check the desired identity that the solution u(t,x) is sufficiently smooth, and vanishes for largex≫1. Step 1. Multiplying the both sides of (1.1) by futand rearranging it one can get: fututt−futuxx+fVu2 t =f1 2d dtu2 t−d dx(futux)+1 2d dtfu2 x−1 2ftu2 x+fVu2 t =1 2d dt[f(u2 t+u2 x)]+(fV−1 2ft)u2 t−1 2ftu2 x−d dx(futux) = 0. Step 2. Multiplying the both sides of (1.1) by guand rearranging it one has: guutt−guuxx+gVutu =gd dtuut−gu2 t−d dx(guux)+gu2 x+gV1 2d dtu2 =d dtguut−gtuut−gu2 t−d dx(guux)+gu2 x+1 2d dt(gVu2)−1 2gtVu2 =d dt(guut)−1 2d dt(gtu2)+1 2gttu2−gu2 t−d dx(guux)+gu2 x+1 2d dt(gVu2)−1 2gtVu2= 0. Step 3. Multiplying the both sides of (1.1) by huxand rearranging it one obtains: huxutt−huxuxx+huxVut =h(d dtuxut)−hd dt(ux)ut−1 2hd dx(u2 x)+hVutux =d dt(huxut)−htuxut−1 2hd dx(u2 t)−1 2d dx(hu2 x)+1 2hxu2 x+hVutux 4=d dt(hutux)−htuxut−1 2d dx(hu2 t)+1 2hxu2 t−1 2d dx(hu2 x)+1 2hxu2 x+hVutux= 0. Step 4. Add all final identities from Step 1 to Step 3 and integrate ove r [0,∞). Then, it follows from the boundary condition (1.3) that one can get: 1 2d dt/integraldisplay∞ 0[f(u2 t+u2 x)+2guut+(gV−gt)u2+2hutux]dx +1 2/integraldisplay∞ 0(2fV−ft−2g+hx)u2 tdx+1 2/integraldisplay∞ 0(2g−ft+hx)u2 xdx +1 2/integraldisplay∞ 0(gtt−gtV)u2dx+/integraldisplay∞ 0(hV−ht)uxutdx−1 2/integraldisplay∞ 0d dx(hu2 x)dx= 0, which implies the desired identity. ✷ Since the finite speed of propagation property can be applied to the solution u(t,x) of the corresponding problem (1 .1)−(1.3), in order to estimate the functions E(t) andF(t) it suffices to consider all the spatial integrand over the closed interv al [0,R+t] (t≥0). Now, because of the assumption ( A) such as V(L2)>0 and the continuity of V(x), there exist constants L1∈[0,L2) andVm>0 such that V(x)≥Vm(x∈[L1,L2]). By using these constants we choose the functions f(t),g(t) andh(t,x) as follow: f(t) =ǫ1(1+t)2, g(t) =ǫ2(1+t), h(t,x) =ǫ3(1+t)φ(x), where the monotone increasing function φ∈C∞([0,∞)) can be defined to satisfy φ(x) = (1+x) (0≤x < L1), 1+L2(L2≤x),(2.1) whereǫi>0 (i= 1,2,3) are some constants determined later on. Lemma 2.3 Letf,gandhbe defined by (2.1). Then one has the following estimates:for each t≥0, (i)/integraldisplay∞ 0(hV−ht)utuxdx≥ −k 2/integraldisplay∞ 0hVu2 tdx−1 2k/integraldisplay∞ 0hVu2 xdx−1 2/integraldisplay∞ 0htu2 tdx−1 2/integraldisplay∞ 0htu2 xdx, (ii)/integraldisplay∞ 0d dx(hu2 x)dx≤0, wherek >0is a constant. Proof.(i): This is easily derived from the following inequality wi th some positive number p: |utux| ≤1 2pu2 t+p 2u2 x. (ii): Because of the finite speed of propagation property, on e has /integraldisplay∞ 0d dx(hu2 x)dx=h(t,∞)ux(t,∞)2−h(t,0)ux(t,0)2 =−h(t,0)ux(t,0)2≤0.✷ Thus, it follows from Lemmas 2.2 and 2.3 that for t≥0 1 2d dt/integraldisplay∞ 0[f(u2 t+u2 x)+2guut+(gV−gt)u2+2hutux]dx 5+1 2/integraldisplay∞ 0(2fV−ft−2g+hx−khV−ht)u2 tdx +1 2/integraldisplay∞ 0(2g−ft+hx−1 khV−ht)u2 xdx+1 2/integraldisplay∞ 0(gtt−gtV)u2dx≤0. (2.2) Furthermore, one can get the estimates below: Lemma 2.4 Let f, g and h be defined by (2.1), and assume (A). If all parameters ǫi(i= 1,2,3) are well-chosen, then there exists a large t0>0such that for all t≥t0≫0,one has (iii)2f(t)V(x)−ft(t)−2g(t)+hx(t,x)−kh(t,x)V(x)−ht(t,x)>0, x∈[0,∞), (iv)2g(t)−ft(t)+hx(t,x)−1 kh(t,x)V(x)−ht(t,x)>0, x∈[0,∞) with some constant k >0. Proof.We first check (iii) by separating the integrand of xinto 3 parts [0 ,L1], [L1,L2], and [L2,+∞). (iii)The case 0 ≤x≤L1: 2fV−ft−2g+hx−khV−ht = 2ǫ1(1+t)2V(x)−2ǫ1(1+t)−2ǫ2(1+t)+ǫ3(1+t)−kǫ3(1+t)(1+x)V(x)−ǫ3(1+x) ≥(1+t)2{2ǫ1−kǫ31+L1 1+t}V(x)+(1+t){ǫ3−2ǫ1−2ǫ2−ǫ31+L1 1+t}. The case L1≤x≤L2: 2fV−ft−2g+hx−khV−ht = 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) ≥(1+t)2{2ǫ1Vm−2ǫ11 1+t−2ǫ21 1+t−kǫ3VM1+L2 1+t−ǫ31+L2 (1+t)2}, whereVM:= sup{V(x)|0≤x≤L2}>0, and we have just used the fact that φ′(x)≥0. The case L2≤x: Here we use the finite speed of propagation property of the so lution below: 2fV−ft−2g+hx−khV−lht = 2ǫ1(1+t)2V(x)−2ǫ1(1+t)−2ǫ2(1+t)−kǫ3(1+t)(1+L2)V(x)−ǫ3(1+L2) ≥2ǫ1(1+t)2V0 1+x−2ǫ1(1+t)−2ǫ2(1+t)−kǫ3(1+t)(1+L2)V1 1+x−ǫ3(1+L2) ≥(1+t)2 1+x{2ǫ1V0−2ǫ11+R+t 1+t−2ǫ21+R+t 1+t−kǫ3(1+L2)V1 1+t−ǫ3(1+L2)1+R+t (1+t)2}. Note that in this region, φ′(x) = 0. Next one can check the condition (iv) similarly to the case (i ii). (iv)The case 0 ≤x≤L1: 2g−ft+hx−1 khV−ht = 2ǫ2(1+t)−2ǫ1(1+t)+ǫ3(1+t)−ǫ3 k(1+t)(1+x)V(x)−ǫ3(1+x) 6≥(1+t){2ǫ2−2ǫ1+ǫ3−ǫ3 k(1+L1)VM−ǫ31+L1 1+t}. The case L1≤x≤L2: 2g−ft+hx−1 khV−ht = 2ǫ2(1+t)−2ǫ1(1+t)+ǫ3(1+t)φ′(x)−ǫ3 k(1+t)φ(x)V(x)−ǫ3φ(x) ≥(1+t){2ǫ2−2ǫ1−ǫ3 k(1+L2)VM−ǫ31+L2 1+t}. The case L2≤x: 2g−ft+hx−1 khV−ht = 2ǫ2(1+t)−2ǫ1(1+t)−ǫ3 k(1+t)(1+L2)V(x)−ǫ3 l(1+L2) ≥(1+t){2ǫ2−2ǫ1−ǫ2 k(1+L2)V1 1+x−ǫ21+L2 1+t} ≥(1+t){2ǫ2−2ǫ1−ǫ3 kV1−ǫ21+L2 1+t}. So, in order to obtain (iii) and (iv) for large t≫1 the following five conditions should be satisfied to ǫi(i= 1,2,3) andk >0: ǫ3−2ǫ1−2ǫ2>0, (2.3) 2ǫ1V0−2ǫ1−2ǫ2>0, (2.4) 2ǫ2−2ǫ1+ǫ3−ǫ3 k(1+L1)VM>0, (2.5) 2ǫ2−2ǫ1−ǫ3 k(1+L2)VM>0, (2.6) 2ǫ2−2ǫ1−ǫ3 kV1>0. (2.7) (2.3) and (2.4) come from the check of (iii), while (2.5)-(2. 7) have its origin in the case when we check (iv) as t→+∞. We need to look for the constants ǫ1,ǫ2,ǫ3,k >0 satisfying these five conditions (2.3)-(2.7) above. First, the condition (2 .5) and (2 .6) and (2 .7) can be unified by the following one condition (2.8): 2ǫ2−2ǫ1+ǫ3−ǫ3 k(1+L1)VM≥2ǫ2−2ǫ1−ǫ3 k(1+L2)VM ≥2ǫ2−2ǫ1−ǫ3 kV∗>0, (2.8) where V∗:= max{(1+L2)VM, V1}>0. 7Thus it suffices to check only 3 conditions (2 .3), (2.4) and (2 .8) above. However, for its purpose it is enough to choose all constants ǫi>0 (i= 1,2,3) andk >0 as follows: ǫ1:= 1, ǫ2:=V0 2, ǫ3:= 2V0, and k:=4V0V∗ V0−2, where the assumption V0>2 is essentially used. Therefore, one has the desired estima tes if one takest≫1 sufficiently large. ✷ Next lemma is a direct consequence of (2.2) and Lemma 2.4. Lemma 2.5 Letu∈X1(0,∞)be the solution to problem (1.1)−(1.3), andf,g,hbe defined by (2.1). Under the condition (A), the following estimate holds true: d dt{f(t)E(t)+g(t)(ut,u)+2(hux,ut)} ≤1 2d dt/integraldisplay∞ 0(gt−gV)u2dx+1 2/integraldisplay∞ 0(gtV−gtt)u2dx(2.9) for allt≥t0, wheret0≫1is a fixed time defined in Lemma 2.4. Then we integrate the both sides of (2 .9) over [t0,t]: f(t)E(t)+g(t)(ut,u)+2(hux,ut) ≤f(t0)E(t0)+g(t0)(ut(t0),ut(t0))+2(h(t0)ux(t0),ut(t0)) +1 2/integraldisplay∞ 0(gt−gV)u2dx−1 2/integraldisplay∞ 0(gt(t0)−g(t0)V)u2(t0)dx+1 2/integraldisplayt t0/integraldisplay∞ 0(gtV−gtt)u2dxds =C+1 2/integraldisplay∞ 0(gt−gV)u2dx+1 2/integraldisplayt t0/integraldisplay∞ 0(gtV−gtt)u2dxds, (2.10) where the constant C >0, which is independent from t≥t0, is defined by C:=f(t0)E(t0)+g(t0)(ut(t0),ut(t0))+2(h(t0)ux(t0),ut(t0)) −1 2/integraldisplay∞ 0(gt(t0)−g(t0)V)u2(t0)dx. Furthermore, we have the following estimates. Lemma 2.6 Letgbe defined by (2.1). The the smooth function gsatisfies the following two estimates: (v)gt−gV(x)≤C1, x∈[0,∞), t≥0, (vi)gtV−gtt≤C2V(x), x∈[0,∞), t≥0, whereCi>0 (i= 1,2)are some constants. Proof.The proof can be easily checked, so we omit it. ✷ Ontheotherhand,weshallpreparethefollowingcruciallem mabasedontheonedimensional Hardy-Sobolev inequality in the half space case, which is st ated in Lemma 2.1. 8Lemma 2.7 Letu∈X1(0,∞)be the solution to problem (1.1)−(1.3). Then it is true that /bardblu(t,·)/bardbl2+/integraldisplayt 0/integraldisplay∞ 0V(x)|u(s,x)|2dxds≤C(/bardblu0/bardbl2+/bardbl(V(·)u0+u1)/bardbl2 1,1/2), (2.11) provided that /bardbl(V(·)u0+u1)/bardbl1,1/2<+∞. Proof.The original idea comes from [6]. We introduce an auxiliary f unction w(t,x) :=/integraldisplayt 0u(s,x)ds. Thenw(t,x) satisfies wtt−wxx+V(x)wt=V(x)u0+u1,(t,x)∈(0,∞)×(0,∞), (2.12) w(0,x) = 0, wt(0,x) =u0(x), x∈(0,∞). (2.13) Multiplying (2 .12) bywtand integrating over [0 ,t]×[0,∞) we get 1 2(/bardblwt(t,·)/bardbl2+/bardblwx(t,·)/bardbl2)+/integraldisplayt 0/bardbl/radicalBig V(·)ws(s,·)/bardbl2ds =1 2/bardblu0/bardbl2+/integraldisplayt 0(V(·)u0+u1,ws)ds. (2.14) Next step is to use Lemma 2.1 to obtain a series of inequalitie s below: /integraldisplayt 0(V(·)u0+u1,ws)ds=/integraldisplayt 0d ds(V(·)u0+u1,w)ds ≤/integraldisplay∞ 0√ 1+x|V(x)u0+u1||w(t,x)|√1+xdx ≤( sup x∈[0,∞)|w(t,x)|√1+x)/bardblV(·)u0+u1/bardbl1,1/2 ≤1 4/bardblwx/bardbl2+/bardblV(·)u0+u1/bardbl2 1,1/2. (2.15) Combining (2 .14) with (2 .15) we can derive 1 2/bardblwt(t,·)/bardbl2+1 4/bardblwx(t,·)/bardbl2+/integraldisplayt 0/integraldisplay∞ 0V(x)wt(s,x)dxds ≤1 2/bardblu0/bardbl2+/bardblV(·)u0+u1/bardbl2 1,1/2. The desired estimate follows from the estimate above and the fact that wt=u.✷ It follows from (2.10) and Lemmas 2.6 and 2.7 that there exist s a constant C >0 such that f(t)E(t)+g(t)(u(t,·),ut(t,·))+2(hux,ut)≤C(t≥t0), (2.16) provided that /bardbl(V(·)u0+u1)/bardbl1,1/2<+∞. Finally, we can derive the following lemma. 9Lemma 2.8 Lethbe defined by (2.1). Then, for all t≥t0≫1it is true that f(t)E(t)+2(hux,ut)≥Cf(t)E(t), whereC >0is a constant. Proof.Indeed, one has f(t)E(t)+2(hux,ut)≥1 2/integraldisplay∞ 0f(t)(u2 t+u2 x)dx−/integraldisplay∞ 0h(t,x)(u2 x+u2 t)dx ≥1 2/integraldisplay∞ 0(f(t)−2h(t,x))(u2 x+u2 t)dx. On the other hand, if necessarily, by choosing t0≫1 further large enough, one can derive the following estimates for t≥t0: f(t)−2h(t,x)≥ǫ1(1+t)2−2ǫ3(1+t)(1+L2) = (1+t)2{ǫ1−2ǫ3(1+L2) 1+t} ≥C(1+t)2≥Cf(t), with some constant C >0. Here we have just used the monotonicity of the function φ(x) closely related with the definition of the function h(t,x).✷ Now we can finalize the proof of Theorem 1.1. Proof of Theorem 1.1. We first note that one can use Lemma 2.7 because one can check /bardbl(V(·)u0+u1)/bardbl1,1/2<+∞under the assumption on the initial data stated in Theorem 1. 1. Thus, by using the Schwarz inequality, (2.16) and Lemma 2.8 w e get C(1+t)2E(t)≤g(t)/bardblu(t,·)/bardbl/bardblut(t,·)/bardbl+C≤Cg(t)/radicalBig E(t)+C, t≥t0. Furthermore, if we set X(t) =/radicalbig E(t) fort∈[0,+∞), then one has f(t)X(t)2−Cg(t)X(t)−C≤0, t≥t0. (2.17) By solving the quadratic inequality (2 .17) forX(t) we have /radicalBig E(t)≤Cg(t)+/radicalbig C2g(t)2+4Cf(t) 2f(t)(t≥t0). This inequality leads to E(t)≤C/parenleftbiggg(t) f(t)/parenrightbigg2 +C/parenleftbigg1 f(t)/parenrightbigg , t≥t0, which implies the desired decay estimates. ✷ Acknowledgment. The work of the first author (R. IKEHATA) was supported in part by Grant- in-Aid for Scientific Research (C) 15K04958 of JSPS. 10References [1] L. Aloui, S. Ibrahim and M. Khenissi, Energy decay for lin ear dissipative wave equations in exterior domains, J. Diff. Eqns 259 (2015), 2061-2079. [2] M. Daoulatli, Energy decay rates for solutions of the wav e equation with linear damping in exterior domain, arXiv:1203.6780v4. [3] R. Ikehata, A remark on a critical exponent for the semili near dissipative wave equation in the one dimensional half space, Diff. Int. Eqns 16, No. 6 (2003) , 727-736. [4] R. Ikehata, Fast decay of solutions for linear wave equat ions with dissipation localized near infinity in an exterior domain, J. Diff. Eqns 188 (2003), 390-40 5. [5] R. Ikehata and Y. Inoue, Total energy decay for semilinea r wave equations with a critical potential type of damping, Nonlinear Anal. 69 (2008), 1396- 1401. [6] R. Ikehata and T. Matsuyama, L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002), 33-42. [7] R. Ikehata, G. Todorova and B. Yordanov, Optimal decay ra te of the energy for linear wave equations with a critical potential, J. Math. Soc. Japan 65, No. 1 (2013), 183-236. [8] G. Lebeau, ´Equations des ondes amorties, Algebraic and geometric meth ods in Math. Physics, A. Boutet de Monvel and V. Marchenko (eds), Kluwer A cademic, The Nether- lands, (1996), 73-109. [9] A. Matsumura, On the asymptotic behavior of solutions of semilinear wave equations, Publ. RIMS Kyoto Univ. 12 (1976), 169-189. [10] K. Mochizuki, Scattering theory for wave equations wit h dissipative terms, Publ. Res. Inst. Math. Sci. 12 (1976), 383-390. [11] K. Mochizuki and H. Nakazawa, Energy decay and asymptot ic behavior of solutions to the wave equations with linear dissipation, Publ. Res. Inst. Ma th. Sci. 32 (1996), 401-414. [12] C. Morawetz, The decay of solutions of the exterior init ial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961), 561-568. [13] M. Nakao, Energy decay for the linear and semilinear wav e equations in exterior domains with some localized dissipations, Math. Z. 238 (2001), 781- 797. [14] K. Nishihara, Lp-Lqestimates to the damped wave equation in 3-dimensional spac e and their application, Math. Z. 244 (2003), 631-649. [15] R. Racke, Decay rates for solutions of damped systems an d generalized Fourier transforms, J. Reine Angew. Math. 412 (1990), 1-19. [16] G. Todorova and B. Yordanov, Weighted L2-estimates of dissipative wave equations with variable coefficients, J. Diff. Eqns 246 (2009), 4497-4518. [17] H. Uesaka, The total energy decay of solutions for the wa ve equation with a dissipative term, J. Math. Kyoto Univ. 20 (1980), 57-65. [18] T. Watanabe, Global existence and decay estimates for q uasilinear wave equations with nonuniform dissipative term, Funk. Ekvac. 58 (2015), 1-42. 11[19] E. Zuazua, Exponential decay for the semilinear wave eq uation with localized damping in unbounded domains, J. Math. Pures Appl. 70 (1991), 513-529. 12

2015-06-16

We derive fast decay estimates of the total energy for wave equations with localized variable damping coefficients, which are dealt with in the one dimensional half line $(0,\infty)$. The variable damping coefficient vanishes near the boundary $x = 0$, and is effective critically near spatial infinity $x = \infty$.

Fast energy decay for wave equations with variable damping coefficients in the 1-D half line

1506.04851v1

arXiv:1104.3002v1 [cond-mat.stat-mech] 15 Apr 2011Lagrangian approach and dissipative magnetic systems Thomas Bose and Steffen Trimper Institute of Physics, Martin-Luther-University, D-06099 Halle, Germany∗ (Dated: September 18, 2018) Abstract A Lagrangian is introduced which includes the coupling betw een magnetic moments mand the degrees of freedom σof a reservoir. In case the system-reservoir coupling break s the time reversal symmetry the magnetic moments perform a damped precession a round an effective field which is self-organized by the mutual interaction of the moments. Th e resulting evolution equation has the form of the Landau-Lifshitz-Gilbert equation. In case the b ath variables are constant vector fields the moments mfulfill the reversible Landau-Lifshitzequation. Applying Noether’s theorem we find conserved quantities under rotation in space and within the configuration space of the moments. PACS numbers: 75.78.-n, 11.10.Ef, 75.10.Hk ∗thomas.bose@physik.uni-halle.de; steffen.trimper@physik.uni-halle.de 1I. INTRODUCTION The dynamics of magnetic systems is described in a wide range of time a nd length scales from a quantum approach up to a macroscopic thermodynamic acce ss. On a coarse-grained mesoscopic level the relevant electronic degrees of freedom are g rouped into effective mag- netic moments. As the consequence the magnetization is characte rized by a spatiotemporal vector field m(r,t). Owing to the mutual interaction between the magnetic moments t hey perform a precession motion around a local effective field giving rise t o the propagation of spin-wave excitations. Due to a system-reservoir coupling the pre cession of the moments should be a damped one. To analyze this situation one has to specify t he coupling between the system and the bath. The most popular approach to incorpora te dissipation is the em- bedding of the relevant system into a quantum-statistical environ ment which is assumed to remain in thermal equilibrium. The reservoir is often represented by harmonic oscillators or spin moments which are analyzed by path integral techniques [1–3]. A specification of the path integral approach for spin systems can be found in [4–7]. A mor e generic description concerning dissipative semiclassical dynamics is presented in [8]. Altho ugh the application of path integrals can be considered as an intuitive formalism analytica l calculations are often impossible and numerical schemes are necessary. In the present p aper we propose an alter- native way to include dissipative effects for mesoscopic magnetic sys tems. On this level the analysis of magnetodynamics is performed properly by applying the L andau-Lifshitz-Gilbert equation designated as LLG [9, 10]. A comprehensive survey of magn etization dynamics is given in [11]. Our investigation can be grouped in the effort to underst and dissipative mech- anisms in magnets. So, a non-linear dissipative model for magnetic sy stems was discussed in [12]. On the relation between fluctuation-dissipation theorems and d amping terms like that one occurring in the LLG was reported in [13]. The dynamical respons e of ferromagnetic shape memory alloy actuators can be modeled by means of a dissipativ e Euler-Lagrange equation as performed in [14]. Likewise, the pinning of magnetic domain walls in multifer- roics is discussed in terms of the EL equations in [15]. An alternative a nsatz is introduced in [16], where a Lagrangian density is obtained based on a projection on to the complex plane. The procedure gives rise to a dynamical equation which is equivalent t o the Landau-Lifshitz equation. Different to the mentioned approaches the present pap er is aimed to derive an equation of motion for a magnetic system on a mesoscopic scale unde r the influence of a 2bath which likewise consists of mesoscopic moments. Following this idea we propose a La- grangian comprising both fields, m(r,t) as the system variables and σas the bath variables. The bath becomes dynamically active by the coupling to the system. I n case the coupling between system and reservoir breaks the time reversal symmetr y the motion of the moments m(r,t) is damped. The Lagrangian is modified in such a manner that dissipatio n can occur. II. THE LAGRANGIAN As indicated we are interested to construct a Lagrangian describin g the motion of a magne- tization vector field within a bath of spins. This reservoir should influe nce the measurable magnetization due to the mutual interaction. Let us formulate the general assumptions for the underlying model. The magnetic order is originated by single magne tic atoms which occupy equivalent crystal positions. Here we refer to a continuou s description in terms of a field vector denoted as m(r,t). Because the ferromagnet is considered below the Curie temperature a sufficient number of microscopic spins preferring a p arallel alignment are in- cluded in m, i.e. the effective magnetic moment is given by m(r,t) =/summationtext iµiwhere the sum is extended over all microscopic moments within a small volume aro und the spatial coordinate rat timet. As each axial vector the moment fulfills m(−t) =−m(t). The bath in which the moments are embedded consists likewise of mesoscopic sp ins. They are denoted asσand are also composed of microscopic moments ηi. This bath moments which play the role of ’virtual’ moments are also axial vectors changing their sig n by time inversion. A further new aspect is that the coupling between the real and the virtual moments is not assumed to be weak. As the result the complete system consists of two subsystems. One of them abbreviated as L1 is occupied exclusively by the real spins with t he moments mand the other one denoted as L2 is occupied by the bath spins σ. The situation is illustrated in FIG. 1. Now let us introduce the action S[{qα}] =/integraldisplay dt/integraldisplay d3xL[{qα}], (1) where the set {qα}consists of the set of both moments σandm. The Lagrange density comprises three terms L[m,˙m,∇m;σ,∇σ] =L(m)[m,˙m,∇m]+L(σ)[∇σ]+L(mσ)[σ,˙m], (2) 3µi−1 µi µi+1 ηj−k ηj ηj+1 ηj+1+kJµηJηηJµµ FIG. 1. (Color online) Schematic illustration of the basic m odel. The red spins represent the magnetic moments µiand refer to the lattice L1 introduced in the text. The green s pin vectors ηibuild the bath lattice L2. Interactions are possible betwee n theµiandηj,µiandµjandηi andηj. The respective coupling strengths correspond to the coupl ing parameters in Eqs. (3)-(5) as follows: Jµη↔J(mσ),Jµµ↔J(m)andJηη↔J(σ). whereL(m)indicates the Lagrangian of the magnetic system, L(σ)represents the reservoir and the interaction term is denoted as L(mσ). To be more specific the magnetic moments of the system interact via exchange coupling defined by the Lagrangia n L(m)=1 2J(m) αβ∂mν ∂xα∂mν ∂xβ+Aν(m) ˙mν, (3) whereJ(m) αβisthecouplingparameter, diagonalintheisotropiccase. Thefirstt ermrepresents the energy density of the magnetic system. Because we are not co nsidering the acceleration of magnetic moments a term of the order ˙m2is missing. Moreover, the magnetic moments performaprecessionaroundaneffective magneticfield, whichisself -organizedbythemutual interaction. Thereforethevectorpotential Adependsonthemoments, i. e. A=A(m(r,t)). The coupling has the same form as the minimal coupling in electrodynam ics. The bath Lagrangian is defined in a similar manner as L(σ)=1 2J(σ) αβ∂σν ∂xα∂σν ∂xβ, (4) with the coupling constant J(σ) αβ. Eventually, the interacting part between system and bath is written as L(mσ)=J(mσ) αβ∂mν ∂xα∂σν ∂xβ+Bν(σ) ˙mν, (5) 4with the coupling strength J(mσ) αβ. The second term is constructed in the same manner as in Eq. (3), where the potential B(σ) will be specified below, see Eq. (8). The dynamics of the bath variable σremains unspecified for the present, i.e. the Lagrangian does not include a term of the form ∝˙σ. Owing to the constraint, introduced in the next section, the dynamically passive bath is sensitive to a change of the system va riablesmin such a manner that small variations of the system variables mare related to small variations of σ. This procedure leads to a coupling between bath and system so tha t the time reversal symmetry is broken. III. RELATION TO THE LANDAU-LIFSHITZ-GILBERT EQUATION In this section we find the equation of motion for the magnetization m(r,t) from Eq. (2) combined with Eqs. (3)-(5). Using the principle of least action it follow s /bracketleftbigg∂L ∂σβ−∂ ∂xα∂L ∂/parenleftBig ∂σβ ∂xα/parenrightBig/bracketrightbigg δσβ+/bracketleftbigg∂L ∂mβ−∂ ∂t∂L ∂˙mβ−∂ ∂xα∂L ∂/parenleftBig ∂mβ ∂xα/parenrightBig/bracketrightbigg δmβ= 0,(6) whereδmβandδσβare the small variations which drive the value for the action out of th e stationary state. In general, one derives a system of coupled par tial differential equations. However, to proceed further let us impose a constraint on the sys tem. A small variation of σβshould be related to a small variation of mβ. Thus, we make the ansatz δσβ=−κδmβ,withκ= const>0. (7) Notice that this condition should be valid only locally but not globally. Ins ofar Eq. (7) is comparable to an anholonom condition in mechanics. Moreover relatio n (7) is in accordance with thebehavior of themoments mandσunder timeinversion. Physically the last relation means that the bath reacts to a change of the system only tempor arily. Because the system- reservoir coupling should typically break the time reversal symmetr y the expansion of the functionBν(σ) in terms of σincludes only odd terms. In lowest order we get from Eq. (5) Bν(σ) =−cσν,withc= const. (8) Due to Eqs. (7) and (8) the second term in Eq. (5) is of the form ∝σ·˙m. Such a term is not invariant under time reversal symmetry t→ −t. As demonstrated below the broken time 5inversion invariance gives rise to damping effects. Inserting Eqs. (7 ) and (8) into Eq. (2) and performing the variation according to Eq. (6) we get 0 =/parenleftbigg∂Aν ∂mβ−∂Aβ ∂mν/parenrightbigg ˙mν+c˙σβ+κc˙mβ −[J(m)−κJ(mσ)]∇2mβ−[J(mσ)−κJ(σ)]∇2σβ.(9) Here we have assumed for simplicity that all coupling tensors Jare diagonal: Jαβ=Jδαβ. The first term on the right hand side in Eq. (9) reminds of the field str ength tensor in electrodynamics [17]. Thus, we rewrite /parenleftbigg∂Aν ∂mβ−∂Aβ ∂mν/parenrightbigg ˙mν≡Fβν˙mν=/bracketleftBig ˙m×(∇m×A(m))/bracketrightBig β. (10) As mentioned above the vector function A(m) is regarded as vector potential which depends on space-time coordinates via the magnetic moment m(r,t). In vector notation the last equation reads ˙m×(∇m×A) = [J(m)−κJ(mσ)]∇2m−κc˙m−c˙σ+[Jmσ−κJ(σ)]∇2σ.(11) If one is interested in weak excited states of a ferromagnet it is rea sonable to assume that the direction of the magnetization in space changes slowly while its abs olute value is fixed, that ism2= 1. Without loss of generality we have set the amplitude of mto unity. In order to proceed it is necessary to specify the condition which should be fu lfilled by the function A(m). Having in mind the LLG then we make the ansatz ∇m×A(m) =gm,g= const. Based on these assumptions we get from Eq. (11) ∂m ∂t=1 g/parenleftBigg m×Heff/parenrightBigg −κc g/parenleftBigg m×∂m ∂t/parenrightBigg . (12) Here the effective field is given by the expression Heff=/parenleftBig J(m)−κJ(mσ)/parenrightBig ∇2m−c∂σ ∂t+/parenleftBig J(mσ)−κJ(σ)+/parenrightBig ∇2σ. (13) Eq. (12) is nothing else than the Gilbert equation [10] by relating the prefactors as follows γ=−1 g, α=−κc g=κcγ, (14) whereγandαare the gyromagnetic ratio and the Gilbert damping parameter, res pectively. Since bothparameters arepositive quantities itfollows that g<0aswell as κc>0. Further, 6Eq. (12) can be converted into the form of the equivalent and widely used Landau-Lifshitz- Gilbert equation which reads ∂m ∂t=−γ (1+α)2(m×Heff)−αγ (1+α2)/bracketleftBig m×(m×Heff)/bracketrightBig , (15) Both quantities γandαare still related to the model parameters by the expressions in Eq. (14) whereas the the effective field Heffis given by Eq. (13). Now we want to analyze this expression and in particular, to assign a physical meaning to the more or less ad hoc introduced quantity σ. In doing so one can distinguish four different cases: (i) The bath is not included which corresponds formally to σis a constant vector depend- ing neither on coordinates nor on time. Then obviously all derivatives with respect to the coordinates and the time of σdisappear in Eq. (13) and consequently, the set of {qα}in Eq. (1) does not include σ. From here we conclude that the variation fulfills δσ= 0 in Eq. (6) which can be easily realized setting κ= 0, cf. Eq. (7). Thus, the effective field in Eq. (13) comprises the pure exchange interaction J(m)between the magnetic moments and the damping term in Eq. (12) is absent due to α= 0 in Eq. (14). A constant bath fieldσlead to the Landau-Lifshitz equation in the exchange interaction ap proach without damping, compare [18]. It describes the precession of magnetic mom ents of an effective field which is self-organized by the mutual interaction of the moments. (ii)σ=σ(t) depends only on the time and not on the spatial coordinates. Rega rding Eq. (13) the effective field is modified by two additional contributions , namely one propor- tional to ∇2m, originated in the exchange interaction of the magnetic moments, a nd the other one ∝˙σ. The latter one could be associated with an external time dependen t field or, ifσpoints into a fixed direction, gives rise to magnetic anisotropy. In th at case the anisotropy axis is spatially constant but the amount of the anisotro py is changing in time. Such a situation could be realized for instance when the ferromagne tic sample is excited by the irradiation with electromagnetic waves. As already mentioned th e exchange coupling J(m)is supplemented by a term −κJ(mσ). In this manner the exchange interaction is influ- enced by the coupling between mandσalthough the spatial dependence of σis not taken into account explicitly. (iii)σ=σ(r) depends only on the spatial coordinates and not on the time. In th is case we first recognize that the coupling strength J(m)in the term ∝ ∇2mis influenced in the same manner as in case when σ=σ(t), see the previous point. Different to the former cases the 7expression ∝ ∇2σbecomes important for the effective field in Eq. (13). The appearan ce of this term suggests that spatial inhomogeneities of the surroundin gs of the magnetic system represented by mhave to be incorporated into the effective field. It seems to be reas onable that the origin of this term is an inherent one and should not be led bac k to external fields. As possible sources we have in mind local varying fields like inner and out er demagnetization fields as well as accessible fields created for instance by different loc al temperatures. (iv)σ=σ(r,t) isthemost general case. Then external aswell asinternal fields arecaptured in the model. Thus, the effective field in Eq. (13) can be rewritten as Heff(r,t) =Hexch(r)+h(r,t), (16) whereHeffconsists of two parts. The term Hexch= (J(m)−κJ(mσ))∇2mis due to the exchange interaction between the magnetic moments whereas h(r,t) represents other possi- ble influences as discussed under the points (ii) and (iii). The function his related to the quantity σby h(r,t) =−c∂σ(r,t) ∂t+/bracketleftbig J(mσ)−κJ(σ)/bracketrightbig ∇2σ(r,t). (17) Remark that the formerly introduced quantity σis related to the physically relevant effec- tive field by the first derivation with respect to the time and the seco nd derivation with respect to the spacial coordinates via Eq. (17). This equation is an inhomogeneous diffusion equation which can be generally solved by means of the expansion into Fourier series and the assumption of accurate initial and boundary conditions which depen d on the actual physical problem. IV. SYMMETRY AND CONSERVATION After regarding the special example of the LLG we proceed with the investigation of more general aspects. The Lagrangian density allows to discuss the beh avior under space-time dependent group transformation. For this purpose we apply Noet her’s theorem [19] to our model. To be more precise we consider the conservation equation [20 ] ∂ ∂Xα/bracketleftbigg/parenleftBig Lδαβ−∂L ∂(∂αΨγ)∂βΨγ/parenrightBig ∆Xβ+∂L ∂(∂αΨγ)∆Ψγ/bracketrightbigg = 0. (18) Here, the expression in the square brackets are the components of the Noether current Iα. The term∂/∂Xαin front ofIαshould be interpreted as an implicit derivative with respect 8to time and three spatial coordinates. The symmetry operations ∆ Xαand ∆Ψ αwill be specified below. With regard to the Lagrangian in Eq. (2) we introduc e the components Ψα= (mx,my,mz,σx,σy,σz) and their partial derivatives with respect to the independent variables∂βΨα=∂Ψα/∂Xβ. Since we examine an Euclidean field theory a distinction between upper and lower indices is not necessary. Eq. (18) can be r ewritten by using Eq. (6). This yields ∂ ∂tL∆t+∂ ∂xαL∆xα+∂L ∂Ψα/parenleftbig ∆Ψα−∂ ∂tΨα∆t−∂ ∂xβΨα∆xβ/parenrightbig = 0. (19) In this equation we distinguish between the time and space variables tandxαexplicitly. Eq. (19) is the basis for the application of the following symmetry ope rations. Now we study the rotation around a certain axis as a relevant one. Here we select for instance the z-axis. Performing a rotation in coordinate space with the infinitesimal angle ∆Θ the change of the xandy-coordinates obeys ∆t= 0,∆xα= ∆Rαβxβ,∆R= 0 ∆Θ −∆Θ 0 . (20) In the same manner one can perform the rotation in the configurat ion space of the moments mandσsymbolized by the before introduced vector Ψ α={mx,my,mz,σx,σy,σz}. The transformation reads ∆Ψ α= ∆Sαβ(∆Φ)Ψ β, where the rotation matrix is a 6 ×6-matrix determined by the rotation angle ∆Φ. Because both rotations in coo rdinate space and configuration space, respectively, are in general independent fr om each other we find two conserved quantities. Using Eq. (19) it results ˆDzL= 0,ˆΓzL= 0. (21) Here the two operators ˆDzandˆΓzare expressed by ˆDz=ˆLz−/parenleftBig ˆLzψα/parenrightBig∂ ∂Ψα, ˆΓz=ˆS(m) z+ˆS(σ) z.(22) The quantity ˆLzis the generator of an infinitesimal rotation around the z-axis in the coor- dinate space ˆLz=y∂ ∂x−x∂ ∂y, (23) 9and therefore, it is identical with the angular momentum operator. The other quantities ˆS(m) zandˆS(σ) zare the corresponding generators in the configuration space of t he moments. They are defined as ˆS(m) z=my∂ ∂mx−mx∂ ∂my, ˆS(σ) z=σy∂ ∂σx−σx∂ ∂σy.(24) These operators reflect the invariance of the total magnetic mom entm+σunder rotation. Moreover the system is invariant under the combined transformat ion expressed by ˆDzand ˆΓz, where ˆDzoffers due to the coupling between system and bath variables as well as the breaking of time reversal invariance a coupling between magnetic mo ments and the angular momentum. V. CONCLUSION In this paper we have presented an approach for a mesoscopic mag netic system with dissi- pation. The Lagrangian consists of two interacting subsystems ch aracterized by the active magnetic moments of the system mand the dynamically inactive moments of the bath denoted as σ. Both systems are in contact so that a small local alteration of the system variables mis related as well to a small change of the bath variables σand vice versa. Due to this constraint we are able to describe the system by a commo n Lagrangian which incorporates both degrees of freedom and their coupling. In case the bath variables are constant then the coupling between both systems is absent and th e whole system decays into two independent subsystems. The magnetic moments mperform a precession around an effective field which is self-organized by the mutual interaction of the moments. If the coupling between both subsystems breaks the time reversal symm etry the related evolution equation of the moments mis associated with the Landau-Lifshitz-Gilbert equation which describes both the precession of magnetic moments as well as their damping. It turned out that the bath variable σcan be linked to the effective magnetic field which drives the motion of the magnetic moments. As consequence the motion of the moments is influenced by the additional bath degrees of freedom. This influence is formula ted mathematically and is described by an inhomogeneous diffusion equation. Finally, we have f ound conservation laws by means of symmetry considerations based on Noether’s theo rem. Aside from the 10expected symmetry transformation in the coordinate space and t he configuration space of the moments, the analysis offers in a non-relativistic Euclidean field th eory an unexpected coupling between both. This point deserves further consideration . Our approach could be also considered as starting point for a further analysis in magnetic a nd multiferroic systems. Especially, we are interested in more refined models which include for in stance higher order couplings or anisotropy in the Lagrangian. In multiferroic systems o ne could study the case that the magnetic and the polar subsystem have their own reservo irs. One of us (T.B.) is grateful to the Research Network ’Nanostructu red Materials’, which is supported by the Saxony-Anhalt State, Germany. 11[1] R. Feynman, A. Hibbs, and D. Styer, Quantum Mechanics and Path Integrals: Emended Edition(Dover Publications, 2010). [2] H. Kleinert, Path integrals in quantum mechanics, statistics, polymer p hysics, and financial markets (World Scientific, 2009). [3] U. Weiss, Quantum dissipative systems , Series in modern condensed matter physics (World Scientific, 1999). [4] L. Schulman, Phys. Rev. 176, 1558 (1968). [5] D. C. Cabra, A. Dobry, A. Greco, and G. L. Rossini, J. Phys. A30, 2699 (1997). [6] V. V. Smirnov, J. Phys. A 32, 1285 (1999). [7] H. Grinberg, Phys. Lett. A 311, 133 (2003). [8] W. Koch, F. Großmann, J. T. Stockburger, and J. Ankerhold , Phys. Rev. Lett. 100, 230402 (2008). [9] L. Landau and E. Lifshitz, Zeitschr. d. Sowj. 8, 153 (1935). [10] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [11] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halp erin, Rev. Mod. Phys. 77, 1375 (2005). [12] P. Durand and I. Paidarov ˜A¡, EPL 89, 67004 (2010). [13] V. L. Safonov and H. N. Bertram, Phys. Rev. B 71, 224402 (2005). [14] P. Weetman and G. Akhras, J. Appl. Phys. 105, 023917 (2009). [15] Z. V. Gareeva and A. K. Zvezdin, EPL 91, 47006 (2010). [16] I. V. Ovchinnikov and K. L. Wang, Phys. Rev. B 82, 024410 (2010). [17] J. Jackson, Classical electrodynamics (Wiley, 1999). [18] L. D. Landau, E. Lifshitz, and L. Pitaevskii, Statistical Physics Part 2: Theory of the Con- densed State (Pergamon Press, Oxford, 1980). [19] E. Noether, Nachr. Ges. Wiss. G¨ ottingen , 235 (1918). [20] E. L. Hill, Rev. Mod. Phys. 23, 253 (1951). 12

2011-04-15

A Lagrangian is introduced which includes the coupling between magnetic moments $\mathbf{m}$ and the degrees of freedom $\boldsymbol{\sigma}$ of a reservoir. In case the system-reservoir coupling breaks the time reversal symmetry the magnetic moments perform a damped precession around an effective field which is self-organized by the mutual interaction of the moments. The resulting evolution equation has the form of the Landau-Lifshitz-Gilbert equation. In case the bath variables are constant vector fields the moments $\mathbf{m}$ fulfill the reversible Landau-Lifshitz equation. Applying Noether's theorem we find conserved quantities under rotation in space and within the configuration space of the moments.

Lagrangian approach and dissipative magnetic systems

1104.3002v1

Inverse problem of recovering the time-dependent damping and nonlinear terms for wave equations Song-Ren Fu Abstract. In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams together with the higher order linearization are respectively used to derive the uniqueness results of recovering the coecients. Key words: semilinear wave equation, damping term, Carleman estimate, higher order linearization, Gaussian beams 1 Introduction Let ( ;g) be a Riemannian manifold of dimension n2 with smooth boundary @ = :LetM= (0;T) and  = (0;T):Assume that ( x;t) = (x1;


;xn;x0=t) are local coordinates on M:The nonlinear wave equation considered in this paper is given by 8 >< >:utt
gu+b(x;t)ut+f(x;t;u ) = 0;(x;t)2M; u(x;t) =h(x;t);(x;t)2; u(0;x) =u0(x); ut(0;x) =u1(x); x2 ;(1.1) wheref: R1C!Cis a smooth function. In the local coordinates,
gu= divgDu= (detg)1 2nX ij=1@xj[(detg)1 2gij@xiu]; where div gandDare the divergence operator and Levi-Civita connection in the metric g, respectively. The main goal in this paper is to study the inverse problem of recovering the time- dependent coecient b(x;t) (damping term), and the nonlinear term f(x;t;z ) by some Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, P. R. China; e-mail: songrenfu@amss.ac.cn. 1arXiv:2212.01815v2 [math.AP] 7 Dec 2022suitable boundary measurements. There are lots of literature concerning the inverse problem of recovering time-dependent or time-independent coecients in PDEs. We will rstly consider the inverse problem of recovering the particular case, where the nonlinear term fis of time-independent. More precisely, we will recovery f(x;z) in (1.1) by means of the Carleman estimates. It is pointed out that, after the methodology created by [6], the Carleman estimates are well used in inverse problems of uniquely determining the time-independent coecients. Secondly, for the time-dependent case, we will recovery b(x;t) andf(x;t;z ) simul- taneously by the higher order linearization method together with constructing some Gaussian beams. We mention that [18] devoted to an inverse boundary problem for a nonlinear parabolic equation, in which the rst order linearization of the DN map was proposed. This approach has been developed and applied in dierent other contexts. For simplicity, we write the inverse problem of recovering f(x;z) as inverse problem (I), recovering b(x;t) andf(x;t;z ) as inverse problem (II). Letmbe a positive integer. We introduce the following energy space Em=m\ k=0Ck([0;T];Hmk( )) with the norm jjujj2 Em= sup 0tTmX k=0jj@k tu(t)jj2 Hmkforu2Em: The well-posedness of system (1.1) is discussed in the appendix of this present paper. 1.1 Recovery of the nonlinear term f(x;z) We consider the following equation. 8 >< >:utt
gu+b(x;t)ut+f(x;u) = 0;(x;t)2 (0;T); u(x;t) =h(x;t);(x;t)2(0;T); u(x;0) =(x); ut(x;0) = 0; x2 :(1.2) Letu=u(f;) be a solution to (1.2) with respect to fand(x):Dene the input-to- output map as f T((x)) =@u(f;)j(0;T); wheredenotes the unit normal eld pointing outside along :For simplicity, we assume that (x)2C1( ):Based on the initial data (x);the boundary data is given by u(x;t)j=h(x;t) =(x)+t2 2[
g(x)f(x;(x))]+mX k=3tk k!@k tu(x;0); x2; m2; 2where @k+2 tu(x;0) =
g(@k tu(x;0))@k t(but)jt=0@k tf(x;u)jt=0; k = 0;1;


: It is easy to see that the compatibility conditions hold for system (1.2) up to order m: Main assumptions. The following are the main assumptions for the inverse problem (I). (A.1)f(x;z) is analytic on Cwith values in C1(M). Moreover, we assume that f(x;0) = 0;andf(k) z(x;0)j 0=f0k(x)j 0;wheref0k(x) is known for each k= 1;2;


; and 0is an arbitrary small neighborhood of inside : (A.2) There exists a non-negative strictly convex function : !R;of classC3 in the metric g:There exists a positive constant on such that (x) satises (i)D2 (x)(X;X )2jXj2 g; x2 ; X2 x: (ii) (x) has no critical point for x2 :Namely, inf x2 jD (x)j>0: (A.3)b(x;t)2C1(M) withb(x;0) = 0:Let >0 be a small constant. Assume thatj(x)j0>0 forx2 ;such that jjhjjHm+1()+jj(x)jjHm+1( ) 2; m>n 2; We give the admissible sets of functions b(x;t) andf(x;z) as follows. Let M0be a positive constant. We dene U1=fb(x;t)2C1(M) :b(x;0) = 0;jjbjjC1(M)M0g; (1.3) U2=ff(x;z)2C1( C);fsatises assumption (A :1); jjf(k) z(x;0)jjL1( )M0; k= 1;2;


g: (1.4) Remark 1.1. Assume that fhas the following Taylor expansion f(x;u) =1X k=0f(k) z(x;0)uk k!: (1.5) Letf1;f22U 2and letuj=uj(x;t;;fj) be the solution to (1.2) with respect to fj forj= 1;2:We see that assumption (A.1) implies that h(x;t;f1) =h(x;t;f2);where h(x;t;fj) is the given boundary data corresponding to the equation ujtt
guj+bujt+fj(x;uj) = 0;forj= 1;2: Assumption (A.2) are usually used in the Carleman estimates, see for example [5, 30]. The existence of convex functions depends on the curvature of ( ;g):Particularly, for the Euclidean case, we can take (x) =jxx0j2withx02Rnn :For general Rie- mannian manifolds, such exists locally. There are a number of non-trivial examples to give such ;see [48, Chapter 2.3]. 3We are now in a position to state the main theorem of recovering f(x;z): Theorem 1.1. Let assumptions (A.1)-(A.3) hold. Assume that b(x;t)2U 1and f1;f22U 2:LetT >T;whereTis given by (2.2). Then f1 T((x)) = f2 T((x)) implies f1(x;z) =f2(x;z);(x;z)2 C: (1.6) Remark 1.2. Assume that 0 such that fx2 :hD ;i0g0: Then the measurement can be replaced by @uj0(0;T): 1.2 Recovery of the time-dependent coecient b(x;t)and the nonlinear termf(x;t;z ) We here consider the nonlinear equation (1.1). Let u(x;t;h;b;f ) be the solution to (1.1) with respect to h;b;f: With the boundary data, we dene the Dirichlet-to-Neumann map as b;f T(h(x;t)) =@u(x;t;h) @ : Before giving the main assumptions for inverse problem (II), we give two denitions as follows. Denition 1.1. A Riemannian manifold ( ;g)is called a simple manifold if it is simply connected, any geodesic in has no conjugate points, and the boundary is strictly convex with respect to the metric g(the second fundamental form is positive for every point on the boundary). Denition 1.2. A compact Riemannian manifold ( ;g)satises the foliation con- dition if there is a smooth strictly convex function, and the boundary is strictly convex with respect to the metric g. The above conditions are crucial in the inverse problems. They are always used as a sucient condition in the geodesic ray transform. The inversion of the geodesic ray transform on with depends on the geometric properties of . See more details in section 4. Similar to [9], let D=f(x;t)2M : dist(x;)<t<Tdist(x;)g be the domain of in uence. It is clear that no information can be obtained about the coecients on MnD;due to the nite speed of propagation of waves. Let T > 2Diam( ) be given, where Diam( ) = supflengths of all geodesics in ( ;g)g<1: 4We dene a subset of Dby E=f(x;t)2M :Dg(x)<t<TDg(x)g; whereDg(x) denotes the length of the longest geodesic passing through the point x2 : The following assumptions are the main assumptions for the inverse problem (II). (B.1)f(x;t;z ) is analytic on Cwith values in C1(M);andf(x;t;0) = 0:We dene an admissible set for fas U3=ff2C1( RC) :fsatises assumption (B :1); suppf(k) z(x;t;0)E;fork= 1;2g: (1.7) (B.2)b(x;t)2C1(M):Similar tof;letb0(x;t) be a known smooth function. We dene an admissible set for bas U4=fb(x;t)2C1(M) : suppb(x;t)Eg: (1.8) (B.3) Assume that the initial data u(x;0) =u0(x);ut(x;0) =u1(x);and (u0(x);u1(x);h(x;t))2Hm+1( )Hm( )Hm+1() satisfying the compatibility conditions (3.30) up to order m>n 2;such that jju0jjHm+1( )+jju1jjHm( )+jjh(x;t)jjHm+1() 2; where>0 is a given small constant. Suppose that ( ;g) is either simple or satises the foliation condition, based on the Gaussian beams, which can allow the existence of conjugate points, we have Theorem 1.2. Assume that uj=u(x;t;bj;fj)solves (1.1) with respect to bjandfj for eachj= 1;2:Let assumptions (B.1), (B.2) and (B.3) hold, and let T >2Diam( ): Suppose that b1;b22U 4; f1;f22U 3andf(2) zis known. Then b1;f1 T(h(x;t)) = b2;f2 T(h(x;t)) implies b1=b2inE;andf1=f2inDC: 1.3 Literature review Inverse problems of PDEs have attracted much attention with lots of literature on in- verse elliptic equations, parabolic equations, hyperbolic equations, and plate equations, etc. For example, for the inverse elliptic problems, the well known Calder on type in- verse problems were investigated in [13, 19, 31, 42] and many subsequent papers. The linear inverse problems of determining either the time-independent coecients or the time-dependent coecients were widely studied. It is known to us that the Carleman estimates method created by [6] have well used in inverse problems to derive the strong 5Lipschit stability results for the time-independent coecients. In this present paper, the Carleman estimate will be used for the recovery of the time-independent nonlin- earityf(x;z). However such method deals with the time-independent case only. Also, there are some other methods for the recovery of the time-independent case (or for the real analytic time-dependent coecients), such as the boundary control (BC) method steaming from [2, 3] together with Tataru's sharp unique continuation theorem [43]. We are not going to list more literature concerning the linear inverse problems. The inverse problems of nonlinear PDEs are much less. Among them, the early works [18, 20, 21] used the linearization procedure to study the recovery of nonlinear terms appearing in elliptic or parabolic equations. It turns out that the nonlinear interaction of waves can generate new waves, which are essential for the nonlinear inverse problems. For instance, [11, 17, 18, 22, 27, 41] devoted to the unique recovery of nonlinear terms or coecients appearing in nonlinear elliptic equations. [8] dealt with the stable recovery of a semilinear term appearing in a parabolic equation, and [26] studied the fractional semilinear Schr odinger equations. For the inverse problems of hyperbolic equations, we refer to [34, 35], which re- spectively concerned the the recovery of a conductivity and quadratic coecients. We mention that [29] devoted to the recovery of nonlinear terms from source-to-solution map, where method of the nonlinear interactions of distorted plane waves, originated from [25], was used. Such method has been successfully used in inverse problems of nonlinear hyperbolic equations, see for example [15, 25, 45, 47]. Similarly, for some semilinear wave equations, instead of the distorted plane waves, Gaussian beams to- gether with the higher order linearization and the stationary phase method are used to recover the coecients. For this method, for instance, we refer to [12, 14, 46]. Among those, [14] studied an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifolds. Both the distorted plane waves and the Gaussian beams were used to derive the uniqueness. In their paper, due to the un- solved inverse problems of recovering the zeroth order term on general manifolds, they assumed the nonlinearity has the following form H(x;z) =1X k=2hk(z)zk; hk2C1: In their paper, the recovery of the quadric term fuuis much complex and interesting. The distorted plane waves were used to construct four future light-like vectors to re- covery the quadric term, but without recovering the zeroth term fu(x;t;0):However, the quadric term is unable to recovery by using the Gaussian beams. It is worth noting that [24] considered the inverse problem of determining a gen- eral nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold (M;g) with boundary of dimension n= 2;3:They determined the expres- 6sionF(t;x;u ) both on the boundary x2Mand inside the manifold x2Mfrom some partial knowledge of the solutions uon the boundary of the time-space cylindrical man- ifold (0;T)Mor on the lateral boundary (0 ;T)@M: Let us also point out that [32] investigated inverse boundary problems associated with a time-dependent semilinear hyperbolic equation with variable coecients. They developed a new method, which combined the observability inequality and a Runge approximation with higher order linearization, to derive the uniqueness of both the initial velocity and the nonlinearity. The measurements they used were either active or passive. For the stability of recover- ing some coecients, [28] devoted to the uniqueness and stability of an inverse problem for a semi-linear wave equation: utt
u+a(x;t)um= 0;where (x;t)2RnRand m2. They used the higher order linearization together with the Radon transform to prove the stability results of recovering a(x;t) by the Dirichlet-to-Neumann map. It is worth mentioning that, instead of dierentiating the nonlinear equation by@m @1


@m; they used the nite dierences operator Dm :We note that, all the above literature concerning the nonlinear wave equations do not consider the recovery of the rst order coecients, such as the damping term. Motivated by these previous works, we study the recovery of the time-dependent damping term and the nonlinearity simultaneously, which can be seen as an extended studying of the existing literature. The rest of this present paper is organized as follows: In Section 2, we prove Theorem 1.1 by the Carleman setimates. We also add some additional contents in this section. Section 4 focuses on the Gaussian beams and proofs of Theorem 1.4. Finally, in the appendix, the well-posedness of the semilinear wave equation (1.1) is discussed. 2 Proofs of Theorem 1.1 and an additional result In this section, we focus on proving Theorem 1.1 and give an additional result of recovering a leading coecient. 2.1 Proofs of Theorem 1.1 LetN1 be a positive inter, and = (1;


;N). Letk(x)6= 0 forx2 ;and (x) =nX k=1kk(x)2Hm+1( ) withjj=NP k=1jkjsuciently small, such that jjh(x;t)jjHm+1()+jj11+


+NNjjHm+1( ) 2: By the same output, we have f1(11+


NN) = f2(11+


NN); 7which gives @jj @1


@N =0f1(11+


NN) =@jj @1


@N =0f2(11+


NN):(2.1) Clearly, f(NP k=1kk) contains more information than ff(k)gk=1;


;N:Indeed, useful information can be extracted from @N @1


@N =0f(NX k=1kk): Let eQ= (T;T);e = (T;T): Extend the domains of ujandb(x;t) to the region eQevenly as usual. Notice that u1(x;t) =u1t=b(x;0) = 0;thenu1ttt(x;0) = 0;which implies that the extension of u1(x;t) (alsouj 1(x;t)) is smooth. Similar to [5, Chapter 5.2], we set (x;t) = (x)t2+0; ' (x;t) =e(x;t);(x;t)2M; whereis a positive constant, 2(0;) andis given by (i) in assumption (A.2). Moreover,0>0 is chose such that (x)>0. Let T=1p max x2 (x)1 2 : (2.2) We assume that T >T:Then we can choose >0 and >0 such that T2>max x2 (x) + 4; T2>max x2 (x) + 4: Thus,(x;t) has the following properties: (1)(x;T)04uniformly for x2 : (2) There exists a small constant ">0 such that (x;t)02for (x;t)2 [T2";T][[T;T+ 2"]: Therefore '(x;t)e(02)=:d1<d 0=:e0uniformly in [T2";T]: We next give a key Carleman estimate for the linear wave operator Lv=vtt
gv+b(x;t)vt+q(x;t)v; whereq;b2L1(M):Let H1 0(eQ) =fu2H1 0( [T;T]) :uj= 0; @l tu(x;T) = 0; l= 0;1g: We have the following from [4]. 8Theorem 2.1. Under assumptions (A.1) and (A.2), there exist constants C= C(M0)>0and>0, such that for any >, there exists s0=s()such that Z eQs[(jDvj+v2 t) +s2v2]e2s'dgdtCZ eQjPvj2e2s'dgdt +CZ j@vj2e2s'd (2.3) holds for all v2H1 0(eQ)ands>s 0>1: Proof of Theorem 1.1. Based on the above linearization procedure and the Carleman estimate, we divide the proofs into two steps. Step 1. First order linearization. Let uj=uj(x;t;fj;)2Em+1be a solution to (1.2) with respect to fjand(x) =11(x) +


+NN(x) forj= 1;2:Then euj=uj(x;t;fj;0) solves 8 >< >:eujtt
geuj+b(x;t)eujt+fj(x;euj) = 0;(x;t)2 (0;T); euj(x;t) = 0;(x;t)2(0;T); euj=eujt(x;0) = 0; x2 ;(2.4) which admits a zero solution euj= 0 sincefj(x;0) = 0 for each j= 1;2:We next linearize the system (1.2) around euj= 0. LetN= 1 and let uj 1=@ @1 =0uj:Thenuj 1satises 8 >>< >>:uj 1tt
guj 1+b(x;t)uj 1t+fju(x;0)uj 1= 0;(x;t)2eQ; uj 1(x;t) =@ @1 =0h:=h1;(x;t)2e; uj 1(x;0) =1(x); uj 1t(x;0) = 0; x2 :(2.5) Setu1=u1 1u2 1andq(x) =f2u(x;0)f1u(x;0);then 8 >< >:u1tt
gu1+b(x;t)u1t+f1u(x;0)u1=q(x)u2 1;(x;t)2eQ; u1(x;t) = 0;(x;t)2e; u1(x;0) =u1t(x;0) = 0; x2 :(2.6) Lety1=u1t:Then 8 >< >:y1tt
gy1+by1t+ (bt+f1u(x;0))y1=q(x)u2 1t;(x;t)2eQ; y1(x;t) = 0;(x;t)2e; y1(x;0) = 0; y1t(x;0) =q(x)1(x); x2 :(2.7) Next, we chose a cut-o function (t)2C1 0([T;T]) satisfying 0(t)1; (t) =( 0; t2[T;T+")[(T";T]; 1; t2[T+ 2";T2"]:(2.8) Let ^y1=(t)y1:Then ^y1tt
g^y1+b^y1t+ (bt+f1u(x;0))^y1=q(x)u2 1t+t(2y1t+by1) +tty1: 9Since the assumption that j1(x)jc1>0;then Z jq(x)j2e2s'(x;0)dgCZ jq(x)1(x)j2e2s'(x;0)dg =ZT 0@ @tZ jy1t(x;t)(t)j2e2s'dtdg =2Z Q(2y1ty1tt+ty2 1t+s't2y2 1t)e2s'dgdt CsZ Q[j(y1)tj2+2 t(y2 1+y2 1t)]e2s'dgdt + 2Z Q2y1ty1tte2s'dgdt: (2.9) We compute the term 2y1ty1tte2s'as follows. 2y1ty1tte2s'=2y1t[
gy1by1t(bt+f1u(x;0))y1+q(x)u2 1t]e2s' =2div (y1te2s'Dy1)1 2(2e2s'jDy1j2)t+ (t+s2't)jDy1j2e2s' 2s2y1thDy1;D'ie2s'2[(bt+f1u(x;0))y1q(x)u2 1t]e2s' 2div (y1te2s'Dy1)1 2(2e2s'jDy1j2)t +Cs[jD(y1)j2+j(y1)tj2+2 t(y2 1t+jDy1j2)]e2s'+2q(x)u2 1te2s': Moreover, for system (3.24), the hyperbolic regularity (see, Lemma A.1 in the ap- pendix) implies that u2 1t2C([T;T];Hm( )) form>n 2:Then the Sobolev embedding theorem shows that u2 1t2L1(eQ):Thus, there exists a positive constant C=C(M0) such that Z Qf2y1t[by1t+ (bt+f1u(x;0))y1q(x)u2 1t]ge2s'dgdt CZ Q[2 ty2 1+j(y1)tj2+q(x)]e2s'dgdt: Notice that1 2(2jDy1j2e2s')jT 0= 0:Therefore, by (2.9) and (2.10), with v= ^y;it follows from the Carleman estimate (2.3) that Z q2e2s'(x;0)dgCsZ Q(^y2+jD^yj2+ ^y2 t)e2s'dgdt +CsZ Q2 t(y2 1+y2 1t+jDy1j2)e2s'dgdt +CZ Qq2e2s'dgdt CZ Qq2e2s'dgdt +Z Q(2 t+2 tt)(y2 1+y2 1t+jDy1j2)e2s'dgdt +CeCsjj@y1jj2 L2(): By the standard energy estimate of system (2.7), there exists a positive constant C= C(T;M 0) such that Z (y2 1+y2 1t+jDy1j2)dgCZ q2dg+CZ j@y1j2d: (2.10) 10Sincet;tt6= 0 in the case where '(x;t)d1;with (2.10), we have Z q2e2s'(x;0)dgCZ Qq2e2s'dgdt +CeCsjj@y1jj2 L2(): (2.11) Moreover, by the Lebesgue's theorem, we have Z Qq2e2s'dgdt =Z q2e2s'(x;0)ZT 0e2s('(x;0'(x;t)))dt=o(1)Z q2e2s'(x;0)dg: Thus, jjqjj2 L2( )CeCsjj@y1jj2 L2()=CeCsjj@(u1 1u2 1)tjj2 L2(): Hence@u1 1=@u2 1on  implies that f1u(x;0) =f2u(x;0) =fu(x;0) forx2 : Step 2. Higher order linearization. Based on Step 1, we set f1u(x;0) =f2u(x;0) = fu(x;0) for simplicity. Let uj 2=@2 @1@2 =0ujforj= 1;2:Then 8 >< >:uj 2tt
guj 2+b(x;t)uj 2t+fu(x;0)uj 2+fjuu(x;0)u1 1u2 1= 0;(x;t)2 (0;T); uj 2(x;t) = 0;(x;t)2(0;T); uj 2(x;0) = 0; uj 2t(x;0) = 0; x2 : (2.12) Hereu1 1andu2 1satisfy 8 >>< >>:uj 1tt
guj 1+b(x;t)uj 1t+fu(x;0)uj 1= 0;(x;t)2 (0;T); uj 1(x;t) =@ @j =0:=hj(x;t);(x;t)2(0;T); uj 1(x;0) =j(x); uj 1t(x;0) = 0; x2 :(2.13) Sety2= (u1 2u2 2)t;then 8 >< >:y2tt
gy2+b(x;t)y2t+ (bt+fu(x;0))y2=F(x;t);(x;t)2 (0;T); y2(x;t) = 0;(x;t)2(0;T); y2(x;0) = 0; y2t(x;0) =(f1uuf2uu)(x;0)1(x)2(x); x2 ;(2.14) whereF(x;t) =(f1uuf2uu)(x;0)(u1 1tu2 1+u1 1u2 1t):By a similar argument as that in Step 1, we have @u1 2=@u2 2on )f1uu(x;0) =f2uu(x;0))u1 2=u2 2:=u2: Suppose that f(N1) 1u(x;0) =f(N1) 2u(x;0); x2 : LetwN j=@N @1


N =0uj:By the recursive assumption, for the expression fj(x;u) =1X k=0f(k) u(x;0)uk k!; 11andvk=@ @k =0u;we know that @N @1


N =0fj(x;u)fu(x;0)wN jf(N) ju(x;0)v1


vN is already known. The above procedure can be proceeded for N-th linearization to obtain f(N) 1u(x;0) =f(N) 2u(x;0); x2 : Up to now, by the analyticity of f;we have f1(x;z) =1X k=1f(k) 1u(x;0)zk k!=1X k=1f(k) 2u(x;0)zk k!=f2(x;z): Thus, the uniqueness result in Theorem 1.1 holds and the proof of Theorem 1.1 is complete. 2.2 An additional result of recovering a leading coecient Based on the Carleman estimate, we introduce here a stability result of recovering a leading coecient. Suppose that there is a leading coecient appearing in the wave equation %(x)utt
gu+but+f(x;t;u ) = 0;(x;t)2M: (2.15) The unique recovery of the mass density %(x) for a wave equation is essential in inverse problems. There are literature concerning this topic, see for example [1, 33, 40]. We give a brief discussion of recovering %(x): Due to the presence of %(x);we consider a new metric ^ g=%g;then
^gu=1 %
gu+n2 2%2hD%;Dui: Let ^Dbe the Levi-Civita connection in the metric ^ g:It is well known that, for any vectorsX;Y ^DXY=DXY+1 2g(Dln%;Y)X+1 2g(Dln%;X)Y+1 2g(X;Y )Dln%: If (x) is strictly convex in the metric ^ g, then one needs ^D2 (X;X ) =%g(^DX^D ;X ) =%[g(DX(%1D );X)] +1 2g(D (ln%)X+X(ln%)D#+X( )D(ln%);X) =D2 (X;X ) +1 2D#(ln%)jXj2 g(2 +1 2D (ln%))jXj2 g =1 %(2 +1 2D (ln%))jXj2 ^g#0jXj2 ^g; 12where#0>0 is a constant. Therefore, we need further assumption on %(x): g(D ;D% )2%(#02): Letuj(x;t) =u(x;t;j) be a solution to (2.15) with respect to %jfor eachj= 1;2:Let w=u1u2;^%=12:Then %2wtt
gw+bwt+cw=^%u1tt; (2.16) wherec(x;t) =R1 0fu(x;t;ru 1+ (1r)u2)dr2L1(M) for suciently smooth u1and u2:Suppose thatj
g(x)f(x;0;(x))j>0 forx2 ;where (uj(x;0);ujt(x;0)) = ((x);0) forj= 1;2:Then a similar argument with the proof of Theorem 1.1 yields the following stability of recovering %(x): jj%1(x)%2(x)jjL2( )Cjj@(u1u2)tjjL2(): Remark 2.1. Clearly, as we have discussed above, the non-degeneracy of initial datau(x;0);which is viewed as the input is needed. Moreover, the existence of some strictly convex functions, which seems not sharp, should be assumed. Such functions guarantee that interior information of solutions to the system arrives at boundary in a nite time. 3 Gaussian beams and proofs of Theorem 1.4 In this section, as usual, the boundary data his the input. A geodesic (t)M is called a null geodesic in the metric g=dt2+g, ifDg __=g(_;_) = 0;whereDgis the connection in the metric g:We intend to construct some Gaussian beams around a null geodesic. For completeness, we give the details of constructing the Gaussian beams. 3.1 Gaussian beams Let(t) be a null geodesic. We rstly introduce the Fermi coordinates in a neighbor- hood of the null geodesic . We follow the constructions in [10], see also [12]. Recall that  1and the functions are extended smoothly to 1:Let(t) = (t; (t))R 1; where (t) is a unit-speed geodesic in the Riemannian manifold ( 1;g):Assume that (t) passes through a point ( t0;x0);wheret02(0;T) and (t0) =x02 :Letjoin two points ( t; (t)) and (t+; (t+)) witht2(0;T) and (t)2:We extendto a larger manifoldM1= (0;T) 1such that (t) is well dened on [ t;t++](0;T) with>0 suciently small. Since the geodesic is parallel along itself, we can choose fe2;


;engsuch that f_ (t0);e2


;engforms an orthonormal basis of x0:Letsdenote the arc length along fromx0:LetEk(s)2 (s)be the parallel transport of ekalong to the point (s): 13We now dene a map F1:Rn+1!M 1 such that F1(y0=t;s=y1;y2


;yn) = (t;exp (s)(y2E2(s) +


+ynEn(s))); where expp(
) denotes the exponential map on 1at the point p. In the new coordinates, we have gj =nX k=1dy2 kand@gij @yk = 0 for 1i;j;kn: On the Lorentzian manifold ( M1;dt2+g);we introduce the well known Fermi coor- dinates near the null geodesic : (t;t++)!M 1as follows. Let a=p 2(t); b =p 2(t++); a 0=p 2(tp 2); b 0=p 2(t++p 2); and z0=1p 2(t+s) +a 2; z 1=1p 2(t+s) +a 2; zj=yjfor 2jn: Then, we have gj= 2dz0dz1+nX k=2dz2 kand@gij @zk = 0 for 0i;j;kn: (3.1) For simplicity, we use the notation z= (z0;z0) = (z0;z1;z00) to denote the so called Fermi coordinates. The following lemma from [10, Lemma 3.1] (see also [12, Lemma 1]) is essential for the construction of Gaussian beams. Lemma 3.1. Let: (t;t++)!M 1be a null geodesic as above. Then there exists a coordinate neighborhood (U;)of(t 2;t++ 2);with the coordinates denoted by (z0;z0)such that V= (U) = (a;b)B(0;); whereB(0;)denotes a ball in Rnwith a small radius : Based on the above coordinates, we will construct some approximate Gauss beams in a neighborhood of by dening V=f(z0;z0)2M 1:z02(a0;b0);jz0j0g with 0<0<suciently small such that the set Vdoes not intersect the sets f0g andfTg : 14We use the shorthand notation Lb;qu=utt
gu+but+qu:We consider the WKB ansatz u=ei'a+r; where >0 is a constant, ris the reminder term. 'andaare called the amplitude and phase respectively. In particular, we will construct '2C1(V) anda2C1 0(V): Directly calculation yields Lb;q(ei'a) =ei'Lb;qa+ 2 i('tathDa;D'ig)ei' + ia('tt
g'+b't)ei'+a2(jD'j2 g'2 t)ei': (3.2) Based on the above equation, we respectively solve The eikonal equation S'=jD'j2 g'2 t=hd';d'ig=nX k;l=0gkl@k'@l'= 0; and the transport equation Tb(a;') =2hd';daig(
g'b't)a = 2('tathDa;D'ig) + ('tt
g'+b't)a= 0: (3.3) To achieve this, we make the following ansatz for ';a; namely '(z0;z0) =NX k=0'k(z0;z0); a =NX k=0k(jz0j 0)ak(z0;z0); ak=NX j=0ak;j(z0;z0): Here'kis a homogeneous polynomial of degree kwith respect to the variables zifor k= 0;1;


;N:In terms of the Fermi coordinates z= (z0;z1;


;zn) forg=dt2+g; we need @jj @z(S')(z0;0) =@jj @zhd';d'igj= 0 for  = (0 ;1;


;n); z 02(a0;b0); wherej0 are integers for 1 jn;andjj=nP j=1jN;Moreover, for k= 1;


;N;we need @jj @zT(a0;')(z0;0) = 0;@jj @z( iT(ak;') +Lb;qak1)(z0;0) = 0; z 02(a0;b0): Construction of the phase. We rstly solve equation (3.4) with jj= 0:That is, nX k;l=0gkl@k'@l'= 0 on: 15By (3.1), this reduces to 2@0'@1'+nX k=2(@k')2= 0: (3.4) Similar, forjj= 1;we have nX kl=0@2 jk'@l'= 0 for 1jn: (3.5) Clearly, equations (3.4) and (3.5) are satised by respectively setting '0= 0; ' 1=z1=1p 2(t+s) +a 2: For the case where jj= 2;we set '2(z0;z0) =nX i;j=1Hij(z0)zizj; (3.6) whereHij=Hjiis a complex-valued matrix such that Im His positive denite. It follows from (3.4) and (3.6) that nX k;l=0(@2 ijgkl@k'@l'+ 2gkl@3 kij'@l'+ 2gkl@2 ki'@2 lj'+ 4@igkl@2 jk'@l')j= 0: (3.7) By the choices of '0;'1and'2;(3.7) implies that (@2 ijg11+ 2g10@3 0ij'+ 2nX k=2@2 ki'@2 kj')j= 0: We nally obtain the following Riccati equation for H(z0);namely, d dz0H+HAH +B= 0; H (s) =H0;with ImH0>0 andz02(a0;b0);(3.8) whereB=1 4@2 ijg11; s=p 2tand the components of A= (Aij) satisfy 8 >< >:A11= 0; Aii= 2; i= 2;


;n; Aij= 0;otherwise: For the above Riccati equation, we have Lemma 3.2. [23, Section 8] The Riccati equation (3.8) admits a unique solution. The solution His symmetric and Im (H(z0))>0for allz02(a0;b0):We haveH(z0) = Z(z0)Y1(z0);where the matrix valued functions Z(z0);Y(z0)solve the rst order linear system dZ dz0=BY;dY dz0=AZ; subject to Y(s) =I; Z (s) =H0: 16Moreover, the matrix Y(z0)is non-degenerate on (a0;b0), and there holds det(ImH(z0))
jdet(Y(z0))j2= det(ImH0): For the case where jj= 3;4


;the polynomials 'jof higher degree are con- structed analogously. We omit the details. Construction of the amplitude. Let us consider the transport equation (3.3). Let jj= 0:It follows from (3.1) that b't=bp 2and
g'=nX k;l=0gklD2 g'(@k;@l) =nX k;l=0gkl@2 kl'= Tr(AH) on: Therefore, the transport equation (3.3) reduces to 2@z0a0;0+ [Tr(AH)bp 2]a0;0= 0; z 02(a0;b0): (3.9) Notice that Tr(AH) = Tr(A(z0)Z(z0)Y1(z0)) = Tr(dY dz0Y1(z0)) =d dz0log(detY(z0)); then a0;0(z0) = (det (Y(z0)))1 2e1 2p 2Rz0 sb(;0)d; z 02(a0;b0) (3.10) is a solution to (3.9). The subsequent terms ak;0can be constructed by solving some linear ODEs of rst order. We refer to [12] for more details. Construction of the remainder terms. By a similar proof with [12, Lemma 2], the Gaussian beam has the following property. Lemma 3.3. Letu=ei'abe an approximate Gaussian beam of order Nalong the null geodesic :Then for all >0 jjLb;qujjHk(M)CK; whereK=N+1k 21: Based on the above lemma and the Sobolev embedding theorem, for sucient large N;the remainder term rsatises the estimate (cf. [14], [36, Proposition 2.2]) jjrjjHk+1(M)CK)jjrjjC(M)Cn+1 22: (3.11) 3.2 Proofs of Theorem 1.4 Let us introduce some basic notations on the geodesic ray transform, which are ex- plicitly discussed in, e.g., [9, 10, 38]. The following contents are mainly from [9], we present here for completeness. 17LetS 2T be the unit sphere bundle of ( ;g), and by (
;x;v) the geodesic with initial data ( x;v)2S :For all (x;v)2S int;we dene the exist times as (x;v) = inffr>0 : (r;x;v)2g: Assume that ( ;g) is simple, then <Diam( ):Dene @S =f(x;v)2S :x2;hv;(x)ig>0g: All geodesics in intcan be parametrized by (
;x;v) for (x;v)2@S :The geodesic ray transform on ( ;g) is dened for f2C1( ) by If(x;v) =Z+(x;v) 0f( (r;x;v))drfor (x;v)2@S : Letbe a null geodesic (also called light ray). By the product structure of the Lorentzian manifold R , we can parametrize the null geodesic as (r;s;x;v ) = (r+s; (r;x;v));8(s;x;v )2R@S : Then, all the null geodesics through(
;s;x;v ) with (s;x;v )2R@S over their maximal intervals [0 ;+(x;v)] can be identied. The so called light ray transform on R can be dened as Gf(s;x;v ) =Z+(x;v) 0f(r+s; (r;x;v))dr8(s;x;v )2R@S : By [9, Proposition 1.3] and the discussions in [10, Section 2.2], together with [37, Theorem 1.6], we have Proposition 3.1. Suppose that either ( ;g)is simple or ( ;g)satises the foliation condition. Let f2C1(M))vanish on the setMnE:ThenGf= 0for all maximal null geodesicD impliesf= 0. Remark 3.1. For the non-simple case, such ray transform has been also discussed e.g., in [39] and references therein. We begin with the rst order linearized equation 8 >< >:Lbj;qjuj=ujtt
guj+bjujt+qjuj= 0;(t;x)2M; uj(t;x) =h1(t;x); (t;x)2; uj(0;x) =ujt(0;x) = 0; x2 ;(3.12) whereqj=fju(x;t;0):By the denition of Lb;q;we have L b;qu=utt
gubut+ (q+bt)u; 18whereL b;qdenotes the formal adjoint of Lb;qwith respect to the L2(M) inner product. The formal adjoint system of (3.12) with j= 1 is given by ( L b1;q1v=vtt
gvb1vt+ (q1+b1t)v= 0;(t;x)2M; v(T;x) =vt(T;x) = 0; x2 ;(3.13) Based on the above constructions of Gaussian beams, we seek such solutions for systems (3.12) and (3.13), respectively. More precisely, we let u2=ei'a1+r1=ei'n 4(jz0j 0)a10;0+r1; v=ei'a2+r2=ei'n 4(jz0j 0)a20;0+r2; where
means the conjugate of
:Thenr1andr2respectively solve 8 >< >:Lb2;q2r1=Lb2;q2(ei'a1) inM; r1= 0 on  ; r1(0) =r1t(0) = 0 in ;(3.14) and 8 >< >:L b1;q1r2=L b1;q1(ei'a2) inM; r2= 0 on  ; r2(T) =r2t(T) = 0 in ;(3.15) According to (3.2), we have Lb2;q2(ei'a1) =ei'[Lb2;q2a1+2(S')a1+ iTb2(a1;')]; and L b1;q1(ei'a2) =ei'[L b1;q1a2+2(S')a2iTb1(a2;')]: As we have discussed in Section 4.1, we choose a10;0(z0) = (detY(z0))1 2e1 2p 2Rz0 sb2(;0)d; z 02(a0;b0); and a20;0(z0) = (detY(z0))1 2e1 2p 2Rz0 sb1(;0)d; z 02(a0;b0): Clearly,L b1;q1(ei'a2) andL b1;q1(ei'a2) are compactly supported in a small tubu- lar region around the null geodesic where the Fermi coordinates are well dened. The following lemma shows that the remainder terms r1andr2vanish as!+1: Lemma 3.4. [10] Let the remainder terms r1andr2respectively solve (3.14) and (3.15). Then rj2C([0;T];H1 0( ))\C1([0;T];L2( ));and lim !1(jjrjjjL2(M)+1jjrjjjH1(M)) = 0;forj= 1;2: 19We are now in a position to prove Theorem 1.4. We only prove Theorem 1.4 for the case where ( M;g) is simple. If ( ;g) satises the foliation condition, as in [44] (see also [14]), for any point q2, there exists a wedge-shaped neighborhood Oq ofqsuch that any geodesic in ( Oq;g) has no conjugate points. Therefore, we can now recover fk z(x;t;0) fork3:Then the folia- tion condition allows a layer stripping scheme to recover the coecients in the whole domain. However, the recovery of fz(x;t;0) andf(2) z(x;t;0) is quite dierent, which needs the inversion of some new ray transforms. Proof of Theorem 1.4. We divide the proof into three steps. Step 1. Leth1=ei'a1jin (3.12). Let w=u1u2; b =b1b2; q =q1q2: Then 8 >< >:wtt
gw+b1wt+q1w=(bu2t+qu2);(t;x)2M; w(t;x) = 0; (t;x)2; w(0;x) =wt(0;x) = 0; x2 :(3.16) Notice that vsolves system (3.13) and  b1;q1(h1) = b2;q2(h1) on  implies @wj= 0; we multiply the rst equation of (3.16) by vand integrate over Mto obtain (Lb1;q1w;v)L2(M)= (w;L b1;q1v)L2(M)= 0)Z M(bu2t+qu2)vdVg= 0: (3.17) HeredVg=jgj1 2dt^dxdenotes the volume form of the metric g=dt2+g:Recalling thatu2=ei'a1+r1andv=ei'a2+r2in a neighborhood of the null geodesic ;we have 0 = iZ Mb'ta1a2e2Im'dVg+Z Mqa1a2e2Im'dVg +Z Mb[a1ta2e2Im'+ ( i'ta1+a1t)r2ei'+ei'a2r1t+r1htr2]dVg +Z Mq(ei'a1r2+ei'a2r1+r1r2)dVg (3.18) It then follows from Lemma 3.4 that lim !1Z Mb'ta1a2e2Im'dVg= 0: (3.19) Since the functions a1;a2are supported in a small tubular neighborhood of the null geodesic, the integrand in (3.19) is supported near :Therefore we can use the Fermi coordinates z= (z0;z1;z00) to compute the limit. Recall that a1a2=n 2a10;0a20;02(jz0j 0) +O(1) =n 2jdetY(z0)j1e1 2p 2Rz0 sb(;0)d+O(1): 20Moreover, by [10, Lemma 3.6], we know that 'tdoes not vanish in V:Notice that b;q= 0 on the setM1nM:Then (3.19) yields lim !1n 2Zb0 a0Z jz0j<02(jz0j 0)e2Im'jdetY(z0)j1b(z0;z0)e1 2p 2Rz0 sb(;0)ddz0^z0= 0: We proceed to calculate the term n 2Z jz0j<02(jz0j 0)b(z0;z0)e2Im'dz0=Z Rne2xTPx(x)dx; where(x) =2(jxj 0)b(z0;x) is a smooth function with compact support B(0;0),P= ImH(z0) is a positive-denite matrix. By the following well-known formula F(e1 2xTPx)() =(2)n 2 (detP)1 2e1 2TP1; whereFdenotes the Fourier transform, we have F(e2xTPx)() =(2)n 2 (detP)1 2(4)n 2e1 8TP1: Equivalently, we have F[e1 2xT(1 4P1)x] =(2)n 2 det (1 4P1)1 2e2xTPx: Since Z RnF[f](x)g(x)dx=Z Rnf(x)F[g](x)dx; forf;g2Lp(Rn); p2(1;+1); we have Z Rne1 8TP1F()d=Z RnF[e1 8xTP1x](x)dx = (2)n 2(4)n 2(detP)1 2Z Rne2xTPx(x)dx: LetP() :Rn!Cbe a measurable function. The well known theory of pseudo- dierential operators tells us F[P(D)u]() =P()^u() =P()F[a]();P()2Sm; whereDis the dierential operator, Smis the symbol class of order m;and P(D)u=1 (2)nZ Rneix
P()^u()d: 21ThereforeZ Rne2hxTPx(x)dx=1 (2)n 2(4h)n 2(detP)1 2Z Rne1 8hTP1F()d =1 (2)n 2(4)n 2(detP)1 2+1X k=01 k!(1 8)kZ Rn(TP1)kF()d =1 (4)n 2(detP)1 2+1X k=01 k!(1 8)k(PP1(D))k(0) =1 (4)n 2(detP)1 2((0) +O(1)); wherePP1(D)() is dened by (TP1)kF() =F[(PP1(D))k()]: By Lemma 3.2, we have jdetY(z0)j1= (det ImH(z0))1 2(det ImH0)1 2 Then lim !1n 2Z jz0j<02(jz0j 0)e2Im'jdetY(z0)j1e1 2p 2Rz0 sb(;0)ddz0 =1 4n(det ImH0)1 2b(z0;0)e1 2p 2Rz0 sb(;0)d: (3.20) Combine (3.19) with (3.20), we nd that 2p 2Zb0 a0d dz0e1 2p 2Rz0 sb(;0)ddz0=Zb0 a0b(z0;0)e1 2p 2Rz0 sb(;0)ddz0= 0: (3.21) Therefore Zb0 a0b(z0;0)dz0= 0; which implies that Z b() = 0 holds for any maximal null geodesic in R . Then Proposition 3.1 is applied to obtain b= 0;which implies that b1=b2inE: We return to the equation (3.18) with b= 0 to have lim !1Z Mqa1a2e2Im'dVg= 0: By a similar argument, this reduces lim !1n 2Zb0 a0Z jz0j<0q2(jz0j 0)jdetY(z0)j1e2Im'dz0^dz0= 0: (3.22) 22Then Z q() = 0 holds for all maximal null geodesics inR :Together with Proposition 3.1, we conclude that q= 0: Step 3. Sincef(2) zis known, we proceed with the third and higher order lineariza- tion. Let W(123)=@3 @1@2@3 =0u(f); W(ij)=@2 @i@j =0u(f);1i;j3: Setm(x;t) =fuuu(x;t;0):Let (3) be the permutation group of f1;2;3g:Then 8 >>< >>:Lb;qW(123)+fuu(x;t;0) 2P 2(3)W((1)(2))v(3)+mv1v2v3= 0 inM; W(123)= 0 on  ; W(123)(x;0) =W(123) t(x;0) = 0 in ;(3.23) where Lb;qW(123)=W(123) tt
gW(123)+bW(123) t+qW(123); and for each k= 1;2;3; vk=@ @k =0u(f) solves 8 >< >:vktt
gvk+bvkt+qvk= 0 inM; vk=hk on ; vk(x;0) =vkt(x;0) = 0 in :(3.24) Letv0solve the following adjoint system of (3.24) 8 >< >:v0tt
gv0bv0t+ (q+bt)v0= 0 inM; v0=h0 on ; v0(x;T) =v0t(x;T) = 0 in :(3.25) Integrating by parts over Myields Z v0@3 @1@2@3 =0(h1+2h2+3h2)d =Z Mmv1v2v3v0+Z Mfuu(x;t;0) 2X 2(3)W((1)(2))v(3)v0dgdt: (3.26) Sincef2uu(x;t;0) is known. Then the Dirichlet to Neumann map  determines Z Mmv1v2v3v0dVg: (3.27) We will use special solutions v1;v2;v3;v0in the above identity. Concretely, we shall use the following Gaussian beam solutions ei'a+rconstructed in section 4.1. 23For givenp= (t0;x0)2E;we choose local coordinates such that gcoincides with the standard Minkowski metric at p:Dene the light cone at pas C(p) =f(t;X)2TpM:t2=jXj2 gg: Similar to [7, Lemma 1] (see also, [14, section 3.2], [12, Section 5]), we can assume without loss of generality that 0;12C(p) satisfying 0= (1;p 12;;0;


;0);  1= (1;1;0;


;0) for some2[0;1]:Takinge>0 small and introduce 2= (1;q 1e2;e;0;


;0);  3= (1;q 1e2;e;0;


;0): By [7, Lemma 1], 0;1are linear-independent, and there are constants k0;k1;k2;k3 such that k00+k11+k22+k33= 0: Denotekto be the null geodesic with cotangent vector kandp:Taking vk=eikk'kak+rkfork= 0;1;2;3 as the Gaussian beams concentrating near the null geodesic k:Notice that the manifold ( ;g) is simple, the null geodesic k(k= 0;1;2;3) can intersect only at p: Insertingvk=eikk'kak+rkinto (3.27), with estimate (3.11), the Dirichlet-to- Neumann map determines n+1 2Z Mmv0v1v2v3dVg =n+1 2Z Mmei(k0'0+k1'1+k2'2+k3'3)a0a1a2a3dVg+O(1): (3.28) Clearly, the product a0a1a2a3is supported in a neighborhood of p:We introduce a lemma to deal with the above integral. Lemma 3.5. [12, Lemma 5]. The function S:=k0'0+k1'1+k2'2+k3'3 is well-dened in a neighborhood of pand (1)S(p) = 0; (2)DgS(p) = 0; (3)ImS(p1)cd(p1;p)forp1in a neighborhood of p;wherec>0is a constant. Based on the above lemma, applying the stationary phase (e.g., see [16, Theorem 7.75]) to (3.28), we cn+1 2Z Mmv0v1v2v3dVg=m(p)(a0a1a2a3)(p) +O(1); 24wherecdenotes some explicit constant. Thus, the Dirichlet-to-Neumann map deter- minesm(p): For the recovery the higher order coecients f(k) u(x;t;0) fork4;we can achieve this by induction. We refer to [14, Section 4] for such an operation and omit the details. Therefore, the proof of Theorem 1.4 is complete. Remark 3.2. We notice that the recovery of qandbis much dierent from that of higher order terms f(k) u(k3). Lemma 3.4 is a special case of Lemma 3.3, and the Gaussian beam of order N= 0 is enough for the linear inverse problem of recovering b andq. It seems that we can not recover fuuby the same method as that for terms f(k) ufor k= 1 ork3. One of the reason is that we can not choose three time-like vectors 0;1;2such that0;1are linear-dependent but 0;1;2are linear-dependent. We mention that, in [46], due to the presence of two dierent matrics gPandgS;the authors have chosen three dierent vectors satisfying the above Lemma 3.5. Therefore, they proved the unique recovery of coecients by the Gaussian beams in place of the distorted plane waves method used in e.g., [14]. Appendix Well-posedness We prove the well-posendess result to (1.1). We begin with the following linear wave equation 8 >< >:utt
gu=FinM; u=h on ; u(x;0) =u0;ut(x;0) =u1(x) in :(3.29) Let (u0;u1)2Hm+1( )Hm( ); h2Hm+1(); F2L1([0;T];Hm( )) with@k tF(x;t)2L1([0;T];Hmk( )) fork= 0;1;


m:Moreover, we assume that the compatibility conditions hold up to order m;which are given by ( h(x;0) =u0(x)j; ht(x;0) =u1(x)j; htt(x;0) = [
gu0(x) +F(x;0)]j; @k th(x;0) =@k tu(x;0)j; k= 3;


;m:(3.30) According to [23, Theorem 2.45], for system 3.29 with the above conditions, we have Lemma 3.6. Letmbe a positive integer and T > 0:Then system (3.29) admits a unique solution u2Em+1and@u2Hm():Moreover, the dependence of uand@u onu0;u1;h;F is continuous in the corresponding spaces, i.e., jjujjEm+1+jj@vjjHm() C(T)(jju0jjHm+1( )+jju1jjHm( )+jjhjjHm+1()+jjFjjXm); (3.31) 25wherejjFjj2 Xm=mP k=0jj@k tFjj2 L1([0;T];Hmk): Assume further that F2Emandb;q2Cm(M) withm >n 2:Assume that the compatibility conditions hold for (3.32) up to order m. Applying lemma 3.6, and by the fact that Emis a Banach algebra when m>n 2;we know that 8 >< >:vtt
gv+bvt+qv=FinM; v=h on ; v(x;0) =v0;vt(x;0) =v1(x) in :(3.32) admits a unique solution v2Em+1with@v2Hm():Moreover, the following energy estimate holds jjvjjEm+1+jj@vjjHm() C(T)(jjv0jjHm+1( )+jjv1jjHm( )+jjhjjHm+1()+jjFjjXm): (3.33) We now in a position to prove the well-posedness of the nonlinear system (1.1) with small initial data ( u0;u1) and small boundary data h: Letvsolve the following non-homogeneous linear wave equation 8 >< >:vtt
gv+bvt+fz(x;t;0)v= 0 inM; v=h on ; v(x;0) =u0;vt(x;0) =u1(x) in :(3.34) ForL>0 given, let Bm+1(L) =f(y1;y2)2Hm+1( )Hm( ) :jjy1jjHm+1+jjy2jjHmLg; Nm+1(L) =fh2Hm+1() :jjhjjHm+1()LgHm+1(): Letu0;u12Bm+1("0=3);andh2Nm+1("0=3) for"0suciently small. Then jjvjjEm+1+jj@vjjHm()C(T)"0: (3.35) For given0>0 suciently small, let Zm+1(0) =f^w2Em+1:jj^wjjEm+10gEm+1: Letwsolve the following homogeneous equation 8 >>< >>:wtt
gw+bwt+fz(x;t;0)w+1P k=2f(k)(x;t;0)(v+ ^w)k k!= 0 inM; w= 0 on  ; w(x;0) =wt(x;0) = 0 in ;(3.36) wherev2Em+1is the solution of (3.34) with estimate (3.35). We dene a map A:Zm+1(0)!Em+1; 26which sends the given ^ w2Zm+1(0) to the solution of (3.36). For any positive integers kandR;one has jjf(k)(x;t;0)jjEm+1k! Rksup jzj=Rjjf(x;t;z )jjEm+1: By the a priori estimate of (3.36) and the property of Banach algebra, we have jjA^wjjEm+1C(T)1X k=21 k!jjf(k)(x;t;0)jjEm+1jjv+ ^wjjk Em+1 C(T)1X k=21 Rksup jzj=Rjjf(x;t;z )jjEm+12k1(jjvjjk Em+1+jj^wjjk Em+1) C(T) Rsup jzj=Rjjf(x;t;z )jjEm+11X k=12k Rk(0k 0+"0"k 0) =2C(T) Rsup jzj=Rjjf(x;t;z )jjEm+1(00 R20+"0"0 R2"0): (3.37) LetR= 2 and"0=01 2small enough such that C(T) sup jzj=Rjjf(x;t;z )jjEm+101 2: Then we have jjA^wjjEm+10; which implies that A:Zm+1(0)!Zm+1(0) is well dened. Let F(x;t;^w) =1X k=2f(k)(x;t;0)(v+ ^w)k k!: Then F(x;t;^w1)F(x;t;^w2) =1X k=2f(k)(x;t;0) (k1)!Z1 0[(v+ ^w1) + (1)(v+ ^w2)]k1d: Therefore, taking 0small enough, we see that jjA^w1A^w2jjEm+1=jjw1w2jjEm+1 C(T)1X k=2k Rksup jzj=Rjjf(x;t;z )jjEm+1(30)k1jj^w1^w2jjEm+1 1 2jj^w1^w2jjEm+1: (3.38) Thus,A:Zm+1(0)!Zm+1(0) is a contraction. The Banach's xed point theorem implies that u=v+w2Em+1is a solution to system (1.1). Moreover, we have jjujjEm+1+jjujjC(M)+jj@ujjHm()C(jju0jjHm+1( )+jju1jjHm( )+jjhjjHm+1()); whereC > 0 is independent of u0;u1andh: 27References [1] L. Beilina, M. Cristofol, S. Li, and M. Yamamoto, Lipschitz stability for an in- verse hyperbolic problem of determining two coecients by a nite number of observations, Inverse Problems 34 (2018) 015001. [2] M. Belishev, An approach to mutidimentional inverse problems for the wave equa- tion, Dokl. Akad. Nauk SSSP, 297 (1987), pp. 524-527 (in Russian). [3] M. Belishev, Y. 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2022-12-04

In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams together with the higher order linearization are respectively used to derive the uniqueness results of recovering the coefficients.

Inverse problem of recovering the time-dependent damping and nonlinear terms for wave equations

2212.01815v2

arXiv:2011.03311v1 [math.NA] 6 Nov 2020A GENERALIZED FINITE ELEMENT METHOD FOR THE STRONGLY DAMPED WAVE EQUATION WITH RAPIDLY VARYING DATA PER LJUNG1, AXEL M ˚ALQVIST1, AND ANNA PERSSON2 Abstract. We propose a generalized finite element method for the strong ly damped wave equation with highly varying coefficients. The pr oposed method is based on the localized orthogonal decomposition introdu ced in [30], and is designed to handle independent variations in both the dam ping and the wave propagation speed respectively. The method does so by a utomatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal orde r is proven in L2(H1)-norm, independent of the derivatives of the coefficients. W e present numerical examples that confirm the theoretical findings. 1.Introduction This paper is devoted to the study of numerical solutions to the str onglydamped waveequation with highly varying coefficients. The equation takes th e generalform (1.1) ¨ u−∇·(A∇˙u+B∇u) =f, on a bounded domain Ω. Here, AandBrepresent the system’s damping and wave propagation respectively, fdenotes the source term, and the solution uis a displacement function. This equation commonly appears in the modellin g of vis- coelastic materials, where the strong damping −∇ ·A∇˙uarises due to the stress being represented as the sum of an elastic part and a viscous part [6 , 13]. Viscoelas- tic materials have several applications in engineering, including noise d ampening, vibration isolation, and shock absorption (see [21] for more applica tions). In partic- ular, in multiscale applications, such as modelling of porous medium or co mposite materials, AandBare both rapidly varying. There have been much recent work regarding strongly damped wav e equations. For instance, well-posedness of the problem is discussed in [7, 20, 2 2], asymptotic behavior in [8, 3, 31, 35] solution blowup in [12, 2], and decay estimates in [18]. In particular, FEM for the strongly damped wave equation has been analyzed in [25] using the Ritz–Volterra projection, and [24] uses the classical Ritz-projection in the homogeneous case with Rayleigh damping. In these papers, co nvergence of optimal order is shown. However, in the case of piecewise linear po lynomials, the convergence relies on at least H2-regularity in space. Consequently, since the Key words and phrases. Strongly damped wave equation, multiscale, localized orth ogonal de- composition, finite element method, reduced basis method. 1Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Gothenburg, Sweden. 2Department of Mathematics, KTH Royal Institute of Technolo gy,, SE-100 44 Stockholm, Sweden. 12 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON H2-norm depends on the derivatives of the coefficients, the error is b ounded by /⌊a∇d⌊lu/⌊a∇d⌊lH2∼max(ε−1 A,ε−1 B) whereεAandεBdenote the scales at which AandBvary respectively. The convergenceorderisthus onlyvalidwhen the mes hwidthhfulfills h <min(εA,εB). In other words, we require a mesh that is fine enough to resolve the variations of AandB, which becomes computationally challenging. This type of difficulty is common for equations with rapidly varying data, an issue for which several numerical methods have been developed (see e.g. [5, 4, 23 , 29, 19]). None of these methods are however applicable to the strongly damped wa ve equation, where two different multiscale coefficients have to be dealt with. In th is paper, we proposea novelmultiscale method based on the localized orthogona ldecomposition (LOD) method. The LOD method is based on the variational multiscale method presen ted in [17]. It was first introduced in [30], and has since then been further d eveloped and analyzed for several types of problems (see e.g. [27, 28, 1, 16, 1 5]). In particular, [26] studies the LOD method for quadratic eigenvalue problems, whic h correspond to time-periodic wave equations with weak damping. The main idea of th e method is based on a decomposition of the solution space into a coarse and a fi ne part. The decomposition is done by defining an interpolant that maps funct ions from an infinite dimensional space into a finite dimensional FE-space. In th is way, the kernel of the interpolant captures the finescale features that t he coarse FE-space misses, and hence defines the finescale space. Subsequently, one may use the or- thogonal complement to this finescale space with respect to a prob lem-dependent Ritz-projection as a modified FE-space. In the case of time-depen dent problems, the LODmethod performs particularlywell in the sense that the mod ified FE-space only needs to be computed once, and can then be re-used in each tim e step. Multiscale methods, as the localized orthogonal decomposition, are usually de- signed to handle problems with a single multiscale coefficient. In this sen se, the strongly damped wave equation is different, as an extra coefficient a ppears due to the strong damping. Hence, one of the main challenges for the nove l method is how to incorporate the finescale behavior of both coefficients in the computation. Nevertheless, it should be noted that existing multiscale methods ar e applicable for some special cases of this equation. An example is the case of Ra yleigh damp- ing where the coefficients are proportional to each other. Other e xamples are the steady state case, the transient phase in which the solution evolve s rapidly in time, as well as the case of weak damping where no spatial derivatives are present on the damping term. In this paper we present a generalized finite element method (GFEM) for solving the strongly damped wave equation. The method uses both the dam ping and diffusion coefficients to construct a generalized finite element space , similar to those in e.g. [30, 28]. The solution is then evaluated in this space, but to acco unt for the time dependence, an additional correction is added to it. However, this correction is evaluated on the fine scale, and thus expensive to compute. To ov ercome this issue, we prove spatial exponential decay for the corrections so that we can restrict the problems to patches in a similar manner as for the modified basis fu nctions in [30]. The effect of the proposed method is that the multiscale basis co mpensates for the damping early on in the simulation when it is dominant and then gr adually starts to compensate for the wave propagation which is dominant a t steady state. This is done seamlessly and automatically by the method. Furthermor e, we proveA GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 3 optimal order convergence in L2(H1)-norm for this method. Following this, we show that it is sufficient to compute the finescale corrections for on ly a few time steps by applying reduced basis (RB) techniques. For related work on RB methods, see e.g. [14, 10, 9], and for an introduction to the topic we refer to [ 33]. The outline of the paper is as follows: In Section 2 we present the wea k formula- tionandclassicalFEMforthestronglydampedwaveequation, along withnecessary assumptions. Section 3 is devoted to the generalized finite element m ethod and its localizationprocedure. In Section 4 errorestimates for the metho d areproven. Sec- tion 5 covers the details of the RB approach, and finally in Section 6 we illustrate numerical examples that confirm the theory derived in this paper. 2.Weak formulation and classical FEM We consider the wave equation with strong damping of the following fo rm ¨u−∇·(A∇˙u+B∇u) =f,in Ω×(0,T], (2.1) u= 0,on Γ×(0,T], (2.2) u(0) =u0,in Ω, (2.3) ˙u(0) =v0in Ω, (2.4) whereT >0 and Ω is a polygonal (or polyhedral) domain in Rd, d= 2,3,and Γ :=∂Ω. The coefficients AandBdescribe the damping and propagation speed respectively,and fdenotesthesourcefunction ofthesystem. Weassume A=A(x), B=B(x) andf=f(x,t), i.e. the multiscale coefficients are independent of time. Denote by H1 0(Ω) the classical Sobolev space with norm /⌊a∇d⌊lv/⌊a∇d⌊l2 H1(Ω)=/⌊a∇d⌊lv/⌊a∇d⌊l2 L2(Ω)+/⌊a∇d⌊l∇v/⌊a∇d⌊l2 L2(Ω) whose functions vanish on Γ. Moreover, let Lp(0,T;B) be the Bochner space with norm /⌊a∇d⌊lv/⌊a∇d⌊lLp(0,T;B)=/parenleftbigg/integraldisplayT 0/⌊a∇d⌊lv/⌊a∇d⌊lp Bdt/parenrightbigg1/p , p∈[1,∞), /⌊a∇d⌊lv/⌊a∇d⌊lL∞(0,T;B)= esssup t∈[0,T]/⌊a∇d⌊lv/⌊a∇d⌊lB, whereBis a Banach space with norm /⌊a∇d⌊l·/⌊a∇d⌊lB. In this paper, following assumptions are made on the data. Assumptions. The damping and propagation coefficients A,B∈L∞(Ω,Rd×d)are symmetric and satisfy 0< α−:= essinf x∈Ωinf v∈Rd\{0}A(x)v·v v·v<esssup x∈Ωsup v∈Rd\{0}A(x)v·v v·v=:α+<∞, 0< β−:= essinf x∈Ωinf v∈Rd\{0}B(x)v·v v·v<esssup x∈Ωsup v∈Rd\{0}B(x)v·v v·v=:β+<∞. In addition, we assume that f∈L∞([0,T];L2(Ω))and˙f∈L2([0,T];L2(Ω)). For the spatial discretization, let {Th}h>0denote a family of shape regular el- ements that form a partition of the domain Ω. For an element K∈ Th, let the4 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON corresponding mesh size be defined as hK:= diam( K), and denote the largest di- ameter of the partition by h:= max K∈ThhK. We now define the classical FE-space using continuous piecewise linear polynomials as Sh:={v∈ C(¯Ω) :v/vextendsingle/vextendsingle Γ= 0,v/vextendsingle/vextendsingle Kis a polynomial of partial degree ≤1,∀K∈ Th}, and letVh=Sh∩H1 0(Ω). The semi-discrete FEM becomes: find uh(t)∈Vhsuch that (2.5) (¨ uh,v)+a(˙uh,v)+b(uh,v) = (f,v),∀v∈Vh, t >0, with initial values uh(0) =uh,0and ˙uh(0) =vh,0whereuh,0,vh,0∈Vhare appropri- ate approximations of u0andv0respectively. Here ( ·,·) denotes the usual L2-inner product, a(·,·) = (A∇·,∇·), andb(·,·) = (B∇·,∇·). For the temporal discretization, let 0 = t0< t1< ... < t N=Tbe a uniform partition with time step tn−tn−1=τ. The time step here is chosen uniformly for simplicity, but the choice of varying time step is still viable. We apply a ba ckward Euler scheme to get the fully discrete system: find un h∈Vhsuch that (2.6) ( ¯∂2 tun h,v)+a(¯∂tun h,v)+b(un h,v) = (fn,v),∀v∈Vh forn≥2. Here, the discrete derivative is defined as ¯∂tun h= (un h−un−1 h)/τ. For results on regularityand errorestimates for the FEM solution o f the strongly damped wave equation, we refer to [24]. Moreover, existence and u niqueness of a solution to (2.6) is guaranteed by Lax–Milgram. In the analysis, we use the notations /⌊a∇d⌊l · /⌊a∇d⌊l2 a:=a(·,·),/⌊a∇d⌊l · /⌊a∇d⌊l2 b:=b(·,·), as well as |||·|||2= ˜a(·,·) :=a(·,·) +τb(·,·), and the fact that these are equivalent with the H1-norm. That is, there exist positive constants Ca,Cb,C˜a,ca,cb,c˜a∈R, such that ca/⌊a∇d⌊lv/⌊a∇d⌊l2 H1≤ /⌊a∇d⌊lv/⌊a∇d⌊l2 a≤Ca/⌊a∇d⌊lv/⌊a∇d⌊l2 H1,∀v∈H1(Ω), cb/⌊a∇d⌊lv/⌊a∇d⌊l2 H1≤ /⌊a∇d⌊lv/⌊a∇d⌊l2 b≤Cb/⌊a∇d⌊lv/⌊a∇d⌊l2 H1,∀v∈H1(Ω), (2.7) c˜a/⌊a∇d⌊lv/⌊a∇d⌊l2 H1≤ |||v|||2≤C˜a/⌊a∇d⌊lv/⌊a∇d⌊l2 H1,∀v∈H1(Ω). Theorem 2.1. The solution un hto(2.6)satisfies the following bounds /⌊a∇d⌊l¯∂tun h/⌊a∇d⌊l2 L2+n/summationdisplay j=2τ/⌊a∇d⌊l¯∂tuj h/⌊a∇d⌊l2 H1+/⌊a∇d⌊lun h/⌊a∇d⌊l2 H1≤Cn/summationdisplay j=2τ/⌊a∇d⌊lfj/⌊a∇d⌊l2 H−1+C(/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 L2+/⌊a∇d⌊lu1 h/⌊a∇d⌊l2 H1),(2.8) n/summationdisplay j=2τ/⌊a∇d⌊l¯∂2 tuj h/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊l2 H1≤Cn/summationdisplay j=2τ/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2+C(/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 H1+/⌊a∇d⌊lu1 h/⌊a∇d⌊l2 H1),. (2.9) forn≥2. Proof.To prove (2.8), choose v=τ¯∂tun hin (2.6) to get τ(¯∂2 tun h,¯∂tun h)+τ/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊l2 a+τb(un h,¯∂tun h) =τ(fn,¯∂tun h). (2.10) Due to Cauchy–Schwarz and Young’s inequality we have the following lo wer bound τ(¯∂2 tun h,¯∂tun h) =/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊l2 L2−(¯∂tun−1 h,¯∂tun h)≥1 2/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊l2 L2−1 2/⌊a∇d⌊l¯∂tun−1 h/⌊a∇d⌊l2 L2, and similarly τb(un h,¯∂tun h)≥1 2/⌊a∇d⌊lun h/⌊a∇d⌊l2 b−1 2/⌊a∇d⌊lun−1 h/⌊a∇d⌊l2 b.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 5 Similar bounds will be used repeatedly throughout the paper. Summin g (2.10) over ngives 1 2/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊l2 L2−1 2/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 L2+n/summationdisplay j=2τ/⌊a∇d⌊l¯∂tuj h/⌊a∇d⌊l2 a+1 2/⌊a∇d⌊lun h/⌊a∇d⌊l2 b−1 2/⌊a∇d⌊lu1 h/⌊a∇d⌊l2 b≤n/summationdisplay j=2τ/⌊a∇d⌊lfj/⌊a∇d⌊lH−1/⌊a∇d⌊l¯∂tuj h/⌊a∇d⌊lH1. Using the equivalence of the norms (2.7), Cauchy–Schwarz and You ng’s (weighted) inequality to subtract/summationtextn j=2τ/⌊a∇d⌊l¯∂tuj h/⌊a∇d⌊l2 H1from both sides, we get exactly (2.8). The proof of (2.9) is similar. We choose v=τ¯∂2 tun hin (2.6) and sum over nto get n/summationdisplay j=2τ/⌊a∇d⌊l¯∂2 tuj h/⌊a∇d⌊l2 L2+1 2/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊l2 a−1 2/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 a+n/summationdisplay j=2τb(uj h,¯∂2 tuj h)≤n/summationdisplay j=2τ/⌊a∇d⌊lfj/⌊a∇d⌊lL2/⌊a∇d⌊l¯∂2 tuj h/⌊a∇d⌊lL2. For the sum involving the bilinear form b(·,·) we use summation by parts to get n/summationdisplay j=2τb(uj h,¯∂2 tuj h) =n/summationdisplay j=3−τb(¯∂tuj h,¯∂tuj−1 h)−b(u2 h,¯∂tu1 h)+b(un h,¯∂tun h). Using (2.8), the equivalence of the norms (2.7), and Young’s weighte d inequality we have/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglen/summationdisplay j=3τb(¯∂tuj h,¯∂tuj−1 h)+b(u2 h,¯∂tu1 h)−b(un h,¯∂tun h)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ≤Cn/summationdisplay j=3τ/⌊a∇d⌊l¯∂tuj h/⌊a∇d⌊l2 H1+C(/⌊a∇d⌊lu2 h/⌊a∇d⌊l2 H1+/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 H1)+C/⌊a∇d⌊lun h/⌊a∇d⌊l2 H1+Cǫ/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊l2 a ≤Cn/summationdisplay j=2τ/⌊a∇d⌊lfj/⌊a∇d⌊l2 H−1+C(/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 H1+/⌊a∇d⌊lu1 h/⌊a∇d⌊l2 H1)+Cǫ/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊l2 a. SinceCǫcan be made arbitrarily small, it can be kicked to the left hand side. Usin g that/⌊a∇d⌊lfj/⌊a∇d⌊l2 H−1≤C/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2we deduce (2.9). /square 3.Generalized finite element method This section is dedicated to the development of a multiscale method ba sed on the framework of the standard LOD. First of all, we introduce some notation for the discretization. Let VHbe a FE-space defined analogously to Vhin previous section, but with larger mesh size H > h. Moreover, we assume that corresponding family of partitions {TH}H>his, in addition to shape-regular, also quasi-uniform. Denote by Nthe set of interior nodes of VHand byλxthe standard hat function forx∈ N, such that VH= span({λx}x∈N). Finally, we make the assumption that This a refinement of TH, such that VH⊆Vh. 3.1.Ideal method. Todefineageneralizedfiniteelementmethod forourproblem, we aim to construct a multiscale space Vmsof the same dimension as VH, but with better approximation properties. For the construction of such a multiscale space, letIH:Vh→VHbe an interpolation operator that has the projection property IH=IH◦IHand satisfies (3.1)H−1 K/⌊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 whereN(K) :={K′∈ TH:K′∩K/\e}atio\slash=∅}.Furthermore, for a shape-regular and quasi-uniform partition, the estimate (3.1) can be summed into the g lobal estimate H−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), whereCγdepends on the interpolation constant CIand the shape regularity pa- rameter defined as γ:= max K∈THγK,whereγK=diam(BK) diam(K). HereBKdenotes the largest ball inside K. A commonly used example of such an interpolant is IH=EH◦ΠH, where Π His the piecewise L2-projection onto P1(TH), the space of functions that are affine on each triangle K∈ TH, andEH:P1(TH)→ VHis an averaging operator that, to each free node x∈ N, assigns the arithmetic mean of corresponding function values on intersecting elements, i.e . (EH(v))(x) =1 card{K∈ TH:x∈K}/summationdisplay K∈TH:x∈Kv/vextendsingle/vextendsingle K(x). For more discussion regarding possible choices of interpolants, see e.g. [11] or [32]. Let the space Vfbe defined by the kernel of the interpolant, i.e. Vf= ker(IH) ={v∈Vh:IHv= 0}. That is, Vfis a finescale space in the sense that it captures the features that are excluded from the coarse FE-space. This consequently leads to th e decomposition Vh=VH⊕Vf, such that every function v∈Vhhas a unique decomposition v=vH+vf, where vH∈VHandvf∈Vf. In the case of the LOD method for the standard wave equation (se e [1]), one considers a Ritz-projection based solely on the B-coefficient to construct a multi- scale space. Instead, the goal is to define a multiscale space based on the inner product a(·,·) +τb(·,·) (for a fixed τ) and add additional correction to account for the time-dependency. This particular choice of scalar product comes from the backward Euler time-stepping formulation and both simplifies the ana lysis and is morenaturalin the implementation. Another possibility is to choose a(·,·) asscalar product. For v∈VH, we consider the Ritz-projection Rf:VH→Vfdefined by a(Rfv,w)+τb(Rfv,w) =a(v,w)+τb(v,w),∀w∈Vf. Using this projection, we may define the multiscale space Vms:=VH−RfVHsuch that (3.2) Vh=Vms⊕Vf,anda(vms,vf)+τb(vms,vf) = 0. Note that dim( Vms) = dim( VH), and hence we can view Vmsas a modified coarse space that contains finescale information of AandB. Next, we may use the Ritz- projection to define the basis functions for the space Vms. Forx∈ N, denote by φx:=Rfλx∈Vfthe solution to the (global) corrector problem (3.3) a(φx,w)+τb(φx,w) =a(λx,w)+τb(λx,w),∀w∈Vf. Wecannowconstructourbasisfor Vmsas{λx−φx}x∈Nwhichincludesthebehavior of the coefficients. For an illustration of the Ritz-projected hat fu nction, as well as the modified basis function for Vms, see Figure 1.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 7 (a)λx−φx. (b)φx. Figure 1. The modified basis function λx−φxand the Ritz- projected hat function φx. We may now formulate our ideal (but impractical) method. Since the s olution space can be decomposed as Vh=Vms⊕Vf, the idea is to solve a coarse scale problem in Vms, and then add additional correction from a problem on the fine scale. The method reads: find un lod=vn+wn, wherevn∈Vmsandwn∈Vfsuch that τ(¯∂2 tvn,z)+a(vn,z)+τb(vn,z) =τ(fn,z)+a(un−1 lod,z),∀z∈Vms, (3.4) a(wn,z)+τb(wn,z) =a(un−1 lod,z), ∀z∈Vf, (3.5) forn≥2 with initial data u0 lod=u0 h∈Vmsandu1 lod=u1 h∈Vms. The initial data is chosen in Vmsto simplify the implementation of the finescale correctors. We further discuss this choice in Section 4.4. Remark 3.1.Note that in (3.5), we do not take neither the source function nor the second derivative into account. This is because we can subtrac t an interpolant within the L2-product, so that corresponding error converges at the same o rder as the method itself. Moreover, the vn-part and wn-part have been excluded from the bilinear form a(·,·)+τb(·,·) in (3.4) and (3.5) respectively, due to the orthogonality between VmsandVf. Note that the multiscale space Vmsis created using (3.3) with small τ. Thus, theA-coefficient dominates the system for short times. Moreover, we n ote from (3.5) that for Nlarge enough, we reach a steady state so that wN≈wN−1and vN≈vN−1. We get for z∈Vf a(wN,z)+τb(wN,z)≈a(uN lod,z) =a(vN,z)+a(wN,z) =−τb(vN,z)+a(wN,z), due to the orthogonality. Hence, by rearranging terms we have th at b(vN,z)+b(wN,z) =b(uN lod,z)≈0, which shows that the solution converges to a state where it is ortho gonal with respect to B. 3.2.Localized method. The method we have considered so far is based on the global projection (3.3) onto the finescale space Vf, which results in a large linear system that is expensive to solve. Moreover, the basis corrector s yield a global support that makes the linear system (3.4) not sparse, but dense . Hence, we wish8 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON to localize the computations onto coarse grid patches in order to yie ld a sparse matrix system. To localize the corrector problem, we first introduce the patches t o which the support of each basis function is to be restricted. For ω⊂Ω, letN(ω) :={K∈ TH:K∩ω/\e}atio\slash=∅}, and define a patch Nk(ω) of size kas N1(ω) :=N(ω), Nk(ω) :=N(Nk−1(ω)),fork≥2. Given these coarse grid patches, we may restrict the finescale spa ceVfto them by defining Vω f,k:={v∈Vf: supp(v)⊆Nk(ω)}, for a subdomain ω⊂Ω. In particular, we will commonly use ω=T∈ THand ω=x∈ N. Next, define the element restricted Ritz-projection RT fsuch that RT fv∈Vfis the solution to the system a(RT fv,z)+τb(RT fv,z) =/integraldisplay T(A+τB)∇v·∇zdx,∀z∈Vf. Note that we may construct the global Ritz-projection as the sum Rfv=/summationdisplay T∈THRT fv. Fork∈N, we may restrict the projection to a patch by letting RT f,k:VH→VT f,kbe such that RT f,kv∈VT f,ksolves a(RT f,kv,z)+τb(RT f,kv,z) =/integraldisplay T(A+τB)∇v·∇zdx,∀z∈VT f,k. By summation we yield the corresponding global version as Rf,kv=/summationdisplay T∈THRT f,kv. Finally, we may construct a localized multiscale space as Vms,k:=VH−Rf,kVH, spanned by {λx−Rf,kλx}x∈N. In order to justify the act of localization, it is required that a corre ctorφx vanishes rapidly outside an area of its corresponding node x. Indeed, the following theorem from [27] shows that the corrector φxsatisfy an exponential decay away from its node, making the localization procedure viable. Theorem 3.2. There exists a constant c≥(8CIγ(2+CI))−1, that only depends on the mesh constant γ, such that for any T∈ THand any v∈H1 0(Ω)the solution φ∈Vfof the variational problem ˜a(φ,w) =/integraldisplay T(˜A∇v)·∇wdx,∀w∈Vf satisfies /⌊a∇d⌊l˜A1/2∇φ/⌊a∇d⌊lL2(Ω\Nk(T))≤√ 2exp/parenleftbig −cα−+τβ− α++τβ+k/parenrightbig /⌊a∇d⌊l˜A1/2∇v/⌊a∇d⌊lL2(T),∀k∈N, where˜A=A+τB.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 9 With the space Vms,kdefined, we are able to localize the computations on the coarse scale system in (3.4) by replacing the multiscale space by its loc alized coun- terpart. It remains to localize the computations of the finescale sy stem in (3.5), which equivalently can be written as a(¯∂twn,z)+b(wn,z) =1 τa(vn−1,z). We replace the right hand side by its localized version vn−1 k∈Vms,kand note that vn−1 k=/summationtext x∈Nαn−1 x(λx−Rf,kλx). Thus, we seek our localized finescale solution as wn k=/summationtext x∈Nwn k,x, wherewn k,x∈Vx f,ksolves (3.6) a(¯∂twn k,x,z)+b(wn k,x,z) =1 τa(αn−1 x(λx−Rf,kλx),z),∀z∈Vx f,k, so that the computation of this equation is localized to a patch surro unding the nodex∈ N. We introduce the functions ξl k,x∈Vx f,kas solution to the parabolic equation (3.7) a(¯∂tξl k,x,z)+b(ξl k,x,z) =a(1 τχ(0,τ)(λx−Rf,kλx),z),∀z∈Vx f,k, with initial value ξ0 k,x= 0, and where χ(0,τ)is an indicator function on the interval (0,τ). We claim that wn k,x=/summationtextn l=1αn−l xξl k,xis the solution to (3.6). This follows as for allz∈Vx f,k a(¯∂twn k,x,z)+b(wn k,x,z) =a(¯∂tn/summationdisplay l=1αn−l xξl k,x,z)+b(n/summationdisplay l=1αn−l xξl k,x,z) =n/summationdisplay l=2αn−l x/parenleftbig a(¯∂tξl k,x,z)+b(ξl k,x,z)/parenrightbig +αn−1 x/parenleftbig a(¯∂tξ1 k,x,z)+b(ξ1 k,x,z)/parenrightbig = 0+a(αn−1 x(λx−Rf,kλx),z). With the localized computations established, the GFEM reads: find un lod,k=vn k+ wn k, wherevn k=/summationtext x∈Nαn x(λx−Rf,kλx)∈Vms,ksolves (3.8)τ(¯∂2 tvn k,z)+a(vn k,z)+τb(vn k,z) =τ(fn,z)+a(un−1 lod,k,z),∀z∈Vms,k, andwn k=/summationtext x∈N/summationtextn l=1αn−l xξl k,x, whereξl k,x∈Vx f,ksolves (3.7). To justify the fact that we localize the finescale equation, we requir e a result similar to that of Theorem 3.2, but for the functions {ξl x}N l=1. We finish this section about localization by proving that these functions satisfy the expo nential decay required for the localization procedure to be viable. Theorem 3.3. For any node x∈ N, letξn x∈Vfbe the solution to a(¯∂tξn x,z)+b(ξn x,z) =a(1 τχ(0,τ)(λx−Rfλx),z),∀z∈Vf, with initial value ξ0 x= 0. Then there exist constants c >0andC >0such that for anyk≥1 /⌊a∇d⌊lξn x/⌊a∇d⌊lH1(Ω\Nk(x))≤Ce−ck/⌊a∇d⌊lλx/⌊a∇d⌊lH1, for sufficiently small time step τ.10 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON Proof.First, we analyze the problem for the first time step, which when mult iplied byτcan be written as (3.9) a(ξ1 x,z)+τb(ξ1 x,z) =a(λx−φx,z),∀z∈Vf, whereφx=Rfλx. We denote ˜ a=a+τbsuch that ˜ a(φx,z) = ˜a(λx,z) for allz∈Vf. Furthermore we use the energy norm |||·|||:=/radicalbig ˜a(·,·), and by |||·|||Dwe denote the restriction of the norm onto a domain D. As seen in the proof of Theorem 4.1 in [27], the result in Theorem 3.2 can be written as |||φx|||Ω\Nk(x)≤Cφµ⌊k/4⌋|||λx|||, for some µ <1. Moreover we define the cut-off function ηk∈VHby ηk:=/braceleftBigg 1,in Ω\Nk+1(x), 0,inNk(x), forx∈ N. Now let ν=ηk−3. Then we have that supp(ν) = Ω\Nk−3(x), supp(∇ν) =Nk−2(x)\Nk−3(x). With this setting, we note that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk≤/integraldisplay Ων˜A∇ξ1 x·∇ξ1 xdx=/integraldisplay Ω˜A∇ξ1 x·∇(νξ1 x)dx−/integraldisplay Ω˜A∇ξ1 x·ξ1 x∇νdx ≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Ω˜A∇ξ1 x·∇(1−IH)(νξ1 x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright =:M1+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Ω˜A∇ξ1 x·∇IH(νξ1 x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =:M2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Ω˜A∇ξ1 x·ξ1 x∇νdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =:M3, wherewe havedenoted ˜A=A+τB. We now proceedto estimate the terms M1,M2 andM3separately. For M1, we use the problem (3.9) with z= (1−IH)(νξ1 x)∈Vf to get M1=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay ΩA∇(λx−φx)·∇(1−IH)(νξ1 x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleτ/integraldisplay Ω\Nk−3B∇(φx)·∇(1−IH)(νξ1 x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, where we have used the ˜ a-orthogonality between VmsandVf, that the integral is zero on supp( λx), and that the support of the remaining integrand is Ω \Nk−3. Thus, we get that M1≤τβ+ α−|||φx|||Ω\Nk−3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Ω\Nk−3≤τβ+ α−Cφµ⌊k−3 4⌋|||λx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Ω\Nk−4. Moreover, by similar calculations as in the proof of Theorem 4.1 in [27], f romM2 andM3we get M2,M3≤˜C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Nk\Nk−4,A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 11 for a constant ˜C >0. In total, for ε∈(0,1), we find that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk≤τβ+ α−Cφµ⌊k−3 4⌋|||λx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Ω\Nk−4+˜C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Nk\Nk−4 ≤(β+Cφ)2 α2 −ετ2µ2⌊k−3 4⌋|||λx|||2+ε/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk−4 +˜C(/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk−4−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk). Letδ:= (ε+˜C)(1+˜C)−1<1, and set κ= max(δ,µ)<1. Then, by rearranging the terms we get the inequality /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk≤(β+Cφ)2 α2 −ε(1+˜C)τ2κ2⌊k−3 4⌋|||λx|||2+κ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk−4. Repeating the estimate, we end up with /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk≤κ⌊k/4⌋/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω+(β+Cφ)2 α2 −ε(1+˜C)|||λx|||2⌊k/4⌋−1/summationdisplay i=0τ2κiκ2⌊k−3−4i 4⌋. We proceed by estimating/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Ω. By choosing z=ξ1 xin (3.9) we get /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2≤ |||λx−φx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ |||λx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle, since |||λx−φx|||2= ˜a(λx−φx,λx−φx)≤ |||λx−φx||||||λx|||. Moreover, for i= 0,1,2,...,⌊k/4⌋−1, we note that κi+2⌊k−3−4i 4⌋≤κ⌊k/4⌋−1+2⌊k−3−4(k/4−1) 4⌋=κ⌊k/4⌋−1+2⌊1 4⌋=κ⌊k/4⌋−1(3.10) so in total we have the estimate /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Ω\Nk≤/radicalbig 1+C0τ2κ1 2⌊k/4⌋|||λx|||,withC0=(β+Cφ)2κ−1 α2 −ε(1+˜C)(⌊k/4⌋−1). Recall that this is for the first time step. In next time step, we cons ider the problem a(ξ2 x,z)+τb(ξ2 x,z) =a(ξ1 x,z),∀z∈Vf. As for the first time step, we split the estimate into the similar integra lsM1,M2, andM3, and get M1≤τβ+ α−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ1 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Ω\Nk−3/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Ω\Nk−3≤τβ+ α−/radicalbig 1+C0τ2κ1 2⌊k−3 4⌋|||λx|||/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Ω\Nk−4, whileM2andM3remain the same. In total, we get the estimate (1+˜C)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk≤β2 + α2 −ετ2(1+C0τ2)κ⌊k−3 4⌋|||λx|||2+(ε+˜C)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk−4. Once again, by letting δ= (ε+˜C)/(1+˜C) and since δ≤κ, we get /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk≤β2 + α2 −ε(1+˜C)τ2(1+C0τ2)κ⌊k−3 4⌋|||λx|||2+κ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Ω\Nk−4 ≤κ⌊k/4⌋|||λx|||2+β2 + α2 −ε(1+˜C)(1+C0τ2)⌊k/4⌋−1/summationdisplay i=0τ2κiκ⌊k−3−4i 4⌋.12 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON Once again we use (3.10) to conclude that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleξ2 x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Ω\Nk≤/radicalbig 1+C1τ2(1+C0τ2)κ1 2⌊k/4⌋|||λx||| =/radicalbig 1+C1τ2+C1τ2C0τ2κ1 2⌊k/4⌋|||λx|||, where C1=β2 +κ−1 α2 −ε(1+˜C)(⌊k/4⌋−1). Inductively, we get for arbitrary time step nthe estimate |||ξn x|||Ω\Nk≤κ1 2⌊k/4⌋|||λx|||/radicaltp/radicalvertex/radicalvertex/radicalbtn−1/summationdisplay i=0(C1τ2)i+(C1τ2)nC0τ2. Sinceκ1 2⌊k/4⌋≤Ce−ckfor some c >0 andC >0, and since the energy norm is equivalent to the H1-norm, the theorem holds. /square Remark 3.4.Note that the constant that appears in the final inequality conver ges to1 1−C1τ2, whichmeansthat theconstantbehavesnicelyforsufficientlysmallt ime steps. More specifically, for time steps τ≤/radicalBigg α2 −εκ(1+˜C) β2 +(⌊k/4⌋−1). 4.Error estimates In this section we derive error estimates of the ideal method (3.4)- (3.5). The additional error due to localization can be controlled in terms of the lo calization parameter k. This is further discussed in Remark 4.9. We begin by considering an auxiliary problem. 4.1.Auxiliary problem. The auxiliary problem is defined as the standard vari- ational formulation for the strongly damped wave equation, but we exclude the second order time derivative. Moreover, we let the starting time t=t0be general and set the time discretization to t=t0< t1< ... < t N=T. Thus, the auxiliary problem is to find Zn h∈Vhforn= 1,...,N, such that (4.1) a(¯∂tZn h,v)+b(Zn h,v) = (fn,v),∀v∈Vh, with initial value Z0 h∈Vms. Equivalently, multiply (4.1) by τand we may consider (4.2) a(Zn h,v)+τb(Zn h,v) =τ(fn,v)+a(Zn−1 h,v),∀v∈Vh. Existence of a solution to this problem is guaranteed by Lax–Milgram. For sim- plicity, we make the assumption that the initial data for the damped w ave equation (2.6) is already in the multiscale space Vms, such that u0 h=u0 lod∈Vms, u1 h=u1 lod∈Vms. For general initial data we refer to Section 4.4 below. Furthermore , to limit the technical details in the proof we have chosen to analyze the error in theL2(H1)- norm instead of the pointwise (in time) H1-norm.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 13 The solution space can be decomposed as Vh=Vms⊕Vf, such that the solution can be written as Zn h=vn+wnwherevn∈Vmsandwn∈Vf. If we insert this into the system in (4.2) and consider test functions z∈Vms, the left hand side becomes a(Zn h,z)+τb(Zn h,z) =a(vn,z)+τb(vn,z), where we have used the orthogonality between VmsandVfwith respect to a(·,·)+ τb(·,·). Likewise, if test functions z∈Vfare considered, the left hand side becomes a(Zn h,z)+τb(Zn h,z) =a(wn,z)+τb(wn,z). With these findings, we define the approximation to the auxiliary prob lem as to findZn lod=vn+wn, wherevn∈Vmsandwn∈Vfsuch that a(vn,z)+τb(vn,z) =τ(fn,z)+a(Zn−1 lod,z),∀z∈Vms, (4.3) a(wn,z)+τb(wn,z) =a(Zn−1 lod,z), ∀z∈Vf, (4.4) with initial data Z0 lod∈Vms. Note that if f= 0, then Zn h=Zn lodfor every n, meaning that the method reproduces Zn hexactly. For the auxiliary problem, we prove the following error estimates. Theorem 4.1. LetZn hbe the solution to (4.1)andZn lodthe solution to (4.3)-(4.4). Assume that Z0 lod−Z0 h= 0, then the error is bounded by /⌊a∇d⌊lZn h−Zn lod/⌊a∇d⌊lH1≤CHn/summationdisplay j=1τ/⌊a∇d⌊lfj/⌊a∇d⌊lL2. (4.5) Iffn∈L2(Ω), forn≥0, then we have n/summationdisplay j=1τ/⌊a∇d⌊lZj h−Zj lod/⌊a∇d⌊l2 L2≤CH2n/summationdisplay j=1τ/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2, (4.6) and iffn=¯∂tgn, for some {gn}N n=0such that gn∈Vh, then n/summationdisplay j=1τ/⌊a∇d⌊lZj h−Zj lod/⌊a∇d⌊l2 L2≤CH2/parenleftBiggn/summationdisplay j=1τ/⌊a∇d⌊lgj/⌊a∇d⌊l2 L2+/⌊a∇d⌊lg0/⌊a∇d⌊l2 L2/parenrightBigg , (4.7) where C does not depend on the variations in AorB. Proof.SinceZn h∈Vhthere are ¯ vn∈Vmsand ¯wn∈Vfsuch that Zn h= ¯vn+ ¯wn. Leten=Zn h−Zn lod, and consider |||en|||2:=a(en,en)+τb(en,en) =τ(fn,en)+a(Zn−1 h,en)−a(vn,en)−τb(vn,en)−a(wn,en)−τb(wn,en). Forvn∈Vmswe have due to the orthogonality and (4.3) a(vn,en)+τb(vn,en) =a(vn,¯vn−vn)+τb(vn,¯vn−vn) =τ(fn,¯vn−vn)+a(Zn−1 lod,¯vn−vn). Similarly, for wn∈Vfwe use the orthogonality and (4.4) to get a(wn,en)+τb(wn,en) =a(Zn−1 lod,¯wn−wn).14 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON Hence, |||en|||2=τ(fn,en)+a(Zn−1 h,en)−τ(fn,¯vn−vn) −a(Zn−1 lod,¯vn−vn)−a(Zn−1 lod,¯wn−wn) =τ(fn,¯wn−wn)+a(Zn−1 h−Zn−1 lod,en). The first term can be bounded by using the interpolation operator IH τ|(fn,¯wn−wn)| ≤τ/⌊a∇d⌊lfn/⌊a∇d⌊lL2/⌊a∇d⌊l¯wn−wn−IH(¯wn−wn)/⌊a∇d⌊lL2 ≤CHτ/⌊a∇d⌊lfn/⌊a∇d⌊lL2/⌊a∇d⌊l¯wn−wn/⌊a∇d⌊lH1 ≤CHτ/⌊a∇d⌊lfn/⌊a∇d⌊lL2/⌊a∇d⌊len/⌊a∇d⌊lH1 ≤CHτ/⌊a∇d⌊lfn/⌊a∇d⌊lL2|||en|||. For the second term we note that Zn−1 h−Zn−1 lod=en−1so that |||en||| ≤CHτ/⌊a∇d⌊lfn/⌊a∇d⌊lL2+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleen−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. Using this bound repeatedly and e0= 0 we get |||en||| ≤CHn/summationdisplay j=1τ/⌊a∇d⌊lfj/⌊a∇d⌊lL2. This concludes the proof since /⌊a∇d⌊len/⌊a∇d⌊lH1≤C|||en|||. To prove the remaining bounds in L2-norm, we define the forward difference operator ˜∂txn= (xn+1−xn)/τand consider the dual problem: find xj h∈Vhfor j=n−1,...,0, such that xn h= 0 and a(−˜∂txj h,z)+b(xj h,z) = (ej+1,z),∀z∈Vh. (4.8) Note that this problem moves backwards in time. By choosing z=xj hin (4.8) and performing a classical energy argument, we deduce /⌊a∇d⌊lxj h/⌊a∇d⌊l2 H1+n/summationdisplay k=jτ/⌊a∇d⌊lxk h/⌊a∇d⌊l2 H1≤Cn/summationdisplay k=j+1τ/⌊a∇d⌊lek/⌊a∇d⌊l2 L2. (4.9) Similarly, by choosing z=−˜∂txj h, we achieve /⌊a∇d⌊lxj h/⌊a∇d⌊l2 H1+n/summationdisplay k=jτ/⌊a∇d⌊l˜∂txk h/⌊a∇d⌊l2 H1≤Cn/summationdisplay k=j+1τ/⌊a∇d⌊lek/⌊a∇d⌊l2 L2. (4.10) Now, use (4.8) to get n/summationdisplay j=1τ/⌊a∇d⌊lej/⌊a∇d⌊l2 L2=n/summationdisplay j=1τa(−˜∂txj−1 h,ej)+τb(xj−1 h,ej). Summation by parts gives n/summationdisplay j=1τ/⌊a∇d⌊lej/⌊a∇d⌊l2 L2=n/summationdisplay j=1τa(−˜∂txj−1 h,ej)+τb(xj−1 h,ej) (4.11) =n/summationdisplay j=1τa(xj−1 h,¯∂tej)+τb(xj−1 h,ej),A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 15 wherewehaveused xn=e0= 0. Furthermore, weusethe equations(4.1) and(4.3), and the orthogonality in (3.2), to show that the following Galerkin ort hogonality holds for zms∈Vms a(¯∂tej,zms)+b(ej,zms) =a(¯∂tZj h,zms)+b(Zj h,zms)−1 τa(vj,zms) (4.12) −b(vj,zms)+1 τa(Zj−1 lod,zms) (4.13) = (fj,zms)−(fj,zms) = 0. Letxj h=xms+xf, for some xms∈Vms, xf∈Vf. Using the orthogonality (4.12) and the equations (4.4) and (4.1) we deduce n/summationdisplay j=1τa(xj−1 h,¯∂tej)+τb(xj−1 h,ej) =n/summationdisplay j=1τa(xj−1 f,¯∂tej)+τb(xj−1 f,ej) =n/summationdisplay j=1τa(xj−1 f,¯∂tZj h)+τb(xj−1 f,Zj h) =n/summationdisplay j=1τ(xj−1 f,fj). Iffj∈L2(Ω), then we may subtract IHxf= 0 and use (3.1) to achieve n/summationdisplay j=1τ(xj−1 f,fj)≤CHn/summationdisplay j=1τ/⌊a∇d⌊lxj−1 f/⌊a∇d⌊lH1/⌊a∇d⌊lfj/⌊a∇d⌊lL2 ≤CH/parenleftBiggn/summationdisplay j=1τ/⌊a∇d⌊lxj−1 f/⌊a∇d⌊l2 H1/parenrightBigg1/2/parenleftBiggn/summationdisplay j=1τ/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2/parenrightBigg1/2 . Note that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexj−1 f/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexj−1 ms/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglexj−1 h/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 . Hence the energy estimate (4.9) can now be used to achieve (4.6). If fj=¯∂tgjone may use summation by parts to achieve n/summationdisplay j=1τ/⌊a∇d⌊lej/⌊a∇d⌊l2 L2=n/summationdisplay j=1τ(xj−1 f,¯∂tgj)≤n/summationdisplay j=1τ(−˜∂txj−1 f,gj)−(x0 f,g0) ≤CHn/summationdisplay j=1τ/⌊a∇d⌊l˜∂txj−1 f/⌊a∇d⌊lH1/⌊a∇d⌊lgj/⌊a∇d⌊lL2+CH/⌊a∇d⌊lx0 f/⌊a∇d⌊lH1/⌊a∇d⌊lg0/⌊a∇d⌊lL2, where we have used xn f=xn h= 0. Using (4.10) we conclude (4.7). /square Remark 4.2.The bound in (4.6) is not of optimal order, but it is useful in the error analysis. The next lemmagiveserrorestimatesforthe discrete time derivativ eofthe error. In the analysis of the (full) damped wave equation we use g=¯∂tuh, see Lemma 4.5. If the initial data is nonzero we expect ¯∂tg1below to be of order t−1inL2-norm. A detailed explanation of this is given below. Hence, we have a blow up clos e to zero due to low regularity of the initial data. Therefore, we need to multip ly the error bytj. This is similar to the parabolic case for nonsmooth initial data see, e.g ., [34].16 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON Lemma 4.3. LetZn hbe the solution to (4.1)andZn lodthe solution to (4.3)-(4.4). AssumeZ0 lod−Z0 h= 0. If¯∂tfn∈L2(Ω), forn≥1, then n/summationdisplay j=2τ/⌊a∇d⌊l¯∂t(Zj h−Zj lod)/⌊a∇d⌊l2 L2≤CH2/parenleftBiggn/summationdisplay j=2τ/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2 L2+/⌊a∇d⌊lf1/⌊a∇d⌊l2 L2/parenrightBigg (4.14) and iffn=¯∂tgn, for some {gn}N n=0, such that gn∈Vh, then n/summationdisplay j=2τ/⌊a∇d⌊l¯∂t(Zj h−Zj lod)/⌊a∇d⌊l2 L2≤CH2/parenleftBiggn/summationdisplay j=2τ/⌊a∇d⌊l¯∂tgj/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂tg1/⌊a∇d⌊l2 L2/parenrightBigg (4.15) and, in addition, the following bound holds n/summationdisplay j=2τt2 j/⌊a∇d⌊l¯∂t(Zj h−Zj lod)/⌊a∇d⌊l2 L2≤CH2/parenleftBiggn/summationdisplay j=2τ/⌊a∇d⌊l¯∂tgj/⌊a∇d⌊l2 L2+t2 1/⌊a∇d⌊l¯∂tg1/⌊a∇d⌊l2 L2/parenrightBigg , (4.16) where C does not depend on the variations in AorB. Proof.The proof of (4.14) is similar to (4.6). Let ej=Zj h−Zj lodand define the dual problem a(−˜∂txj h,z)+b(xj h,z) = (¯∂tej+1,z),∀z∈Vh, j=n−1,...,0, (4.17) withxn= 0. Choosing z=¯∂tej+1and performing summation by parts we deduce n/summationdisplay j=2τ/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2 L2=n/summationdisplay j=2τa(−˜∂txj−1,¯∂tej)+τb(xj−1 h,¯∂tej) (4.18) =n/summationdisplay j=2τa(xj−1 h,¯∂2 tej)+τb(xj−1 h,¯∂tej)+a(x1 h,¯∂te1), where we used that xn= 0. Following the same argument as for (4.7), but with a difference quotient, we arrive at n/summationdisplay j=2/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2 L2=n/summationdisplay j=2τa(xj−1 h,¯∂2 tej)+τb(xj−1 h,¯∂tej)+a(x1 h,¯∂te1) =n/summationdisplay j=2τ(xj−1 f,¯∂tfj)+a(x1 h,¯∂te1). Usinge0= 0, we deduce a(x1 h,¯∂te1) =1 τa(x1 h,e1)≤C τH/⌊a∇d⌊lx1 h/⌊a∇d⌊lH1τ/⌊a∇d⌊lf1/⌊a∇d⌊lL2≤CH/⌊a∇d⌊lx1 h/⌊a∇d⌊lH1/⌊a∇d⌊lf1/⌊a∇d⌊lL2, and with ¯∂tfj∈L2(Ω) we get n/summationdisplay j=2/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2 L2≤CHn/summationdisplay j=2τ/⌊a∇d⌊lxj−1 f/⌊a∇d⌊lH1/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊lL2+CH/⌊a∇d⌊lx1 h/⌊a∇d⌊lH1/⌊a∇d⌊lf1/⌊a∇d⌊lL2, and (4.14) follows by using an energy estimate of xj hsimilar to (4.9), but with ¯∂tej on the right hand side. Iffj=¯∂tgjwe proceed as for (4.7) to achieve n/summationdisplay j=2τ(xj−1 f,¯∂tfj)≤CHn/summationdisplay j=2τ/⌊a∇d⌊l˜∂txj−1 f/⌊a∇d⌊lH1/⌊a∇d⌊l¯∂tgj/⌊a∇d⌊lL2+CH/⌊a∇d⌊lx1 f/⌊a∇d⌊lH1/⌊a∇d⌊l¯∂tg1/⌊a∇d⌊lL2A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 17 and (4.15) follows by using energy estimates similar to (4.10). For (4.16) we consider the dual problem a(−˜∂txj h,z)+b(xj h,z) = (tj+1¯∂tej+1,z),∀z∈Vh, j=n−1,...,0. (4.19) A simple energy estimate shows /⌊a∇d⌊lxj h/⌊a∇d⌊l2 H1+n−1/summationdisplay k=jτ/⌊a∇d⌊l˜∂txk h/⌊a∇d⌊l2 H1≤Cn−1/summationdisplay k=jτt2 k+1/⌊a∇d⌊l¯∂tek+1/⌊a∇d⌊l2 L2, j= 0,...,n−1. (4.20) Now choosing z=tj+1¯∂tej+1in (4.19) and performing summation by parts gives n/summationdisplay j=2τt2 j/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2 L2=n/summationdisplay j=2τa(−˜∂txj−1 h,tj¯∂tej)+τb(xj−1 h,tj¯∂tej) (4.21) =n/summationdisplay j=2/parenleftbig τa(xj−1 h,tj¯∂2 tej)+τb(xj−1 h,tj¯∂tej) +a(xj−1 h,(tj−tj−1)¯∂tej−1)/parenrightbig +a(x1 h,t1¯∂te1). The first two terms of the sum can be handled similarly to (4.15), n/summationdisplay j=2τa(xj−1 h,tj¯∂2 tej)+τb(xj−1 h,tj¯∂tej) =n/summationdisplay j=2τ(xj−1 f,tj¯∂tfj). Now, using summation by parts we achieve n/summationdisplay j=2τ(xj−1 f,tj¯∂tfj) =n/summationdisplay j=2τ/parenleftbig (−˜∂txj−1 f,tjfj)+(xj−1 f,fj)/parenrightbig +(x1 f,t1f1) ≤CH/parenleftBiggn/summationdisplay j=2τ/parenleftbig tj/⌊a∇d⌊l˜∂txj−1 f/⌊a∇d⌊lH1/⌊a∇d⌊lfj/⌊a∇d⌊lL2+/⌊a∇d⌊lxj−1 f/⌊a∇d⌊lH1/⌊a∇d⌊lfj/⌊a∇d⌊lL2/parenrightbig +t1/⌊a∇d⌊lx1 f/⌊a∇d⌊lH1/⌊a∇d⌊lf1/⌊a∇d⌊lL2/parenrightBigg where we can use (4.20). Note that in the first term we can use the ( crude) bound t2 j≤t2 nand let the constant Cdepend on T. We get n/summationdisplay j=2τ(xj−1 f,tj¯∂tfj)≤CH/parenleftBiggn/summationdisplay j=1τt2 j/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2 L2/parenrightBigg1/2/parenleftBigg/parenleftBiggn/summationdisplay j=2τ/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2/parenrightBigg1/2 +t1/⌊a∇d⌊lf1/⌊a∇d⌊lL2/parenrightBigg . Forthe thirdterm in(4.21), weuse tj−tj−1=τandonceagainperformsummation by parts to get n/summationdisplay j=2τa(xj−1 h,¯∂tej−1) =n/summationdisplay j=2τa(−˜∂txj−1 h,ej−1), where we have used xn h=e0= 0. Combining (4.20) and (4.5) we get n/summationdisplay j=2τa(−˜∂txj−1 h,ej−1)≤Cmax j=1,...,n/⌊a∇d⌊lej/⌊a∇d⌊lH1/parenleftBiggn/summationdisplay j=2τ/parenrightBigg1/2/parenleftBiggn/summationdisplay j=2τ/⌊a∇d⌊l˜∂txj−1 h/⌊a∇d⌊l2 H1/parenrightBigg1/2 ≤CHn/summationdisplay j=1τ/⌊a∇d⌊lfj/⌊a∇d⌊lL2/parenleftBiggn/summationdisplay j=1τt2 j/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2 L2/parenrightBigg1/2 .18 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON For the last term in (4.21) we use (4.20) and (4.5) for n= 1 to achieve a(x1 h,t1¯∂te1) =a(x1 h,e1)≤CH/parenleftBiggn/summationdisplay j=1τt2 j/⌊a∇d⌊l¯∂tej/⌊a∇d⌊l2 L2/parenrightBigg1/2 t1/⌊a∇d⌊lf1/⌊a∇d⌊lL2, and (4.16) follows by letting fj=¯∂tgj. /square 4.2.The damped wave equation. For the error analysis of the full damped wave equation we shall make use of the projection corresponding t o the auxiliary problem. For un h∈Vh, letXn=Xn v+Xn w∈VhwithXn v∈VmsandXn w∈Vfsuch that a(Xn v−un h,z)+τb(Xn v−un h,z) =a(Xn−1−un−1 h,z),∀z∈Vms, (4.22) a(Xn w,z)+τb(Xn w,z) =a(Xn−1,z), ∀z∈Vf. (4.23) Note that since un hsolves (2.6), the system (4.22)-(4.23) is equivalent to a(Xn v,z)+τb(Xn v,z) =τ(fn−¯∂2 tun h,z)+a(Xn−1,z),∀z∈Vms, (4.24) a(Xn w,z)+τb(Xn w,z) =a(Xn−1,z), ∀z∈Vf. (4.25) That is, we may view un handXnas the solution and approximationto the auxiliary problem with source data fn−¯∂2 tun h. We deduce following lemma. Lemma 4.4. Letun hbe the solution to (2.6)andXnthe solution to (4.22)-(4.23). The error satisfies the following bounds /⌊a∇d⌊lXn−un h/⌊a∇d⌊lH1≤CHn/summationdisplay j=2τ/⌊a∇d⌊lfj−¯∂2 tuj h/⌊a∇d⌊lL2, n≥2, (4.26) n/summationdisplay j=2τ/⌊a∇d⌊lXn−un h/⌊a∇d⌊l2 L2≤CH2/parenleftBiggn/summationdisplay j=2τ(/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂tuj h/⌊a∇d⌊l2 L2)+/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 L2/parenrightBigg , n≥2,(4.27) where C does not depend on the variations in AorB. Proof.We let the auxiliaryproblem(4.1) startat t1. In this case e0=u1 h−u1 ms= 0, sinceu1 ms=u1 h∈Vms. The bound (4.26) now follows directly from (4.5) with fn−¯∂2 tun has right hand side. The second bound (4.27) follows from (4.6) and (4 .7) withfn∈L2(Ω) and gn=¯∂tun+1 h. /square In a similar way me may deduce bounds for the (discrete) time derivat ive of the error. As a direct consequence of Lemma 4.3, we get the following re sult.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 19 Lemma 4.5. Letun hbe the solution to (2.6)andXnthe solution to (4.22)-(4.23). The following bounds hold n/summationdisplay j=3τ/⌊a∇d⌊l¯∂t(Xj−uj h)/⌊a∇d⌊l2 L2(4.28) ≤CH2/parenleftBiggn/summationdisplay j=3τ(/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂2 tuj h/⌊a∇d⌊l2 L2)+/⌊a∇d⌊lf2/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂2 tu2 h/⌊a∇d⌊l2 L2/parenrightBigg , n/summationdisplay j=3τt2 j/⌊a∇d⌊l¯∂t(Xj−uj h)/⌊a∇d⌊l2 L2(4.29) ≤CH2/parenleftBiggn/summationdisplay j=3τ(/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂2 tuj h/⌊a∇d⌊l2 L2)+t2 2/⌊a∇d⌊lf2/⌊a∇d⌊l2 L2+t2 2/⌊a∇d⌊l¯∂2 tu2 h/⌊a∇d⌊l2 L2/parenrightBigg , where C does not depend on the variations in AorB. Lemma 4.6. Letun handun lodbe the solutions to (2.6)and(3.4)-(3.5), respectively. Assume that u0=u1= 0. The error is bounded by n/summationdisplay j=2τ/⌊a∇d⌊luj lod−uj h/⌊a∇d⌊l2 H1≤CH2/parenleftBiggn/summationdisplay j=1τ(/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2 L2)+ max j=1,...,n/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2/parenrightBigg , forn≥2, where C does not depend on the variations in AorB. Proof.Begin by splitting the error into two contributions un lod−un h=un lod−Xn+Xn−un h=:θn+ρn, whereXnis the solution to the simplified problem in(4.22)-(4.23). By Lemma 4.4 ρnis bounded by /⌊a∇d⌊lρn/⌊a∇d⌊lH1≤CHn/summationdisplay j=2τ/parenleftbig /⌊a∇d⌊lfj/⌊a∇d⌊lL2+/⌊a∇d⌊l¯∂2 tuj h/⌊a∇d⌊lL2/parenrightbig , and we can now apply the energy bound (2.9). It remains to bound θn. Recall that for anyz∈Vhwe have z=zms+zffor some zms∈Vmsandzf∈Vf. Using that un lod=vn+wnsatisfies (3.4) and the orthogonality (3.2) we get (¯∂2 tun lod,zms)+a(¯∂tun lod,zms)+b(un lod,zms) = (fn,zms)+(¯∂2 twn,zms). Similarly, due to (3.5) and the orthogonality, (¯∂2 tun lod,zf)+a(¯∂tun lod,zf)+b(un lod,zf) = (¯∂2 tun lod,zf). ForXnwe use (4.22)-(4.23) and the orthogonality to deduce (¯∂2 tXn,z)+a(¯∂tXn,z)+b(Xn,z) = (¯∂2 tXn,z)+(fn−¯∂2 tun h,zms), z∈Vh, Hence,θnsatisfies (¯∂2 tθn,z)+a(¯∂tθn,z)+b(θn,z) = (−¯∂2 tρn,z)−(¯∂2 tun h,zf) +(¯∂2 tun lod,zf)+(¯∂2 twn,zms), z∈Vh,20 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON withθ0=θ1= 0, since u0 lod=u0 h=X0andu1 lod=u1 h=X1. Let˜θn=/summationtextn j=2τθj. Multiplying by τand summing over ngives (¯∂tθn,z)+a(θn,z)+b(˜θn,z)≤(−¯∂tρn,z)−(¯∂tun h−¯∂tu1 h,zf)(4.30) +(¯∂tun lod−¯∂tu1 lod,zf)+(¯∂twn−¯∂tw1,zms), where we have used that θ1=θ0=ρ1=ρ0= 0. Using the interpolant IHwe deduce (¯∂tun h,zf)+(¯∂tun lod,zf)+(¯∂twn,zms)≤CH(/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊lL2+/⌊a∇d⌊l¯∂tun lod/⌊a∇d⌊lL2)/⌊a∇d⌊lz/⌊a∇d⌊lH1 +CH/⌊a∇d⌊l¯∂tun lod/⌊a∇d⌊lH1/⌊a∇d⌊lzms/⌊a∇d⌊lL2, for 1≤n≤N. Letα(n) =/⌊a∇d⌊l¯∂tun h/⌊a∇d⌊lL2+/⌊a∇d⌊l¯∂tun lod/⌊a∇d⌊lH1. Since/⌊a∇d⌊lzms/⌊a∇d⌊lL2≤C/⌊a∇d⌊lz/⌊a∇d⌊lH1and α(1) = 0 due to the vanishing initial data, we get (¯∂tθn,z)+a(θn,z)+b(˜θn,z)≤(−¯∂tρn,z)+CHα(n)/⌊a∇d⌊lz/⌊a∇d⌊lH1, z∈Vh. Now, choose z=θn=¯∂t˜θnin (4.30). We get 1 2/⌊a∇d⌊lθn/⌊a∇d⌊l2 L2−1 2/⌊a∇d⌊lθn−1/⌊a∇d⌊l2 L2+τ/⌊a∇d⌊lθn/⌊a∇d⌊l2 a+1 2/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 b−1 2/⌊a∇d⌊l˜θn−1/⌊a∇d⌊l2 b ≤τ/⌊a∇d⌊l¯∂tρn/⌊a∇d⌊lL2/⌊a∇d⌊lθn/⌊a∇d⌊lL2+CHτα(n)/⌊a∇d⌊lθn/⌊a∇d⌊lH1. Summing over ngives /⌊a∇d⌊lθn/⌊a∇d⌊l2 L2+n/summationdisplay j=2τ/⌊a∇d⌊lθj/⌊a∇d⌊l2 H1+/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 H1≤n/summationdisplay j=2τ/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊lL2/⌊a∇d⌊lθj/⌊a∇d⌊lL2+CHn/summationdisplay j=2τα(j)/⌊a∇d⌊lθj/⌊a∇d⌊lH1. Nowusing that /⌊a∇d⌊lθn/⌊a∇d⌊lL2≤ /⌊a∇d⌊lθn/⌊a∇d⌊lH1and Young’sweighted inequality, θjcan be kicked back to the left hand side. We deduce n/summationdisplay j=2τ/⌊a∇d⌊lθj/⌊a∇d⌊l2 H1≤Cn/summationdisplay j=2τ/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊l2 L2+CH2n/summationdisplay j=2τα(j)2. Using Lemma 4.5 we have n/summationdisplay j=2τ/⌊a∇d⌊lθj/⌊a∇d⌊l2 H1≤CH2/parenleftBiggn/summationdisplay j=2τ(/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂2 tuj h/⌊a∇d⌊l2 L2)+/⌊a∇d⌊l¯∂2 tu2 h/⌊a∇d⌊l2 L2/parenrightBigg +CH2n/summationdisplay j=2τα(j)2. To bound /⌊a∇d⌊l¯∂2 tu2 h/⌊a∇d⌊l2 L2, we consider (2.6) for n= 2 and choose v=¯∂2 tu2 h, which gives (¯∂2 tu2 h,¯∂2 tu2 h)+a(¯∂tu2 h,¯∂2 tu2 h)+b(u2 h,¯∂2 tu2 h) = (¯∂tf2,¯∂2 tu2 h). Due to the vanishing initial data ¯∂tu2 h=τ−1u2 hand¯∂2 tu2 h=τ−2u2 h. We get /⌊a∇d⌊l¯∂2 tu2 h/⌊a∇d⌊l2 L2+1 τ3/⌊a∇d⌊lu2 h/⌊a∇d⌊l2 a+1 τ2/⌊a∇d⌊lu2 h/⌊a∇d⌊l2 b= (f2,¯∂2 tu2 h), (4.31) and we deduce /⌊a∇d⌊l¯∂2 tu2 h/⌊a∇d⌊l2 L2≤C/⌊a∇d⌊lf2/⌊a∇d⌊l2 L2. All terms, except/summationtextn j=2τ/⌊a∇d⌊l¯∂tuj lod/⌊a∇d⌊l2 H1that appears in/summationtextn j=2α2(j), can now be bounded by using the regularity in Theorem 2.1. To bound/summationtextn j=2τ/⌊a∇d⌊l¯∂tuj lod/⌊a∇d⌊l2 H1weA GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 21 choosez=¯∂tvnandz=¯∂twnin (3.4) and (3.5) respectively. Adding the two equations and using the orthogonality between VmsandVfwe achieve (¯∂2 tvn,¯∂tvn)+a(¯∂tun lod,¯∂tun lod)+b(un lod,¯∂tun lod) = (fn,¯∂tvn) ≤Cǫ/⌊a∇d⌊lfn/⌊a∇d⌊l2 L2+ǫ/⌊a∇d⌊l¯∂tvn/⌊a∇d⌊l2 L2. Note 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 lod/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/⌊a∇d⌊l¯∂tun lod/⌊a∇d⌊la, so we may choose ǫsmall enough such that/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle¯∂tun lod/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecan be kicked to the left hand side. As in the proof of Theorem 2.1 we may now deduce /⌊a∇d⌊l¯∂tvn/⌊a∇d⌊l2 L2+n/summationdisplay j=2τ/⌊a∇d⌊l¯∂tuj lod/⌊a∇d⌊l2 H1+/⌊a∇d⌊lun lod/⌊a∇d⌊l2 H1≤C/parenleftBiggn/summationdisplay j=2/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 L2+/⌊a∇d⌊lu1 h/⌊a∇d⌊l2 H1/parenrightBigg ,(4.32) where we have used that v1=u1 lod=u1 h∈Vms. However, we have assumed vanishing initial data so these terms disappear. The lemma follows. /square Lemma 4.7. Letun handun lodbe the solutions to (2.6)and(3.4)-(3.5), respectively. Assume that f= 0. The error is bounded by n/summationdisplay j=2τt2 j/⌊a∇d⌊luj lod−uj h/⌊a∇d⌊l2 H1≤CH2(/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 H1+/⌊a∇d⌊lu1 h/⌊a∇d⌊l2 H1+/⌊a∇d⌊lu0 h/⌊a∇d⌊l2 H1), n≥2, where C does not depend on the variations in AorB. Proof.We follow the steps in the proof of Lemma 4.6 to equation (4.30). Note that /⌊a∇d⌊lρn/⌊a∇d⌊lH1can be bounded by Lemma 4.4 and the energy bound in (2.9) with f= 0. Now, let ˜θn=/summationtextn j=2τθj. Choose z=θn=¯∂t˜θnin (4.30) and multiply by τt2 n. We get t2 n 2/⌊a∇d⌊lθn/⌊a∇d⌊l2 L2−t2 n−1 2/⌊a∇d⌊lθn−1/⌊a∇d⌊l2 L2+τt2 n/⌊a∇d⌊lθn/⌊a∇d⌊l2 a+t2 n 2/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 b−t2 n−1 2/⌊a∇d⌊l˜θn−1/⌊a∇d⌊l2 b ≤τt2 n/⌊a∇d⌊l¯∂tρn/⌊a∇d⌊lL2/⌊a∇d⌊lθn/⌊a∇d⌊lL2+CHt2 nτ(α(n)+α(1))/⌊a∇d⌊lθn/⌊a∇d⌊lH1 +(t2 n−t2 n−1) 2/⌊a∇d⌊lθn−1/⌊a∇d⌊l2 L2+(t2 n−t2 n−1) 2/⌊a∇d⌊l˜θn−1/⌊a∇d⌊l2 b. Summing over nand using t2 n−t2 n−1≤2τtngives t2 n/⌊a∇d⌊lθn/⌊a∇d⌊l2 L2+n/summationdisplay j=2τt2 j/⌊a∇d⌊lθj/⌊a∇d⌊l2 H1+t2 n/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 H1≤Cn/summationdisplay j=2τt2 j/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊lL2/⌊a∇d⌊lθj/⌊a∇d⌊lL2(4.33) +CHn/summationdisplay j=2τt2 j(α(j)+α(1))/⌊a∇d⌊lθj/⌊a∇d⌊lH1+Cn/summationdisplay j=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2 L2+Cn/summationdisplay j=2τtj/⌊a∇d⌊l˜θj/⌊a∇d⌊l2 b. From 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 andtj/⌊a∇d⌊lθj/⌊a∇d⌊lH1to the left hand side. The remaining two sums needs to be bounded by other energy estimates.22 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON Multiply (4.30) by τand sum over nto get (θn,z)+a(˜θn,z)+b/parenleftBiggn/summationdisplay j=2τ˜θj,z/parenrightBigg ≤(ρn,z)−(un h−u1 h,zf)+(un lod−u1 lod,zf)(4.34) +(wn−w1,zms)+tn((¯∂tu1 h,zf)−(¯∂tu1 lod,zf)−(¯∂tw1,zms)). where we have used θ1=ρ1= 0. As in the proof of Lemma 4.6 we get (un h,zf)+(un lod,zf)+(wn,zms)≤CH(/⌊a∇d⌊lun h/⌊a∇d⌊lL2+/⌊a∇d⌊lun lod/⌊a∇d⌊lL2)/⌊a∇d⌊lz/⌊a∇d⌊lH1 +CH/⌊a∇d⌊lun lod/⌊a∇d⌊lH1/⌊a∇d⌊lzms/⌊a∇d⌊lL2, for 1≤n≤N. Letβ(n) =/⌊a∇d⌊lun h/⌊a∇d⌊lL2+/⌊a∇d⌊lun lod/⌊a∇d⌊lH1. Choose z=˜θn=¯∂t/summationtextn j=1τ˜θj. Similar to above energy estimates, we get /⌊a∇d⌊l˜θn/⌊a∇d⌊l2 L2+n/summationdisplay j=2τ/⌊a∇d⌊l˜θj/⌊a∇d⌊l2 a+/⌊a∇d⌊ln/summationdisplay j=2τ˜θj/⌊a∇d⌊l2 b≤n/summationdisplay j=2τ/⌊a∇d⌊lρj/⌊a∇d⌊l2 L2+CH2n/summationdisplay j=2τ(β(j) +β(1)+α(1))2.(4.35) Since/summationtextn j=2τtj/⌊a∇d⌊l˜θj|2 b≤C(tn)/summationtextn j=2τ/⌊a∇d⌊l˜θj/⌊a∇d⌊l2 awe may use (4.35) in (4.33). This gives t2 n/⌊a∇d⌊lθn/⌊a∇d⌊l2 L2+n/summationdisplay j=2τt2 j/⌊a∇d⌊lθj/⌊a∇d⌊l2 H1+t2 n/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 H1≤Cn/summationdisplay j=2τ(t2 j/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊l2 L2+/⌊a∇d⌊lρj/⌊a∇d⌊l2 L2)(4.36) +Cn/summationdisplay j=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2 L2+CH2n/summationdisplay j=2τ(t2 j(α(j)+α(1))2+(β(j)+β(1)+α(1))2). It remains to bound C/summationtextn j=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2 L2. For this purpose, choose z=θn=¯∂t˜θnin (4.34). Multiply by tnτand sum over nto achieve n/summationdisplay j=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2 L2+tn/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 a+n/summationdisplay j=2tjτb/parenleftBiggj/summationdisplay k=2τ˜θk,¯∂t˜θj/parenrightBigg ≤n/summationdisplay j=1τtj/⌊a∇d⌊lρj/⌊a∇d⌊l2 L2+n/summationdisplay j=2τ/⌊a∇d⌊l˜θj/⌊a∇d⌊l2 a(4.37) +CHn/summationdisplay j=2τtj(β(j)+β(1)+α(1))/⌊a∇d⌊lθj/⌊a∇d⌊lH1. Note that /⌊a∇d⌊lθj/⌊a∇d⌊lH1in the last sum in only present in the right hand side. The second term on the right hand side is bounded by (4.35). For the ter m involving the bilinear form b(·,·) we use summation by parts to get −n/summationdisplay j=2τb/parenleftBigg tjj/summationdisplay k=2τ˜θk,¯∂tj/summationdisplay k=2τθk/parenrightBigg ≤n/summationdisplay j=2τb/parenleftBigg tj˜θj+j/summationdisplay k=2τ˜θk,˜θj−1/parenrightBigg −b/parenleftBigg tnn/summationdisplay j=2τ˜θj,˜θn/parenrightBigg ≤Cn/summationdisplay j=2τtj/⌊a∇d⌊l˜θj/⌊a∇d⌊l2 b+C/⌊a∇d⌊ln/summationdisplay j=2τ˜θj/⌊a∇d⌊l2 b+Cǫt2 n/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 H1.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 23 Here the constant Cǫcan be made arbitrarily small due to Young’s weighted in- equality. The first two terms can be bounded by (4.35). Thus, (4.37 ) becomes n/summationdisplay j=2τtj/⌊a∇d⌊lθj/⌊a∇d⌊l2 L2+tn/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 a≤n/summationdisplay j=1τ(tj/⌊a∇d⌊lρj/⌊a∇d⌊l2 L2+/⌊a∇d⌊lρj/⌊a∇d⌊l2 L2)+CHn/summationdisplay j=2τtj(β(j)+β(1) +α(1))/⌊a∇d⌊lθj/⌊a∇d⌊lH1+CH2n/summationdisplay j=2τ(β(j) +β(1)+α(1))2. Using this in (4.36) we arrive at t2 n/⌊a∇d⌊lθn/⌊a∇d⌊l2 L2+n/summationdisplay j=2τt2 j/⌊a∇d⌊lθj/⌊a∇d⌊l2 H1+t2 n/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 H1≤Cn/summationdisplay j=2τ(t2 j/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊l2 L2+tj/⌊a∇d⌊lρj/⌊a∇d⌊l2 L2) +CH2n/summationdisplay j=2τ(t2 j(α(j)+α(1))2+(β(j)+β(1)+α(1))2) (4.38) +CHn/summationdisplay j=2τtj(β(j)+β(1)+α(1))/⌊a∇d⌊lθj/⌊a∇d⌊lH1+Cǫt2 n/⌊a∇d⌊l˜θn/⌊a∇d⌊l2 H1. Using Lemma 4.5 and Lemma 4.4 with f= 0 we deduce for the first two terms in (4.38) Cn/summationdisplay j=2τ(t2 j/⌊a∇d⌊l¯∂tρj/⌊a∇d⌊l2 L2+tj/⌊a∇d⌊lρj/⌊a∇d⌊l2 L2)≤CH2/parenleftBiggn/summationdisplay j=2τ/⌊a∇d⌊l¯∂2 tuj h/⌊a∇d⌊l2 L2+t2 2/⌊a∇d⌊l¯∂2 tu2 h/⌊a∇d⌊l2 L2/parenrightBigg , where we can use (2.9) for n= 2 and f= 0 to bound ¯∂2 tu2 h. We get t2 2/⌊a∇d⌊l¯∂2 tu2 h/⌊a∇d⌊l2 L2≤Cτ(/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 H1+/⌊a∇d⌊lu1 h/⌊a∇d⌊l2 H1). For the remaining terms in (4.38) we note that tj/⌊a∇d⌊lθj/⌊a∇d⌊lH1now may be kicked to left hand side using Cauchy–Schwarz and Young’s weighted inequality . The term involving Cǫcan also be moved to the left hand side. All terms involving α(j) and β(j) can be bounded by (2.8) and (4.32). This finishes the proof after u sing the regularity in Theorem 2.1 with f= 0. /square 4.3.Error bound for the ideal method. We get the final result by combining the two previous lemmas. Corollary 4.8. Letun handun lodbe the solutions to (2.6)and(3.4)-(3.5), respec- tively. The solutions can be split into un h=un h,1+un h,2andun lod=un lod,1+un lod,2, where the first part has vanishing initial data, and the secon d part a vanishing right hand side. The error is bounded by n/summationdisplay j=2τ/⌊a∇d⌊luj h,1−uj lod,1/⌊a∇d⌊l2 H1≤CH2/parenleftBiggn/summationdisplay j=1τ(/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2+/⌊a∇d⌊l¯∂tfj/⌊a∇d⌊l2 L2)+ max j=1,...,n/⌊a∇d⌊lfj/⌊a∇d⌊l2 L2/parenrightBigg , and n/summationdisplay j=2τt2 j/⌊a∇d⌊luj h,2−uj lod,2/⌊a∇d⌊l2 H1≤CH2(/⌊a∇d⌊l¯∂tu1 h/⌊a∇d⌊l2 H1+/⌊a∇d⌊lu1 h/⌊a∇d⌊l2 H1+/⌊a∇d⌊lu0 h/⌊a∇d⌊l2 H1), forn≥2.24 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON Proof.This is a direct consequence of Lemma 4.6 and Lemma 4.7 together with the fact that the problem is linear so the error can be split into two co ntributions satisfying the conditions of each lemma. /square Remark 4.9.The result from Corollary 4.8 is derived for the ideal method pre- sented in (3.4)-(3.5). The GFEM in (3.7)-(3.8) will yield yet another er ror from the localization procedure. However, due to the exponential decay in T heorem 3.2 and Theorem 3.3, it holds for the choice k≈ |log(H)|that the perturbation from the ideal method is of higher order and the derived result in Corollary 4.8 is still valid. For the details regarding the error from the localization procedure , we refer to [30]. 4.4.Initial data. For general initial data u0 h,u1 h∈Vhwe consider the projections Rmsu0 handRmsu1 h, whereRms=I−Rfis the Ritz-projection onto Vms. Letvbe the differencebetween twosolutionsto the dampedwaveequationw ith the different initial data. From (2.8) it follows that /⌊a∇d⌊lv/⌊a∇d⌊l2 H1≤C(/⌊a∇d⌊l¯∂t(u1 h−Rmsu1 h)/⌊a∇d⌊l2 L2+/⌊a∇d⌊lu1 h−Rmsu1 h/⌊a∇d⌊l2 b), where we have chosen to keep the b-norm. For the first term we may use the interpolant IHto achieve H. For the second term use /⌊a∇d⌊lu1 h−Rmsu1 h/⌊a∇d⌊l2 b≤β+ α−+τβ−(/⌊a∇d⌊lu1 h−Rmsu1 h/⌊a∇d⌊l2 a+τ/⌊a∇d⌊lu1 h−Rmsu1 h/⌊a∇d⌊l2 b), (4.39) If the initial data fulfills the following condition for some g∈L2(Ω) a(u1 h,v)+τb(u1 h,v) = (g,v),∀v∈Vh, (4.40) then we may deduce /⌊a∇d⌊lu1 h−Rmsu1 h/⌊a∇d⌊l2 a+τ/⌊a∇d⌊lu1 h−Rmsu1 h/⌊a∇d⌊l2 b= (g,u1 h−Rmsu1 h)≤CH/⌊a∇d⌊lg/⌊a∇d⌊lL2/⌊a∇d⌊lu1 h−Rmsu1 h/⌊a∇d⌊lH1. Hence, the error introduced by the projection of the initial data is of order H. The condition (4.40) appears when applying the LOD method to classic al wave equations, see [1], where it is referred to as “well prepared data” . We note in our case that if Bis small compared to A, that is if the damping is strong, then the constant in (4.39) is small. In some sense, this means that the condit ion in (4.40) is of “less importance”, which is consistent with the fact that strong damping reduces the impact of the initial data over time. 5.Reduced basis method The GFEM as it is currently stated requires us to solve the system in ( 3.7) for eachcoarsenodeineachtimestep, i.e. Nnumberoftimes. Wewillalterthemethod by applying a reduced basis method, such that it will suffice to find the solutions forM < N time steps, and compute the remaining in a significantly cheaper and efficient way. First of all, we note how the system (3.7) that ξn k,xsolves resembles a parabolic type equation with no source term. That is, the solution will decay ex ponentially until it is completely vanished. An example of how ξn k,xvanish with increasing n can be seen in Figure 2, where the coefficients are given as A(x) =/parenleftBig 2−sin/parenleftbig2πx εA/parenrightbig/parenrightBig−1 andB(x) =/parenleftBig 2−cos/parenleftbig2πx εB/parenrightbig/parenrightBig−1 , withεA= 2−4andεB= 2−6.A GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 25 In Figure2it isalsoseenhowthe solutionsdecaywith asimilarshapethr oughall time steps. This gives the idea that it is possible to only evaluate the so lutions for a few time steps, and utilize these solutions to find the remaining ones . This idea can be further investigated by storing the solutions {ξn k,x}N n=1and analyzing the corresponding singular values. The singular values are plotted and s een in Figure 3. It is seen how the values decrease rapidly, and that most of the v alues lie on machine precision level. In practice, this means that the information in{ξn k,x}N n=1 can be extracted from only a few ξn k,x’s. We use this property to decrease the computational complexity by means of a reduced basis method. We r emark that singular value decomposition is not used for the method itself, but is m erely used as a tool to analyze the possibility of applying reduced basis methods . 0.3 0 .4 0 .5 0 .6 0 .7−0.0010−0.00050.00000.00050.0010n= 1 n= 25 n= 50 n= 100 n= 150 n= 200 Figure 2. The behavior of the correction functions ξn k,xwith in- creasing n. The time step is τ= 0.01 andkis here chosen so that the support covers the entire interval. 0 20 40 60 80 100 n10−1710−1510−1310−1110−910−710−510−310−1 Figure 3. The singular values obtained when performing a sin- gular value decomposition of the matrix created by storing the finescale corrections {ξn k,x}N n=1withN= 100.26 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON The main idea behind reduced basis methods is to find an approximate s olu- tion in a low-dimensional space VRB M,k,x, which is created using a number of al- ready computed solutions. More precisely, to construct a basis fo r this space, one first computes Msolutions {ξm k,x}M m=1, whereM < N. By orthonormalizing these solutions using e.g. Gram–Schmidt orthonormalization, we yield a set o f vectors {ζm k,x}M m=1, called the reduced basis. Consequently, the reduced basis space be- comesVRB M,k,x= span({ζm k,x}M m=1) for each node x∈ N. With this space created, the procedure of finding {ξn k,x}N n=1is now reduced to finding {ξn k,x}M n=1, and then approximate the remaining solutions by {ξn,rb k,x}N n=M+1⊂VRB M,k,x. The matrix sys- tem to solve for a solution in VRB M,k,xis of dimension M×M, so when Mis chosen small, the last N−Msolutions are significantly cheaper to compute, which solves the issue of computing Nproblems on the finescale space. When constructing the reduced basis {ζm k,x}M m=1, it is important to be aware of the fact that the solution corrections {ξn k,x}N n=1all show very similar behavior. In practice, this implies that many of the ξn k,x’s are linearly dependent, hence causing floating point errors to become of significant size in the RB-space VRB M,k,x. To work around this issue, one may include a relative tolerance level that rem oves a vector from the basis if it is too close to being linearly dependent to one of the previously orthonormalized vectors. One may moreover use this tolerance lev el as a criterion for the amount of solutions, M, to pre-compute. That is, once the first vector is removedfromthe orthonormalizationprocess, then the RB-spac econtainssufficient information and no more solutions need to be added. In total, the novel method first requires that we solve NHnumber of systems on the localized fine scale in order to construct the multiscale space Vms,k. Moreover, we require to solve a localized fine system NHtimes for Mtime steps to create the RB-space VRB M,k,xfor each coarse node x∈ N. By utilizing the RB-space, the remaining N−Mfinescale corrections are then solved for in an M×Mmatrix system, and we yield the sought solution uN,rb lod,kby computing a matrix system on the coarse grid with the multiscale space Vms,k. 6.Numerical examples In this section we present numerical examples that illustrate the pe rformance of the established theory. For all examples, we consider the domain to be the unit square Ω = [0 ,1]×[0,1]. The coefficients A(x,y) andB(x,y) used in all examples aregeneratedrandomlywith valuesin the interval [10−1,103], and examples ofsuch areseen in Figure 4. Moreover,as initial value for eachexample we se tu0=u1= 0, and the source function is given by f= 1. The first example is used to show how the performance is effected by the lo- calization parameter k. Here, we evaluate the solution on the full grid, un lod, and compare it with the localized solution, un lod,k, askvaries. For the example the time step τ= 0.02 was used and final time was set to T= 1. The fine and coarse meshes were set to h= 2−7andH= 2−4respectively, and we let k= 2,3,...,7. The relative error between the functions can be seen in Figure 5. He re we can see how the error decays exponentially as kincreases, verifying the theoretical findings regarding the localization procedure. For the second example, the performance of the GFEM in (3.7)-(3.8 ) depending on the coarse mesh width His shown. For this example, the fine mesh width isA GENERALIZED FEM FOR THE STRONGLY DAMPED WAVE EQUATION 27 200400600800 (a)A(x,y). 200400600800 (b)B(x,y). Figure 4. The two different coefficients used for the numerical examples. The contrast is α+/α−=β+/β−= 104. 2 3 4 5 6 k10−510−410−310−210−1 Figure 5. Relative H1-error/⌊a∇d⌊lun lod−un lod,k/⌊a∇d⌊lH1//⌊a∇d⌊lun lod/⌊a∇d⌊lH1between the non-localized and localized method, plotted against the layer numberk. set toh= 2−8, and for each coarse mesh width the localization parameter is set to k= log2(1/H). Moreover, the time step is set to τ= 0.02 (for the GFEM as well as the reference solution) and the solution is evaluated at T= 1. To compute the error, we use a FEM solution on the fine mesh as a reference solution . The error as a function of 1 /Hcan be seen in Figure 6. Here it is seen how the error for the novel method decays faster than linearly, confirming the error es timates derived in Section 4. For comparison, Figure 6 also shows the error of the st andard FEM solution, as well as the solution using the standard LOD method with c orrection solely on AandBrespectively, i.e. corrections based on the bilinear forms a(·,·) andb(·,·) respectively and without finescale correctors. As expected, th e error of these methods stay at a constant level through all coarse grid siz es. At last, we compute the solution where the system (3.7) is computed using the reduced basis approach. For this example, we let the number of pre -computed solutions Mvary, and see how the error between the solutions un lod,kandun,rb lod,k28 PER LJUNG, AXEL M ˚ALQVIST, AND ANNA PERSSON 2122232425 1/H10−210−1 New method FEM LOD (A) LOD (B) O(H) Figure 6. Relative H1-error/⌊a∇d⌊lun ref−un lod,k/⌊a∇d⌊lH1//⌊a∇d⌊lun ref/⌊a∇d⌊lH1between the referencesolutionandthe approximatesolutioncomputed with the proposed method (without reduced basis computations). 5 10 15 20 M10−410−310−210−1100 Figure 7. Relative H1-error/⌊a∇d⌊lun lod,k−un,rb lod,k/⌊a∇d⌊lH1//⌊a∇d⌊lun lod,k/⌊a∇d⌊lH1be- tween the solution with and without the reduced basis approach, plotted against the number of pre-computed solutions. behaves. 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In Building bridges: connections and challenges in modern app roaches to numerical partial differential equations , volume 114 of Lect. Notes Comput. Sci. Eng. , pages 341–367. Springer, [Cham], 2016. [33] A. Quarteroni, A. Manzoni, and F. Negri. Reduced basis m ethods for partial differential equations. an introduction. 2016. [34] V. Thom´ ee. Galerkin Finite Element Methods for Parabolic Problems . Springer Series in Computational Mathematics. Springer-Verlag, Berlin, sec ond edition, 2006. [35] G. F. Webb. Existence and asymptotic behavior for a stro ngly damped nonlinear wave equa- tion.Canadian Journal of Mathematics , 32(3):631–643, 1980.

2020-11-06

We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced and is designed to handle independent variations in both the damping and the wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal order is proven in $L_2(H^1)$-norm, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.

A generalized finite element method for the strongly damped wave equation with rapidly varying data

2011.03311v1

1 Low magnetic damping of ferrimagnetic GdFeCo alloys Duck-Ho Kim1†*, Takaya Okuno1†, Se Kwon Kim2, Se-Hyeok Oh3, Tomoe Nishimura1, Yuushou Hirata1, Yasuhiro Futakawa4, Hiroki Yoshikawa4, Arata Tsukamoto4, Yaroslav Tserkovnyak2, Yoichi Shiota1, Takahiro Moriyama1, Kab-Jin Kim5, Kyung-Jin Lee3,6,7, and Teruo Ono1,8* 1Institute for Chemical Research, Kyoto University, Uji, Kyoto 6 11-0011, Japan 2Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 3Department of Nano-Semiconductor and Engineering, Korea Univers ity, Seoul 02841, Republic of Korea 4College of Science and Technology, Nihon University, Funabashi, Chiba 274-8501, Japan 5Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea 6Department of Materials Science & Engineering, Korea University , Seoul 02841, Republic of Korea 7KU-KIST Graduate School of Converging Science and Technology, K orea University, Seoul 02841, Republic of Korea 8Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan † These authors contributed equally to this work. * E-mail: kim.duckho.23z@st.kyoto-u.ac.jp , ono@scl.kyoto-u.ac.jp 2 We investigate the Gilbert damping parameter for rare earth (RE)– transition metal (TM) ferrimagnets over a wide temperature rang e. Extracted from the field-driven magnetic domain-wall mobility, was as low as 7.2 × 10-3 and was almost constant across the angular momentum compensation temperature 𝑻𝐀, starkly contrasting previous predictions that should diverge at 𝑻𝐀 due to vanishing total angular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to the total angular momentum but is dominated by electron scatter ing at the Fermi level where the TM has a dominant damping role. 3 Magnetic damping, commonly described by the Gilbert damping par ameter, represents the magnetization relaxation phenomenon, describing how quickly magnetization spins reach equilibrium [1–3]. Understanding the fundamental or igin of the damping as well as searching for low damping materials has been a central theme of magnetism research. Several theoretical models for magnetic damping have been propo sed [4–11] and compared with experiments [12–20]. Ultra-low damping was predicted in fe rromagnetic alloys using a linear response damping model [11] and was demonstrated experim entally for CoFe alloys [20]. However, the majority of these studies have focused only on ferromagnetic systems. Antiferromagnets, which have alt ernating orientations of their neighboring magnetic moments, have recently received considerable attention because of their potential importance for spintronic applications [21– 30]. Antiferromagnetic spin sys tems can have much faster spin dynamics than their ferromagnetic counterparts, which is a dvantageous in spintronic applications [21, 25, 31–39]. However, the manipulation and con trol of antiferromagnets is challenging because the net magnetic moment is effectively zero . Recently, antiferromagnetic spin dynamics have been successfully demonstrated using the mag netic domain-wall (DW) dynamics in ferrimagnets with finite magnetization in the vicin ity of the angular momentum compensation temperature, at which the net angular momentum van ishes [38]. This field- driven antiferromagnetic spin dyn amics is possible because the time evolution of the magnetization is governed by the commutation relation of the an gular momentum rather than the commutation relation of the magnetic moment. Motivated by the aforementioned result, in this letter, we inve stigate the magnetic damping of ferrimagnets across th e angular momentum compensatio n temperature, which will allow us to understand magnetic damping in antiferromagnet ically coupled system. We 4 selected rare earth (RE)–transition metal (TM) ferrimagnets for the material platforms because they have an angular momentum compensation temperature 𝑇 w h e r e antiferromagnetic spin dynamics are achieved [38, 40, 41]. The magnetic-field-driven DW motion was explored over a wide range of temperatures including 𝑇, and the Gilbert damping parameter was extracted from the measured DW mobility a t each temperature by employing the collective coordina te model initially developed f or ferrimagnetic spin dynamics [38]. Contrary to the previous prediction that the Gil bert damping parameter would diverge at 𝑇 due to the vanishing of the total angular momentum [42, 43], w e found that the Gilbert damping parameter remained nearly constant over a wide range of temperatures across 𝑇 with the estimated value as low as 7.2 × 10-3, which was similar to the reported values of TM-only ferromagnets [20]. These results suggested th at Gilbert damping was mainly governed by electron scattering at the Fermi level, and hence, the 4f electron of the R 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 magnetic damping of RE–TM ferrimagnets. For this study, we prepared perpendicularly magnetized ferrimag netic GdFeCo films in which the Gd and FeCo moments were coupled antiferromagnetic ally. Specifically, the films were 5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/100-nm SiN on an intrinsic Si substrate. The GdFeCo films were then patterned into 5-µm-wide and 500-µm-long microwires with a Hall cross structure using electron beam lithography and Ar ion mill ing. For current injection, 100-nm Au/5-nm Ti electrodes were stacked on the wire. A Hall b ar was designed to detect the DW velocity via the anomalous Hall effect (AHE). We measured the magnetic DW motion using a real-time DW detecti on technique [38, 40, 41, 44, 45] [see Fig. 1(a) for a schematic]. We first appli ed a magnetic field of –200 mT 5 to saturate the magnetization al ong the –z direction. Subsequen tly, a constant perpendicular magnetic field 𝜇𝐻, which was lower than the coercive field, was applied along +z direction. Next, a d.c. current was applied along the wire to measure the anomalous Hall voltage. Then, a current pulse (12 V , 100 ns) was injected through the writing line to nucleate the DW in the wire. The created DW was moved along the wire and passed throug h the Hall bar because of the presence of 𝜇𝐻. The DW arrival time was detected by monitoring the change in the Hall voltage using a real-time oscillo scope. The DW velocity could t hen be calculated from the arrival time and the travel dis tance between the writing line a nd Hall bar (500 µm). Figure 1(b) shows the averaged DW velocity 〈𝑣〉 as a function of the perpendicular magnetic field 𝜇𝐻 for several temperatures 𝑇∗. Here, we used the d.c. current density of |𝐽|ൌ1.3×1010 A / m2 to measure the AHE change due to DW motion. Note that 𝑇∗ i s a n elevated temperature that considers Joule heating by d.c. curre nt [46]. To eliminate the undesired current-induced spin-transfer-torque effect, we avera ged the DW velocity for 𝐽 and –𝐽, i.e., 〈𝑣〉ൌሾ𝑣ሺ𝐽ሻ𝑣ሺെ𝐽ሻሿ/2. Figure 1(b) shows that 〈𝑣〉 increases linearly with 𝜇𝐻 for all 𝑇∗. Such linear behavior can be described by 〈𝑣〉ൌ𝜇ሾ𝜇𝐻െ𝜇 𝐻ሿ, where 𝜇 is the DW mobility and 𝜇𝐻 is the correction field, which generally arises from imperfections in the sample or complexities of the internal DW structure [47, 48]. We note that 𝜇𝐻 can also depend on the temperature dependence of the magnetic properties of ferrimagnets [45]. Figure 1(c) shows 𝜇 as a function of 𝑇∗ at several current densities (|𝐽|ൌ1.3, 1.7, and 2.0 ×1010 A / m2). A sharp peak clearly occurs for 𝜇 a t 𝑇∗ൌ241.5 K irrespective of |𝐽|. The drastic increase of 𝜇 is evidence of antiferromagnetic spin dynamics at 𝑇, as demonstrated in our pre vious report [38, 40, 41]. The obtained DW mobility was theoretically analyzed as follows. The DW velocity 6 of ferrimagnets in the precessional regime is given by [38, 39] 𝑉 ൌ 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ ሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶ𝜇𝐻, ሺ1ሻ where 𝑉 is the DW velocity, 𝜆 is the DW width, 𝜇𝐻 is the perpendicular magnetic field, 𝛼 is the Gilbert damping parameter, 𝑀 and 𝑠 are the magnetization and the spin angular momentum of one sublattice, respectively. The spin angular mome ntum densities are given by 𝑠ൌ𝑀 /𝛾 [49], where 𝛾ൌ𝑔 𝜇/ℏ is the gyromagnetic ratio of lattice 𝑖, 𝑔 i s t h e Landé g factor of lattice 𝑖, 𝜇 is the Bohr magneton, and ℏ is the reduced Plank’s constant. The 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 assume that it is the same, which can be considered as the aver age value of the damping parameters for the two sublattices weighted by the spin angular momentum density. We note that this assumption does not alter our main conclusion: low da mping and its insensitivity to the temperature. Equation (1) gives the DW mobility 𝜇 a s 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ/ ሼሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶሽ, which can be rearranged as 𝜇 ሺ𝑠ଵ𝑠 ଶሻଶ𝛼ଶെ𝜆ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ𝛼𝜇 ሺ𝑠ଵെ𝑠 ଶሻଶൌ 0 ሺ2ሻ Using Eq. (2) to find the solution of 𝛼, we find 𝛼 േൌ𝜆ሺ𝑀ଵെ𝑀 ଶሻേඥሾ𝜆ଶሺ𝑀ଵെ𝑀 ଶሻଶെ4 𝜇ଶሺ𝑠ଵെ𝑠 ଶሻଶሿ 2𝜇ሺ𝑠ଵ𝑠 ଶሻ. ሺ3ሻ Equation (3) allows us to estimate 𝛼 for the given 𝜇. We note that for each value of 𝜇, 𝛼 ca 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 these two solutions is physically sound, which can be obtained using the following energy dissipation analysis. 7 The energy dissipation (per unit cross section) through the DW dynamics is given by 𝑃ൌ2 𝛼 ሺ 𝑠 ଵ𝑠 ଶሻ𝑉ଶ/𝜆 2𝛼ሺ𝑠 ଵ𝑠 ଶሻ 𝜆Ωଶ [38, 39], where Ω is the angular velocity of the DW. The first and the second terms represent the energy dissipa tion through the translational and angular motion of the DW, respectively. In the precessional regime, the angular velocity is proportional to the translational velocity: Ωൌ ሺ𝑠ଵെ𝑠 ଶሻ𝑉/𝛼ሺ𝑠 ଵ𝑠 ଶሻ𝜆. Replacing Ω b y the previous expression yields 𝑃ൌ𝜂 𝑉ଶ w h e r e 𝜂ൌ2 ሺ 𝑀 ଵെ𝑀 ଶሻ/𝜇 is the viscous coefficient for the DW motion: 𝜂 ൌ2 𝜆ቊ𝛼ሺ𝑠ଵ𝑠 ଶሻ ሺ𝑠ଵെ𝑠 ଶሻଶ 𝛼ሺ𝑠ଵ𝑠 ଶሻቋ . ሺ4ሻ The first and the second terms in parenthesis capture the contr ibutions to the energy dissipation from the translational and angular dynamics of the DW, respectively. The two solutions for the Gilbert damping parameter, 𝛼ା and 𝛼ି, can yield the same viscous coefficient 𝜂. The case of the equal solutions, 𝛼ାൌ𝛼 ି, corresponds to the situation when the two contributions are identical: 𝛼േൌሺ 𝑠 ଵെ𝑠 ଶሻ/ሺ𝑠ଵ𝑠 ଶሻ. For the larger solution 𝛼ൌ 𝛼ା, the energy dissipation is dominated by the first term, i.e., through the translational DW motion, which should be the case in the vicinity of 𝑇 where the net spin density ሺ𝑠ଵെ𝑠 ଶሻ is small and thus the angular velocity is negligible. For examp le, at exact 𝑇, the larger solution 𝛼ା is the only possible solution because the smaller solution is zero, 𝛼ିൌ0, and thus unphysical. For the smaller solution 𝛼ൌ𝛼 ି, the dissipation is dominated by the second term, i.e., through the precessional motion, which should descr ibe cases away from 𝑇. Therefore, in the subsequent analysis, we chose the larger solu tion 𝛼ା in the vicinity of 𝑇 and the smaller solution 𝛼ି far away from 𝑇 and connected the solution continuously in between. 8 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 measuring the net magnetic moment of GdFeCo film, |𝑀୬ୣ୲|, for various temperatures. Because 𝑀୬ୣ୲ includes contributions from both the Gd and FeCo sub-moments, the sub- magnetic moments, 𝑀ଵ a n d 𝑀ଶ, could be decoupled based on the power law criticality [see details in refs. 38, 40]. The spin angular momentums, 𝑠ଵ and 𝑠ଶ, were calculated using the known Landé g factor of FeCo and Gd (the Landé g factor of FeCo is 2.2 and that of Gd is 2.0) [50–52]. Figures 2(a)–(c) show the temperature-dependent DW mobility 𝜇, sub-magnetic moment 𝑀, and sub-angular momentum 𝑠, respectively. Here, we used the relative temperature defined as ∆𝑇 ൌ 𝑇∗െ𝑇 to investigate the Gilbert damping near 𝑇. The Gilbert damping parameter 𝛼 was obtained based on Eq. (3) and the information in Fig. 2(a)–(c). Figure 2(d) shows the resulting values of 𝛼േ as a function of ∆𝑇. For ∆𝑇ଵ൏ ∆𝑇 ൏ ∆𝑇 ଶ, 𝛼ା is nearly constant, while 𝛼ି varies significantly. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 ∆𝑇ଶ, on the other hand, 𝛼ି is almost constant, while 𝛼ା varies significantly. At ∆𝑇 ൌ ∆𝑇 ଵ and ∆𝑇 ൌ ∆𝑇 ଶ, the two solutions are equal, corresponding to the aforementio ned case when the energy dissipation through the translational and angular mo tion of the DW are identical. The proper damping solution can be selected by following the gu ideline obtained from the above analysis. For ∆𝑇ଵ൏∆ 𝑇൏∆ 𝑇 ଶ, which includes 𝑇, the energy dissipation should be dominated by the translational motion, and thus 𝛼ା is a physical solution. Note also that 𝛼ି becomes zero at 𝑇, which results in infinite DW mobility in contradiction with the experimental observation. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 ∆𝑇 ଶ, where the energy dissipation is dominated by the angular motion of the DW, 𝛼ି is the physical solution. 9 Figure 3 shows the resultant Gil bert damping parameter in all t ested temperature ranges. The Gilbert damping parameter was almost constant acros s 𝑇 with 𝛼ൌ7.2 × 10-3 (see the dotted line in Fig. 3). This result is in stark contra st to the previous prediction. In ref. [42], Stanciu et al. investigated the temperature dependence of the effective Gilb ert damping parameter based on a ferromagnet-based model and found that the damping diverged at 𝑇. Because they analyzed the magnetic resonance in ferrimagnetic m aterials based on a ferromagnet-based model, which cannot describe the antiferromag netic dynamics at 𝑇 a t which the angular momentum vanis hes, it exhibits unphysical res ults. However, our theoretical analysis for field-driven ferromagnetic DW motion b ased on the collective coordinate approach can properly describe both the antiferromag netic dynamics in the vicinity of 𝑇 and the ferromagnetic dynamics away from 𝑇 [38]. Therefore, the unphysical divergence of the Gilbert damping parameter at 𝑇 is absent in our analysis. Our results, namely the insensitivity of damping to the compens ation condition and its low value, have important implications not only for fundame ntal physics but also for technological applications. From the viewpoint of fundamental p hysics, nearly constant damping across 𝑇 indicates that the damping is almost independent of the total angular momentum and is mostly determined by electron spin scattering n ear the Fermi level. Specifically, our results suggest that the 4f electrons of RE e lements, which lie in a band far below the Fermi level, do not play an important role in the mag netic damping of RE-TM ferrimagnets, whereas the 3d and 4s bands of TM elements have a governing role in magnetic damping. This result is consistent with the recently reported t heoretical and experimental results in FeCo alloys [20]. From the viewpoint of practical ap plication, we note that the estimated damping of 𝛼ൌ7.2 × 10-3 is the upper limit, as the damping estimated from DW 10 dynamics is usually overestimated due to disorders [53]. The ob tained value of the Gilbert damping parameter is consistent with our preliminary ferromagne t i c r e s o n a n c e ( F M R ) measurements. The experimental results from FMR measurements an d the corresponding theoretical analysis will be publ ished elsewhere. This low valu e of the Gilbert damping parameter 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 high-speed magnetic devices such as spin-transfer-torque magnet ic random-access memory and terahertz magnetic oscillators. In conclusion, we investigated the field-driven magnetic DW mot ion in ferrimagnetic G 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 damping parameter from the DW mobility. The estimated Gilbert d amping parameter was as low as 7.2 × 10-3 and almost constant over the temperature range including 𝑇, which is in stark contrast to the previous prediction in that the Gilbert d amping parameter would diverge at 𝑇 due to the vanishing total angular momentum. 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Their directions are separate ly handled through the signs in the equations of motion. [50] C. Kittel, Phys. Rev. 76, 743 (1949). [51] G. G. Scott, Rev. Mod. Phys. 34, 102 (1962). [52] B. I. Min and Y.-R. Jang, J. Phys. Condens. Matter 3, 5131 (1991). [53] H. Min, R. D. McMichael, M. J. Donahue, J. Miltat, and M. D. Stiles, Phys. Rev. Lett. 104, 217201 (2010). 16 Figure Captions Figure 1(a) Schematic illustration of the GdFeCo microwire devi ce. (b) The averaged DW velocity 〈𝑣〉 as a function of the perpendicular magnetic field 𝜇𝐻 for several temperatures 𝑇∗ (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 mobility 𝜇 as a function of 𝑇∗ at several current densities ( |𝐽|ൌ1.3, 1.7, and 2.0 ×1010 A/m2). Figure 2 The temperature-dependent (a) DW mobility 𝜇, (b) sub-magnetic moment 𝑀, and (c) sub-angular momentum 𝑠. Here, we use the relative temperature defined as ∆𝑇 ൌ 𝑇∗െ 𝑇. (d) The Gilbert damping parameter 𝛼േ as a function of ∆𝑇. Here, we use 𝜆ൌ15 nm for proper solutions of Eq. (3). Figure 3 The resultant Gil bert damping parameter 𝛼 in all tested temperature ranges. 17 Acknowledgements This work was supported by the JSPS KAKENHI (Grant Numbers 15H0 5702, 26103002, and 26103004), Collaborative Research Program of the Institute for Chemical Research, Kyoto University, and R & D project for ICT Key Technology of MEXT fr om the Japan Society for the Promotion of Science (JSPS). This work was partly supported by The Cooperative Research Project Program of the Research Institute of Electrica l Communication, Tohoku University. D.H.K. was supported as an Overseas Researcher unde r the Postdoctoral Fellowship of JSPS (Grant Number P16314). S.H.O. and K.J.L. wer e supported by the National Research Foundation of Korea (NRF-2015M3D1A1070465, 20 17R1A2B2006119) and the KIST Institutional Program (Project No. 2V05750). S.K.K . was supported by the Army Research Office under Contract No. W911NF-14-1-0016. K.J.K . was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIP) (No. 2017R1C1B2009686). Competing financial interests The authors declare no competing financial interests. 200 225 250 275 3000.00.51.01.52.0 1.3 1.7 2.0 [104 m/sT] T* [K]J [1010 A/m2]0 50 100 1500.00.51.01.5 202 222 242 262 282<v> [km/s] 0H [mT]T* [K] Figure 1b ca Writing line ܫ ܸ ߤܪ y xz-60 -40 -20 0 20 40 60 801.52.02.53.0 s1 s2s [10-6 Js/m3] T [K]-60 -40 -20 0 20 40 60 800.00.51.01.52.0 [104 m/sT] T [K] -60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100 + - T [K]T1T2-60 -40 -20 0 20 40 60 800.30.40.50.6 M1 M2M [MA/m] T [K]a b c d Figure 2Figure 3-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100 T [K]

2018-06-13

We investigate the Gilbert damping parameter for rare earth (RE)-transition metal (TM) ferrimagnets over a wide temperature range. Extracted from the field-driven magnetic domain-wall mobility, the Gilbert damping parameter was as low as 0.0072 and was almost constant across the angular momentum compensation temperature, starkly contrasting previous predictions that the Gilbert damping parameter should diverge at the angular momentum compensation temperature due to vanishing total angular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to the total angular momentum but is dominated by electron scattering at the Fermi level where the TM has a dominant damping role.

Low magnetic damping of ferrimagnetic GdFeCo alloys

1806.04881v1

arXiv:1412.3783v1 [cond-mat.mtrl-sci] 11 Dec 2014Deviation From the Landau-Lifshitz-Gilbert equation in th e Inertial regime of the Magnetization E. Olive and Y. Lansac GREMAN, UMR 7347, Universit´ e Fran¸ cois Rabelais-CNRS, Pa rc de Grandmont, 37200 Tours, France M. Meyer, M. Hayoun, and J.-E. Wegrowe Laboratoire des Solides Irradi´ es, ´Ecole Polytechnique, CEA-DSM, CNRS, F-91128 Palaiseau, Fr ance (Dated: February 7, 2018) We investigate in details the inertial dynamics of a uniform magnetization in the ferromagnetic resonance (FMR) context. Analytical predictions and numer ical simulations of the complete equa- tions within the Inertial Landau-Lifshitz-Gilbert (ILLG) model are presented. In addition to the usual precession resonance, the inertial model gives a seco nd resonance peak associated to the nuta- tion dynamics provided that the damping is not too large. The analytical resolution of the equations of motion yields both the precession and nutation angular fr equencies. They are function of the in- ertial dynamics characteristic time τ, the dimensionless damping αand the static magnetic field H. A scaling function with respect to ατγHis found for the nutation angular frequency, also valid for the precession angular frequency when ατγH≫1. Beyond the direct measurement of the nutation resonance peak, we show that the inertial dynamics of the mag netization has measurable effects on both the width and the angular frequency of the precession re sonance peak when varying the applied static field. These predictions could be used to experimenta lly identify the inertial dynamics of the magnetization proposed in the ILLG model. PACS numbers: I. INTRODUCTION The Landau-Lifshitz-Gilbert (LLG) equation is a ki- netic equation that does not contain acceleration terms, i.e. that does not contain inertia. The corresponding trajectory is reduced to a damped precession around the axis defined by the effective field. The measurement of this precession is usually performed by the mean of ferromagnetic resonance (FMR). The power absorbed by the system is then measured at steady state while adding an oscillatory field to the effective field, and tuning the frequency close to the resonance frequency. However, the validity of the LLG equation is limited to large time scales1, or low frequency regimes (similarly to the Debye model of electric dipoles2). Indeed, the precession with damping described by the LLG equation is a diffusion process in a field of force, for which the angular momentum has reached equilibrium. Accordingly, if the measurements are performed at fast enough time scales, or high enough frequencies, inertial terms should be expected to play a role in the dynamics, which is no longer reduced to a damped precession3–9. A nutation dynamics is therefore expected, giving a second resonant peak at the nutation frequency, and this new absorption should be measurable with dedicated spectroscopy (e.g. using infrared spectroscopy). Despite its fundamental importance, a systematic experimental investigation of possible inertial effects of the uniform magnetization has however been overlooked. In order to evidence experimentally the consequences of inertia in the dynamics of a uniform magnetization, it is first necessary to establish the characteristics that would allow to discriminate inertia from spuriouseffects in spectroscopy experiments. We propose in this paper some simple theoretical and numerical tools than can be used by experimentalists in order to evidence unambiguouslytheeffectsofinertiaofthemagnetization. The LLG equation reads : dM dt=γM×/bracketleftbigg Heff−ηdM dt/bracketrightbigg (1) whereMisthemagnetization, Hefftheeffectivemagnetic field,ηthe Gilbert damping, and γthe gyromagnetic ra- tio. If the description is extended to the fast degrees of freedom (i.e. the degrees of freedom that includes the time derivative of the angularmomentum), a supplemen- tary inertial term should be added with the correspond- ingrelaxationtime τ. FromthisInertialLandau-Lifshitz- Gilbert (ILLG) model, the new equation reads3–7: dM dt=γM×/bracketleftbigg Heff−η/parenleftbiggdM dt+τd2M dt2/parenrightbigg/bracketrightbigg (2) One of the main consequences of the new dynami- cal equation is the emergence of the second resonance peak associated to the nutation at high frequencies, as reported in our previous study7. In the literature the nutation dynamics of magnetic moments has been in- vestigated using various theoretical approaches though not yet evidenced experimentally. B¨ ottcher and Henk studied the significance of nutation in magnetization dy- namics of nanostructures such as a chain of Fe atoms, and Co islands on Cu(111)8. They found that the nu- tation is significant on the femtosecond time scale with a typical damping constant of 0.01 up to 0.1. Moreover,2 they concluded that nutation shows up preferably in low- dimensional systems but with a small amplitude with respect to the precession. Zhu et al.predicted a nuta- tion dynamics for a single spin embedded in the tunnel- ing barrierbetween twosuperconductors10. This unusual spin dynamics is caused by coupling to a Josephson cur- rent. They argue that this prediction might be directly tested for macroscopic spin clusters. The nutation is also involved in the dynamics of a single spin embedded in the tunnel junction between ferromagnets in the pres- ence of an alternating current11. In an atomistic frame- work, Bhattacharjee et al.showed that first-principle techniques used to calculate the Gilbert damping factor may be extended to calculate the moment of inertia ten- sor associated to the nutation9. Our previous work7was focussed on the short time nutation dynamics generated by the ILLG equation, and was limited to fixed values of the inertial characteristic time scale τ, the dimensionless damping αand the static fieldH. In this paper we present a combined analytical and numerical simulation study of the ILLG equation with new results. In particular we derive analytical re- sults in the small inclination limit that can be used in ferromagnetic resonance (FMR) experiments, and which allow to predict both the precession and nutation reso- nance angular frequencies. We also investigate the ILLG equation while varying the three parameters α,τandH, and scaling functions are found. Finally, we present im- portant indications for experimental investigations of the inertial dynamics of the magnetization. Indeed, a conse- quence of the ILLG equation is the displacement of the well-known FMR peak combined with a modified shape with respect to that given by the LLG equation. This displacement could not be without consequences on the determination of the gyromagnetic factor γby ferromag- netic resonance. The paper is organized as follows. In section II we show analytical solutions of the precession and nutation dynamics for the uniform magnetization in a static ap- plied field H. The small inclination limit is investigated in order to reproduce the usual experimental FMR con- text. In section III we describe the numerical simula- tions of the magnetization inertial dynamics in both a static anda smallperpendicular sinusoidalmagneticfield (Heff=H+h⊥(ω)). The resonance curves are computed and, provided that the damping is not too large, a nu- tation resonance peak appears in addition to the usual ferromagnetic resonance peak associated to the magne- tization precession. In section IV the behavior of the ILLG equation is investigated in details while varying the characteristic time τof the inertial dynamics, the di- mensionless damping αand the static field H. A very good agreement is found between the analytical and nu- merical simulation results, and a scaling function with respect to ατγHis found. In section V we propose ex- periments in the FMR context that should evidence the inertial dynamics of the magnetization described in the ILLGmodel. Inparticular,whenthestaticfieldisvaried,the ILLG precession resonance peak has different behav- iors compared to the usual LLG precession peak with shifted resonance angular frequency and modified shape. We show that the differences between LLG and ILLG precession peaks are more pronounced in large damping materials and increase with the static field. Finally, we derive the conclusions in section VI. II. ANALYTICAL SOLUTIONS FOR THE ILLG EQUATION The magnetization position is described in spherical coordinates ( Ms,θ,φ), where Msis the radius coordinate fixed at a constant value for the uniformly magnetized body,θis the inclination and φis the azimuthal angle. In a static magnetic field Hˆ zapplied in the zdirection, i.e.H=H(cosθer−sinθeθ) in the spherical basis (er,eθ,eφ), Eq. (2) gives the following system : ¨θ=−1 τ˙θ−1 τ1˙φsinθ+˙φ2sinθcosθ −ω2 τ1sinθ (3a) ¨φsinθ=1 τ1˙θ−1 τ˙φsinθ−2˙φ˙θcosθ (3b) where the characteristic times are τandτ1=ατ,ω2= γHis the Larmor angular frequency, and α=γηMsis the dimensionless damping. Usingthedimensionlesstime t′=t/τ, Eqs. (3)become θ′′=−θ′−/tildewideτ1φ′sinθ+φ′2sinθcosθ −/tildewideω2/tildewideτ1sinθ (4a) φ′′sinθ=/tildewideτ1θ′−φ′sinθ−2φ′θ′cosθ(4b) where θ′=dθ/dt′, θ′′=d2θ/dt′2, φ′=dφ/dt′, φ′′=d2φ/dt′2, and /tildewideτ1=τ τ1=1 α /tildewideω2=ω2τ=τγH In the following subsections we extract analytical re- sults that can be used to predict the positions in the angular frequency domain of the precession and nutation resonance peaks. We will consider the small inclination limit which holds in the FMR context.3 A. Precession : exact and approximate solutions To determine the precession dynamics of the iner- tial model we search for the long time scale solution φ′(t′) =φ′ prec, whereφ′ precis the constant precession ve- locity. Since the damping progressivelyshifts the magne- tization to the zaxis, we investigate the small inclination limit where φ′(t′) =φ′ precshould hold. With sin θ∼θ and cosθ∼1, Eqs. (4) therefore reads : θ′′+θ′+/tildewideω2 0θ= 0 (5a) φ′ prec=/tildewideτ1θ′ θ+2θ′(5b) where the natural angular frequency of the overdamped harmonic oscillator θ(t′) defined by Eq. (5a) is given by /tildewideω0=/radicalBig /tildewideτ1(φ′prec+/tildewideω2)−φ′2prec (6) The characteristic equation associated to the differential equation Eq. (5a) is β2+β+/tildewideω2 0= 0 which gives in the aperiodic regime the two solutions β±=−1±/radicalbig 1−4/tildewideω2 0 2(7) Since|β+|<|β−|, the inclination of the magnetization behaves at long time scales as θ(t′)∼eβ+t′, which inserted in Eq. (5b) gives φ′ prec=/tildewideτ1β+ 1+2β+(8) In original time units, the precession velocity ˙φprecis therefore the solution of ˙φprec=β+(˙φprec) ατ/parenleftBig 1+2β+(˙φprec)/parenrightBig (9) where the function β+(˙φprec) is given by β+(˙φprec) =−1+/radicalbigg 1−4τ/parenleftBig˙φprec+γH α−τ˙φ2prec/parenrightBig 2(10) Equation 9 may be numerically solved to extract the precession velocity, and therefore the precession reso- nance peak when a sinusoidal magnetic field h⊥(ω) is superimposed perpendicular to the static field Hˆ z. Forτ≪10−11sandα≤0.1, theprecessionvelocity ˙φprec for small applied static fields may be accurately evalu- ated from a quadratic equation : in this case /tildewideω2 0≪1 and Eq. (7) leads to β+≈ −/tildewideω2 0. Eq. (8) therefore gives a cubic equation in φ′ precwhere the cubic term −2αφ′3 precis negligeable. In this case the solution of the resulting quadratic equation is in original time units ˙φprec=−b−/radicalbig b2+12τγH/α 6τ(11) withb= 2τγH−α−1/α. We choose the negative so- lution of the quadratic equation in order to agree with the negative velocity ˙φLLG=−γH/(1+α2) given by the LLG model. B. Nutation : angular frequency Unlike the precession, the nutation properties should bederivedconsideringintermediatetime scaleswherethe precession has not yet reached a constant velocity. Eqs. (4) should therefore be reconsidered. To derive the nuta- tion properties, it is convenient to examine the angular velocityθ′. For simplicity we note θ′=/tildewideωθandφ′=/tildewideωφ. Eqs. (4) therefore rewrite /tildewideω′ θ=−/tildewideωθ−/tildewideτ1/tildewideωφsinθ+/tildewideω2 φsinθcosθ −/tildewideω2/tildewideτ1sinθ (12a) /tildewideω′ φsinθ=/tildewideτ1/tildewideωθ−/tildewideωφsinθ−2/tildewideωφ/tildewideωθcosθ(12b) We derive Eq. (12a) with respect to time t′which gives /tildewideω′′ θ=−/tildewideω′ θ+(2/tildewideωφcosθ−/tildewideτ1)/tildewideω′ φsinθ−/tildewideτ1/tildewideωφ/tildewideωθcosθ +/tildewideω2 φ/tildewideωθ(cos2θ−sin2θ)−/tildewideω2/tildewideτ1/tildewideωθcosθ where the term /tildewideω′ φsinθmay be replaced with the expres- sion in Eq. (12b). We therefore obtain /tildewideω′′ θ+/tildewideω′ θ+/parenleftbig /tildewideτ2 1+/tildewideω2/tildewideτ1cosθ/parenrightbig /tildewideωθ= /tildewideτ1/tildewideωφsinθ+3/tildewideτ1/tildewideωφ/tildewideωθcosθ−2/tildewideω2 φcosθsinθ −(3cos2θ+sin2θ)/tildewideω2 φ/tildewideωθ (13) Eq. (13)shouldbecloselyrelatedtothenutationdynam- ics since it describes the /tildewideωθoscillator. This assumption will be confirmed in section IVA2 for a broad range of parameters. Eq. (13) defines the damped oscillator /tildewideωθ which is non-linearly coupled to the /tildewideωφoscillator. This expression shows that, in the absence of coupling and in the smallinclination limit θ≪1rad, the/tildewideωθoscillatoros- cillatesatthenaturalangularfrequency/radicalbig /tildewideτ2 1+/tildewideω2/tildewideτ1. We therefore deduce an approximate expression for the nu- tation angularfrequency in the weak coupling case which is given by the expression /tildewideωweak nu=/radicalBig /tildewideτ2 1+/tildewideω2/tildewideτ1 (14) which in original time units gives ωweak nu=√1+ατγH ατ(15)4 From Eq. (15) we deduce the following asymptotic behaviors : when τ≪1/αγHthenωweak nu∼1/ατ, and whenτ≫1/αγHthenωweak nu∼1/√ατ. Because of the non-linear coupling terms in the right- hand side of Eq. (13), the true position of the nuta- tion resonancepeak in FMR experiments may differ from the approximate angular frequency defined by Eq. (15). Howeverthe simulation of the resonancecurves with a si- nusoidal magnetic field h⊥(ω) superimposed perpendic- ular to the static field Hˆ zwill show in section IVA2 that the non-linear coupling terms only slightly shift the nutation resonance peak from the approximate angular frequency. III. NUMERICAL SIMULATIONS OF THE RESONANCE CURVES IN THE ILLG MODEL We apply a fixed magnetic field H=Hˆ zalong the zdirection, and a small sinusoidal magnetic field h⊥= h⊥cosωtˆ xin thexdirection. In the spherical basis the components of the total magnetic field Heff=H+h⊥in Eq. (2) are Heff r=Hcosθ+h⊥sinθcosφcosωt Heff θ=−Hsinθ+h⊥cosθcosφcosωt Heff φ=−h⊥sinφcosωt. which lead to the following dynamical equations for the spherical angles ( θ,φ) of the magnetization ¨θ=−1 τ˙θ−1 τ1˙φsinθ+˙φ2sinθcosθ −ω2 τ1sinθ+ω3 τ1cosθcosφcosωt(16a) ¨φsinθ=1 τ1˙θ−1 τ˙φsinθ−2˙φ˙θcosθ −ω3 τ1sinφcosωt (16b) whereω3=γh⊥is the angular frequency associated to the sinusoidal field. Using the dimensionless time t′=t/τ, Eqs. (16) be- come θ′′=−θ′−/tildewideτ1φ′sinθ+φ′2sinθcosθ −/tildewideω2/tildewideτ1sinθ+/tildewideω3/tildewideτ1cosθcosφcos/tildewideωt′(17a) φ′′sinθ=/tildewideτ1θ′−φ′sinθ−2φ′θ′cosθ −/tildewideω3/tildewideτ1sinφcos/tildewideωt′(17b) where θ′=dθ/dt′, θ′′=d2θ/dt′2, φ′=dφ/dt′, φ′′=d2φ/dt′2,and /tildewideτ1=τ τ1=1 α /tildewideω2=ω2τ=τγH /tildewideω3=ω3τ=τγh⊥ /tildewideω=ωτ We useγ= 1011rad.s−1.T−1, and we vary the charac- teristic time τfor three different values of the dimension- less damping α= 0.1, 0.01 and 0 .5. We investigate sev- eralvalues of the static magnetic field from H= 0.2Tup toH= 200T. We numerically integrate Eqs. (17) using either a double precision second order Runge-Kutta algo- rithm or a double precision five order Gear algorithm12. Typically, we use time steps 10−7< δt′<10−3depend- ing on the values of τandω. The resonance curves are obtained by investigating the magnetization response to the small oscillating field h⊥(ω) =h⊥cosωtˆ xapplied perpendicular to the static fieldH=Hˆ z. We analyse the permanent dynami- cal regime where the magnetization components oscil- late around well defined mean values. For fixed values of the oscillating field angular frequency ωand oscil- lating field amplitude h⊥, we compute the mean value < M⊥>(averaged over time) of the transverse magneti- zationM⊥(t) =/radicalBig M2x(t)+M2y(t), fromwhichweextract for fixed values of ωthe transverse susceptibility defined byχ⊥=d < M ⊥> /dh ⊥. We choose values of the oscillating field amplitude h⊥= 10−1,10−2,10−3,10−4 and 10−5T, and we plot < M⊥>with respect to h⊥ for each ω. As an example, we show the case α= 0.1, τ= 2×10−10s,H= 2Tandω= 1.2×1011rad.s−1. The inset of Fig. 1 shows that the response is linear < M⊥>=χ⊥h⊥wherefrom we extract the transverse susceptibility χ⊥using a linear fitting. We repeat the same procedure for each oscillating field angular fre- quencyωwhich gives the resonance curve χ⊥(ω) of the transverse susceptibility shown in Fig. 1. Two peaks clearly appear, the usual FMR peak associated to the precession velocity, and the nutation peak associated to the nutation dynamics originatingfrom the inertial term. IV. RESULTS A. Effects of τ We now examine the ILLG model when varying the characteristic time τ. For different values of the parame- terτ, we show in Fig. 2 the typical profiles of the trans- verse susceptibility χ⊥versus the angular frequency ωof the applied oscillating field. The four resonance curves plotted in figure 2 are obtained by numerical simulations withH= 2Tandα= 0.1. They show how the nu- tation resonance peak position depend on the value of τ. Asτis increased, the nutation peak moves towards5 01×10112×10113×1011 ω (rad.s-1)1234567χ⊥10-510-410-310-2 h⊥10-510-410-310-210-1< Μ⊥>Precession peak Nutation peak Figure 1: Resonance curves of the transverse susceptibil- ityχ⊥(ω) with respect to the oscillating field angular fre- quencyω. The resonance curves are computed within the ILLG model with τ= 2×10−10s, for dimensionless damping α= 0.1 and for an applied static field H= 2T. Two reso- nance peaks are observed : the precession resonance at lower angular frequency which is the usual FMR and the nutation resonance at higher angular frequency. Inset : Example of the calculation of χ⊥such that < M⊥>=χ⊥h⊥obtained for ω= 1.2×1011rad.s−1. the precession peak with an increasing intensity which is an order of magnitude smaller than the precession one forτ= 10−11s. Note that the transverse susceptibil- ity at the resonance follows a power law of the form χ⊥(ωILLG nu)∝1/ωILLG nu, whereωILLG nuis defined as the nu- tation resonance angular frequency. A similar power law is reported for the precession peak obtained for different static fields H(see section IVB). We now compare the analytical and numerical sim- ulation results concerning the positions in the angular frequency domain of both the precession and nutation resonance peaks. 1. Precession peak We define ωprec=|˙φprec|as the angular frequency of the precession. When computed from the exact expres- sions (9) and (10) we will refer to ωexact prec, and when com- puted from the approximate expression (11) we will refer toωapprox prec. Finally, we will denote by ωILLG precthe angular frequency of the precession resonance peak obtained in the numerical simulations of the ILLG model. Eq. (9) maybeeasilynumericallysolvedtofindthesolution ˙φprec for several values of αandτ. The behavior with respect toτofωprecobtained either analytically or from the sim- ulated FMR curves is shown in Fig. 3. There is an excel-10 -4 10 -3 10 -2 10 -1 10 0 10 11 10 12 10 13 10 14 10 15 χ ⊥ ω ( rad.s -1 )τ=1×10 -113UHFHVVLRQ 1utation τ=1×10 -12 τ=1×10 -13 τ=1×10 -14 Figure 2: Resonance curves of the transverse susceptibilit y showing the displacement of the nutation peak caused by the variation of τ:τ= 10−11s (open circles), 10−12s (filled circles), 10−13s (crosses), and 10−14s (open squares). These curves are simulated using the ILLG model with α= 0.1, andH= 2T. Note that the precession peak positions are only slightly affected. The dotted line shows the power law fitted on χ⊥∝1/ωILLG nu, whereωILLG nuis the resonance angular frequency of the nutation. lent agreement between the analytical prediction ωexact prec and the precession resonance peak ωILLG precobtained in nu- mericalsimulations. WealsoshowinFig.3theprecession angular frequency ωapprox prec. Forτ <10−11sandα= 0.1, it nicely agrees with the exact analytical value and with thenumericalsimulationresults, but the approximateso- lution becomes no longer valid for τ >10−11s. To quan- tify the validity of the approximate solution we compute, forτ= 10−12sand for three different dampings, the relative difference δana prec=ωapprox prec−ωexact prec ωexactprec×100 We show in the inset of Fig. 3 the evolution of δana precwith respect to the applied static field H. ForH <20T the relative difference remains less than 0 .1% for small damping α= 0.01, and remains less than 3% for mod- erate damping α= 0.1. For large damping α= 0.5 the approximate solution remains valid for small fields, but for 12T < H < 20Tthe error becomes larger than 10%. 2. Nutation peak Figure 4 displays both the analytical prediction of the nutation angular frequency ωweak nugiven by Eq. (15) and the angular frequency ωIILG nuof the nutation resonance6 10-1610-1510-1410-1310-1210-1110-1010-9 τ (s)01×10112×1011ωprec (rad.s-1) 0 5 10 15 20 25 30 H (T)10-410-310-210-1100101102δprecana (%)α=0.5 α=0.1 α=0.01 Figure 3: (Color online) Comparaison of the analytical and numerical simulation results for the precession angular fr e- quency obtained for α= 0.1 andH= 2T. Filled circles (black) are the precession angular frequency ωexact prec, open cir- cles (red) are the position of the precession resonance peak s ωILLG prec, stars (orange) are the approximate precession angular frequencies ωapprox precvalid for small values of τ. Thedashed line (black) is the LLG precession angular frequency, i.e.without inertial term. Inset : relative difference δana precfor three differ- ent dampings. obtained in the numerical simulations. The agreement is excellent for τ <10−11s, and indicates that the non- linearcouplingtermsofEq. (13)donotsignificantlyshift the angular frequency of the nutation resonance from the approximate angular frequency ωweak nu. On the contrary, in the range 10−11s < τ < 10−8s, the simulated nuta- tion resonance angular frequency is slightly higher than ωweak nu, as shown in the upper inset of Fig. 4. In the lower inset ofFig. 4 we show the relative difference δnubetween the approximate nutation angular frequency ωweak nuand the nutation resonance angular frequency ωILLG nuof the numerical simulations, i. e. δnu=ωILLG nu−ωweak nu ωILLGnu×100 We therefore see that in the range 10−11s < τ <10−8s, the approximate nutation angular frequency remains less than 15% close to the simulated nutation resonance angular frequency. B. Scaling and overview of the ILLG equation In the preceding section we investigated the behav- ior of the ILLG model when varying the characteristic time scale τwhich drives the inertial dynamics. We also10-1610-1510-1410-1310-1210-1110-1010-9 τ (s)1011101210131014101510161017ωnu (rad.s-1) 10-1110-1010-9 τ (s)10111012ωnu (rad.s-1) 10-1310-1210-1110-1010-910-8 τ (s)010 δnu (%) Figure 4: (Color online) Comparaison of the analytical and numerical simulation results for the nutation angular fre- quency obtained for α= 0.1 andH= 2T. Filled cir- cles (black) are the approximate nutation angular frequenc ies ωweak nuand open circles (red) are the positions ωILLG nuof the simulated nutation resonance peaks. Upper inset : enlarge- ment showing the effect of the non-linear coupling terms of Eq. (13). Lower inset : relative difference δnubetween ωweak nu andωILLG nu. vary the static field Hand the dimensionless damping α. Increasing Hmoves both the precession and nuta- tion resonance peaks to higher angular frequencies, with smaller and broadened peaks, while increasing the di- mensionless damping moves both peaks to lower angular frequencies with still smaller and broadened peaks. Note that the ILLG precession resonances obtained when the static field His varied show that the transverse suscep- tibility follows a power law χ⊥∝1/ωIILG prec(not shown). This law is the same as the one resulting from the LLG model13. Eq. (15) suggests a scaling function ωnu γH=√1+x x wherex=ατγH. Scaling curves obtained for different values of τ,αandHare shown in the inset of Fig. 5 where both the precession and nutation resonance an- gular frequencies are dispayed with respect to ατγH. Fig. 5 is an enlargement of the intermediate region of the inset where we added the points obtained by the nu- merical simulations for H= 2Tandα= 0.1. The two asymptotic behaviors of the nutation are highlighted with the dashed lines in agreement with Eq. (15) : whenατγH≪1 thenωweak nu/γH= 1/ατγH, and when ατγH≫1 thenωweak nu/γH= 1/√ατγH. Remarquably, we see that the precession peak position divided by γH also scales as ωprec/γH∼1/√ατγHwhenατγH≫1.7 0.01 0.1 1 10 100 ατγH0.010.1110100ωnu/γH , ωprec/γH 10-610-410-210010210-2100102104106 1/ατγH 1/(ατγH)1/2 Figure 5: (Color online) Scaling curves : nutation ωnuand precession ωprecpeak positions in the angular frequency do- main divided by γHwith respect to ατγH. Open circles (red) are the nutation and precession resonance peak posi- tions obtained in the numerical simulations for α= 0.1 and H= 2T. Other points are ωweak nucomputed from Eq. (15), andωexact preccomputed from Eq. (9). Different values of the static field Hand the dimensionless damping αare reported : H= 0.2Tandα= 0.1 (red open diamonds), H= 2T(blue open squares for α= 0.1 and blue filled squares for α= 0.01), H= 20Tandα= 0.1 (green open triangles), H= 200T andα= 0.1 (black crosses). The dashed lines are the two asymptotic behaviors of the nutation in agreement with Eq. (15). Inset : same scaling curves (without red open circles) displayed on larger scales. The two asymptotic behaviors intersect at ατγH= 1 andω/γH= 1. This point corresponds to the max- imum value of the LLG precession angular frequency ωLLG/γH= 1/(1 +α2) which is obtained in the limit case of no damping α= 0. The inset of Figure 5 indicates that only one resonance peakisexpectedwhen ατγH→ ∞. Inthiscase,boththe nutation and the precession contribute to a unique peak. On the contrary, for finite ατγHthey remain separated. There are two different well-defined peaks in the investi- gated range ( ατγH≤100). For ατγH≪1 the preces- sion peak is close to the usual LLG precession peak, and the nutation peak shifts rapidly ( ωweak nu/γH∼1/ατγH) to high angular frequencies. In other words, the nutation oscillator defined by Eq. (13) is independent of the pre- cession for ατγH≪1, whereas both synchronize at the same frequency for ατγH→ ∞. Accurate predictions about the precession and nutation peak positions in the angular frequency domain can be made, as long as the non-linear coupling terms of Eq. (13) remain weak or compensate each other.V. TOWARDS EXPERIMENTAL EVIDENCE OF THE INERTIAL DYNAMICS OF THE MAGNETIZATION Throughout the preceding sections we studied the new properties of the inertial dynamics of the magnetization within the ILLG model. We specifically considered the FMR framework where a small perpendicular sinusoidal field is applied implying that the small inclination limit holds. We now focus on possible simple experiments in such FMR framework that should highlight the inertial dynamics of the magnetization. The first direct evidence would of course be the measure of the nutation resonance peak at frequencies larger than the precession resonance peak. Since the expected nuta- tion resonance peak is given by Eq. (15), the evolution with the static field Hmay be used to discriminate the nutation resonnce peak from possible other higher fre- quency peaks. However the amplitude of the nutation resonance peak is smallerthan forthe precessionpeak, and itmaybe tricky in unfavorable situations to measure such a peak, for ex- ample in materials with small characteristic time τ. Fur- thermore, for large dimensionless damping αboth peaks have smaller amplitude and are rounded. It may even appear that the nutation resonance peak of the magneti- zationinthe ILLGmodel disappearsforalargedamping, likethe resonantpeakofthe classicaldrivendamped har- monic oscillator. For exampleFig. 6showsthat for mate- rials with a large damping ( α= 0.5) the resonance peaks are smaller and rounded compared to smaller damping (α= 0.1), and the nutation resonance peak disappears forH≤5T. It is therefore necessary to find measurable characteris- ticsofthemagnetizationinertialdynamicsotherthanthe direct measure of the nutation resonance peak. Actually, we show in the following that beyond the nutation reso- nance peak, the inertial dynamics has measurable effects on the precession resonance peak. Indeed, as shown in Fig. 7, the shape of the precession peak and its position intheangularfrequencydomainaremodifiedbytheiner- tial dynamics. And the effects are shown to be more pro- nounced for large damping materials and for large static magnetic fields H. To show these effects we compare the precession resonance angular frequencies ωILLG precandωLLG prec obtained in the numerical simulations of both the ILLG and non-inertial LLG models. We use two different di- mensionless damping α= 0.1 andα= 0.5, and vary the amplitude Hof the static magnetic field. For the ILLG model, we choose, as in Ref. 4, a rough estimation of the characteristic time scale τ= 10−12s. A. Angular frequency of the precession resonance peak Wefirstlookatthepositionoftheprecessionresonance peakin the angularfrequencydomain. Fig.8(a)and 8(b)8 02×10124×10126×1012 ω (rad.s-1)1×10-21×10-11×100χ⊥α=0.5 Figure 6: (Color online) Resonance curves of the transverse susceptibility χ⊥(ω) with respect to the oscillating field angu- lar frequency ω. The resonance curves are computed within the ILLG model with τ= 10−12s, for a large dimension- less damping α= 0.5 and for various applied static fields : H= 2T(black filled circles), H= 5T(green filled trian- gles),H= 20T(blue open circles) and H= 50T(red open triangles). Both resonance peaks clearly appear for H= 20T andH= 50T. ForH= 5TandH= 2Tthe nutation reso- nance peak dissapears due to the large damping. 00.10.20.3 1×10 12 2×10 12 3×10 12 20 T 00.20.40.6 0 1×10 12 2×10 12 10 T2 T 0123 0 1×10 12 00.050.1 3×10 12 4×10 12 5×10 12 6×10 12 50 T2 T 2 T χ ⊥ILLG LLG ILLG LLG χ ⊥ ILLG LLG ILLG LLG ω (rad.s -1 ) ω (rad.s -1 ) Figure 7: Precession resonance curves of the transverse sus - ceptibility simulated for differentvalues ofthestatic fiel dH= 2T, 10T, 20T, and 50 T. ILLG model (filled circles) and non-inertial LLGmodel (opencircles). γ= 1011rad.s−1.T−1, α= 0.1, andτ= 10−12s. display the evolution of the resonance angular frequencyωprecwith respect to Hobtained for α= 0.1 andα= 0.5 within the numerical simulations of both the ILLG and LLG models. As expected the resonance angular 0 10 20 30 40 50 60 H (T)0123456ωprec (1012 rad.s-1) 0 10 20 30 40 50 60 H (T)0123456ωprec (1012 rad.s-1) 0 10 20 30 40 50 60 H (T)01020304050δ prec (%)a) b) α=0.1 c)α=0.5 α=0.5 α=0.1LLGLLG ILLG ILLG Figure 8: (Color online) (a) and (b) Precession resonance an - gular frequency with respect to the applied static field. Re- sults obtainedin thenumerical simulations of theILLGmode l (withτ= 10−12s) and non-inertial LLG model, for dimen- sionless damping (a) α= 0.1 (blue open circles for LLG and red filled circles for ILLG) and (b) α= 0.5 (blue open squares for LLG and red filled squares for ILLG). (c) Relative differ- enceδprecbetween LLG and ILLG precession resonance an- gular frequencies for α= 0.1 (green filled circles) and α= 0.5 (green filled squares). frequency of the LLG precession is linear with Hsince ωLLG prec=γH/(1+α2) whereas the behavior is not linear in Hfor the ILLG model. In Fig. 8(c) we plot the relative difference δprec=ωLLG prec−ωILLG prec ωLLGprec×100 betweenbothresonanceangularfrequencies. Therelative distance between both precession peaks increases with H and with the dimensionless damping α. B. Width of the precession resonance peak We now examine the evolution with Hof the shape of the precession resonance peak obtained in the sim- ulations of the ILLG and LLG models. For α= 0.1, the full width at half maximum (FWHM) is shown in Fig. 9(a) while Fig. 9(b) displays the FWHM divided by the resonance angular frequency. For large damping α= 0.5 we change the criterion since the reduced amplitude of the resonant peak does not allow anymore to compute the FWHM. We therefore compute the bandwith defined by the width of the peak at Amax/√ 29 0 10 20 30 40 50 60 H (T)00.10.20.3FWHM / ωprec 0 10 20 30 40 50 60 H (T)0.00.51.01.52.0FWHM (1012 rad.s-1) 0 10 20 30 40 50 60 H (T)00.51Bandwidth / ωprec 0 10 20 30 40 50 60 H (T)012345Bandwidth (1012 rad.s-1)a) b) d)c)α=0.1 α=0.1 α=0.5 α=0.5LLGLLG LLGLLGILLGILLG ILLGILLG Figure 9: (Color online) (a) Full width at half maximum (FWHM) for the precession resonance peak for α= 0.1 within the LLG (blue open circles) and the ILLG (red filled circles) models. (b)FWHMdividedeither by ωLLG prec(blueopencircles) or byωILLG prec(red filled circles). (c) Bandwidth of the preces- sion resonance peak for α= 0.5 within the LLG (blue open squares)andILLG(redfilledsquares)models. (d)Bandwidth divided either by ωLLG prec(blue open squares) or by ωILLG prec(red filled squares). The numerical simulations of the ILLG model are computed withτ= 10−12s whereAmaxis the maximum value of the peak. The bandwidth for α= 0.5 is shown in Fig. 9(c) and the bandwidth divided by the resonance angular frequency is plotted in Fig. 9(d). The numerical simulations of the ILLG and LLG models lead to different behaviors for the shape of the precession resonance peak. In the LLG model the FWHM and the bandwidth exhibit a linear evolution with the applied static field which results in a constant evolution when divided by the resonance angular frequency. Very different behaviors are observed within the ILLG model where no linear evolution of the FWHM or the bandwidth is measured. Figs. 8 and 9 show that high applied static fields in large damping materials produce large differences be- tween the positions and shapes of the precession reso- nance peaks originating from the LLG and ILLG mod- els. Therefore, applying high static fields in large damp- ing materials better allows to differentiate the precession peak originating from the ILLG and LLG models. Although the theory is clear and allows in principle to differentiate inertialfromnon-inertialdynamicswhen ex- aminingboth precessionresonancepeaks, the experimen- tal investigations are rather more complex. Indeed, the experimental demonstration of inertial effects first ne- cessitate to identify and control the different contribu-tions to the effective field (anisotropy, dipolar interac- tion, magnetostriction, ...) other than the applied static field. VI. CONCLUSION The magnetization dynamics in the ILLG model that takes into account inertial effects has been studied from both analytical and numerical points ofview. Within the FMR context, a nutation resonance peak is expected in addition to the usual precession resonance peak. Analytical solutions of the inertial precession and nuta- tion angular frequencies are presented. The analytical solutions nicely agree with the numerical simulations of the resonance curves in a broad range of parameters. At first, we investigated the effects of the time scale τ which drives the additional inertial term introduced in Eq. (2)comparedtotheusualLLGequationEq. (1). We also varied the dimensionless damping αand the static magnetic field H, and a scaling function with respect to ατγHis found for the nutation angular frequency. Re- marquably, the same scaling holds for the precession an- gular frequency when ατγH≫1. In the second part of the paper we focussed on the sig- natures of the inertial dynamics which could be detected experimentallywithintheFMRcontext. Weshowedthat beyondthemeasureofthenutationresonancepeakwhich would be a direct signature of the inertial dynamics, the precession is modified by inertia and the ILLG preces- sion resonance peak is different from the usual LLG pre- cession peak. Indeed, whereas a linear evolution with respect to His expected for the LLG precession reso- nance angular frequency, the ILLG precession resonance angular frequency is clearly non-linear. Furthermore, the shape of the precession resonance peak is different in the LLG and ILLG models. Again, the width variation of the precession resonance peak is non-linear in the ILLG dynamics as opposed to the linear evolution with Hin the LLG dynamics. We also showed that the difference between both LLG and ILLG precession peaks is more pronounced when the damping is increased and when τ is increased. For example the discrepancy between the LLG and ILLG precession resonance angular frequencies atH= 20Tforτ= 1psis expected to be of the order of 20% for α= 0.1 and 30% for α= 0.5. Therefore, large damping materials are better candidates to experimen- tallyevidencetheinertialdynamicsofthemagnetization. Finally, a specific behavior of the amplitude of the mag- netic susceptibility as a function of the nutation reso- nance angular frequency ωnuis predicted, of the form χ⊥(ωnu)∝ω−1 nu(analogousto that ofthe usual FMR sus- ceptibility). This law could be a useful criterion in order to discriminate the nutation peak among the other exci- tations that could also occur close to the infrared region (100 GHz up to 100 THz) in a ferromagnetic material.10 1W.F. Brown Thermal Fluctuations of a Single-Domain Particle, Phys. Rev. 130, 1677 (1963). 2R. Kubo, M. Toda, N. Hashitzume, Statistical physics II, Nonequilibrium Statistical Mechanics , Springer Series in Solid-State Sciences 31, Berlin 1991 (second edition), Ed. P. Fulde, Chap 3, Paragraph 3.4.3, p. 131. 3M.-C. Ciornei, Role of magnetic inertia in damped macrospin dynamics , Ph. D. thesis, Ecole Polytechnique, Palaiseau France 2010. 4M.-C. Ciornei, J. M. Rub´ ı, and J.-E. Wegrowe, Magnetiza- tion dynamics in the inertial regime : Nutation predicted at short time scales , Phys. Rev. B 83, 020410(R) (2011). 5M. F¨ ahnle, D. Steiauf, and Ch. Illg, Generalized Gilbert equation including inertial damping : Derivation from an extended breathing Fermi surface model , Phys. Rev. B 84, 172403 (2011). 6J.-E. Wegrowe, C. Ciornei Magnetization dynamics, Gyro- magnetic Relation, and Inertial Effects , Am J. Phys. 80, 607 (2012). 7E. Olive, Y. Lansac, and J.-E. wegrowe, Beyond ferromag- netic resonance : the inertial regime of the magnetization ,Appl. Phys. Lett. 100, 192407 (2012). 8D. B¨ ottcher, and J. Henk Significance of nutation in mag- netization dynamics of nanostructures , Phys. Rev. B 86, 020404(R) (2012). 9S. Bhattacharjee, L. Nordstr¨ om, and J. Fransson Atomistic spin dynamic method with both damping and moment of inertia effects included from first principles , Phys. Rev. Lett.108, 057204 (2012). 10J.-X. Zhu, Z. Nussinov, A. Shnirman, and A. V. Balatsky Novel spin dynamics in a Josephson junction , Phys. Rev. Lett.92, 107001 (2004). 11J. Fransson, and J. Xi. Zhu Spin dynamics in a tunnel junction between ferromagnets , New J. Phys. 10, 013017 (2008). 12C. W. Gear, Numerical initial value problems in ordinary differential equations , Prentice Hall, Englewood Cliffs (N. J.) 1971. 13A. G. Gurevich and G. A. Melkov, Magnetization Oscilla- tion and Waves , CRC Press, 1996, p. 19.

2014-12-11

We investigate in details the inertial dynamics of a uniform magnetization in the ferromagnetic resonance (FMR) context. Analytical predictions and numerical simulations of the complete equations within the Inertial Landau-Lifshitz-Gilbert (ILLG) model are presented. In addition to the usual precession resonance, the inertial model gives a second resonance peak associated to the nutation dynamics provided that the damping is not too large. The analytical resolution of the equations of motion yields both the precession and nutation angular frequencies. They are function of the inertial dynamics characteristic time $\tau$, the dimensionless damping $\alpha$ and the static magnetic field $H$. A scaling function with respect to $\alpha\tau\gamma H$ is found for the nutation angular frequency, also valid for the precession angular frequency when $\alpha\tau\gamma H\gg 1$. Beyond the direct measurement of the nutation resonance peak, we show that the inertial dynamics of the magnetization has measurable effects on both the width and the angular frequency of the precession resonance peak when varying the applied static field. These predictions could be used to experimentally identify the inertial dynamics of the magnetization proposed in the ILLG model.

Deviation From the Landau-Lifshitz-Gilbert equation in the Inertial regime of the Magnetization

1412.3783v1

arXiv:1408.2160v1 [math.AP] 9 Aug 2014LOCAL EXISTENCE RESULTS FOR THE WESTERVELT EQUATION WITH NONLINEAR DAMPING AND NEUMANN AS WELL AS ABSORBING BOUNDARY CONDITIONS VANJA NIKOLI ´C Abstract. We investigate the Westervelt equation with several versio ns of nonlinear damping and lower order damping terms and Neumann as well as ab- sorbingboundary conditions. We prove localintimeexisten ce ofweaksolutions under the assumption that the initial and boundary data are s ufficiently small. Additionally, we prove local well-posedness in the case of s patially varying L∞coefficients, a model relevant in high intensity focused ultr asound (HIFU) applications. 1.Introduction High intensity focused ultrasound (HIFU) is crucial in many medical a nd in- dustrial applications including lithotripsy, thermotherapy, ultraso und cleaning or welding and sonochemistry. Widely used mathematical model for non linear wave propagation is the Westervelt equation, which can either be written in terms of the acoustic pressure p (1−2kp)ptt−c2∆p−b∆pt= 2k(pt)2, (1.1) or in terms of the acoustic velocity potential ψ (1−2˜kψt)ψtt−c2∆ψ−b∆ψt= 0, (1.2) with̺ψt=p. Here,cdenotes the speed and bthe diffusivity of sound, k=βa/λ, βa= 1+B/(2A),B/Arepresents the parameter of nonlinearity, ̺is the mass den- sity,λ=̺c2is the bulk modulus and ˜k=̺k. For a detailed derivation of (1.1) and (1.2) we refer the reader to [4], [9], [13]. Well-posedness and exponential decay of small and H2−spatially regular solu- tionsisestablishedforthe Westerveltequationwith homogeneous[ 6] andinhomoge- neous [7] Dirichlet and Neumann [8] boundary conditions as well as with boundary instead of interior damping [5]. A significant task in the analysis of the Westervelt equation is avoiding degener- acyofthe coefficient 1 −2kpforthe secondtime derivative pttin (1.1) and, similarly, of the term 1 −2˜kψtin the formulation (1.2). At the same time, in applications the existence of spatially less regular solutions is important, e.g. in th e coupling of acoustic with acoustic or elastic regions with different material para meters. In [2], 2010Mathematics Subject Classification. Primary: 35L05; Secondary: 35L20. Key words and phrases. nonlinear acoustics, Westervelt’s equation, local existe nce. Research supported by the Austrian Science Fund (FWF): P249 70. 12 V. NIKOLI ´C Brunnhuber, Kaltenbacher and Radu treated this issue by introdu cing nonlinear damping terms to the Westervelt equation and considering the follow ing equations (1−2ku)utt−c2∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig = 2k(ut)2,(1.3) (1−2ku)utt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut= 2k(ut)2, (1.4) utt−c2 1−2˜kut∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig = 0, (1.5) with homogeneous Dirichlet boundary data. First two equations are derived from the Westervelt equation in the acoustic pressure formulation (1.1) , while the third equation comes from the acoustic potential formulation (1.2) (with the notation changed to p→u,ψ→u). Added nonlinear damping terms make obtaining L∞(0,T;L∞(Ω)) estimate on u(ut) possible, without the need to estimate ∆ u (∆ut) and thus refraining from too high regularity. The central aim of the present paper is to investigate this relaxatio n of regu- larity by nonlinear damping, but equipped with practically relevant abs orbing and Neumann boundary data. This is motivated by many applications of hig h intensity focused ultrasoundwherethe need formorerealisticboundaryco nditions is evident. E.g. in lithotripsy one faces the problem of a physically unbounded dom ain, as typ- ical in acoustics, which should be truncated for numerical computa tions. Absorbing boundary conditions are then used to avoid reflections on the artifi cial boundary ˆΓ of the computational domain. Ultrasound excitation, e.g. by piezoelectric transducers, can be m odeled by Neu- mann boundary conditions on the rest of the boundary Γ = ∂Ω\ˆΓ. In our case, the design of the nonlinear absorbing and inhomogeneo us Neumann boundary conditions is influenced by the presence of the nonlinear s trong damping in the equations. We will study initial boundary value problems of the f ollowing type: (1−2ku)utt−c2∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig +βut = 2k(ut)2in Ω×(0,T], c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gon Γ×(0,T], αut+c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0},(1.6) (1−2ku)utt−c2∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig +γ|ut|q−1ut = 2k(ut)2in Ω×(0,T], c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gon Γ×(0,T], αut+c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0},(1.7)LOCAL EXISTENCE RESULTS 3 (1−2ku)utt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut+βut = 2k(ut)2in Ω×(0,T], c2∂u ∂n+c2ε|∇u|q−1∂u ∂n+b∂ut ∂n=gon Γ×(0,T], αut+c2∂u ∂n+c2ε|∇u|q−1∂u ∂n+b∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0},(1.8) utt−c2 1−2˜kut∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig +γ|ut|q−1ut = 0 in Ω ×(0,T], c2 1−2˜kut∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gon Γ×(0,T], αut+c2 1−2˜kut∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0}.(1.9) Note that in the case of b= 0,α=cand˜k= 0 the absorbing conditions prescribed in (1.6)-(1.9) would reduce to the standard linear absor bing boundary conditions of the form ut+c∂u ∂n= 0. In the equations, we assume that the parameters βandγare nonnegative; the caseβ=γ= 0 reduces them to (1.3)-(1.4). Another task of the present pap er is to investigate possible introduction of these lower order linearandnonlinear damping terms to the equations (1.3)-(1.4), this becomes beneficial when d eriving energy estimates. Additionally, in the context ofHIFU devices based on the acoustic len s immersed in a fluid medium, a problem of Westervelt’s equation coupled with other equations or with jumping coefficients arises. We will treat acoustic-acoustic c oupling which can be modeled by Westervelt’s equation in the pressure formulation with spatially varying coefficients (see [1] for the linear case and [2] for the nonlin ear case with homogeneous Dirichlet boundary conditions): 1 λ(x)(1−2k(x)u)utt−div(1 ̺(x)∇u)−div/parenleftBig b(x)(((1−δ(x))+δ(x)|∇ut|q−1)∇ut/parenrightBig =2k(x) λ(x)(ut)2in Ω×(0,T], 1 ̺(x)∂u ∂n+b(x)((1−δ(x))+δ(x)|∇ut|q−1)∂ut ∂n=gon Γ×(0,T], α(x)ut+1 ̺(x)∂u ∂n+b(x)((1−δ(x))+δ(x)|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0}. (1.10) 1.1.Notations and Preliminaries. We assume Ω ⊂Rd,d∈ {1,2,3}to be an open, connected, bounded set with Lipschitz boundary; ∂Ω is assumed to be a disjoint union of Γ and ˆΓ. We denote by nthe outward unit normal vector. Wewillstudytheproblemswithstrongdamping b>0andwithc2>0,δ∈(0,1), ε>0 andk,˜k∈R. Our results will hold for αassumed to be nonegative; the case α= 0 reduces (1.6)-(1.9) to problems with only Neuman boundary cond itions. Note that, in general, we will assume that q≥1, but this condition will have to be strenghtened at several instances to assure well-posedne ss of (1.6), (1.7) and4 V. NIKOLI ´C existence results for (1.8) and (1.9). We will often make use of the c ontinuous embeddings H1(Ω)֒→L4(Ω),with the norm CΩ H1,L4,and W1,q+1(Ω)֒→L∞(Ω),with the norm CΩ W1,q+1,L∞, with the latter being valid for q+1>d. In Section 2 and 5 we will need to employ the embedding Lq+1(Ω)֒→L4(Ω), which holds true for q≥3. We denote with Ctr 1the norm of the trace mapping Tr:W1,q+1(Ω)→W1−1 q+1,q+1(Γ), and withCtr 2the norm of the trace mapping tr:H1(Ω)→H−1/2(Γ) (withCtr 1= Ctr 2forq= 1). Throughout the paper we assume t∈[0,T], whereTis a finite time horizon. 1.2.Outline of the paper. The rest of the paper is organized as follows. Subsec- tion 1.3 contains the derivation of L∞-bounds on uandutas well as several useful inequalities that will be employed in the paper. In Section 2, we start by looking at a linearized version of (1.6) and (1 .7) with β=γ= 0, with nonlinearity appearing only through damping, and show local well-posedness. Then we discuss linearized versions of (1.6) and (1.7 ) withβ,γ >0. By employing the result for the linearized version we proceed to prov e local well- posedness for (1.6) and (1.7). Section 3 deals with the short time well-posedness of the acoustic-a coustic cou- pling modeled by (1.10). In Section 4 and 5 we consider (1.8) and (1.9), respectively. We again begin by investigating the linearized versions of the problems at hand for β= 0 andγ= 0 re- spectively, and continue with introducing lower order damping terms and the proof of local existence of solutions. 1.3.Inequalities. In the case of problems with inhomogeneous Neumann bound- ary data it is often necessary to employ Poincar´ e’s inequality valid fo r functions in W1,q+1(Ω). We recall such inequality (cf. Theorem 12.23, [10]), namely that there exists a constant CP>0 depending on qand Ω such that |ϕ−1 |Ω|/integraldisplay Ωϕdx|Lq+1(Ω)≤CP|∇ϕ|Lq+1(Ω), (1.11) for allϕ∈W1,q+1(Ω). The nonlinear damping term appearing in the equations (1.6)-(1.9) will enable us to avoid degeneracy of the coefficients 1 −2kuand 1−2kutby deriving L∞ estimates on uandut. From (1.11) we can obtain |u(t)|W1,q+1(Ω)≤(1+CP)|∇u(t)|Lq+1(Ω)+CΩ 1/vextendsingle/vextendsingle/vextendsingle/integraldisplay Ωu(t)dx/vextendsingle/vextendsingle/vextendsingle,(1.12) and by replacing uwithutalso |ut(t)|W1,q+1(Ω)≤(1+CP)|∇ut(t)|Lq+1(Ω)+CΩ 2|ut(t)|L2(Ω), (1.13) whereCΩ 1=|Ω|−q q+1andCΩ 2=|Ω|−q−1 2(q+1). From (1.13), by making use of the embedding W1,q+1(Ω)֒→L∞(Ω),q>d−1, weobtain anL∞estimate on ut /ba∇dblut/ba∇dblL∞(0,T;L∞(Ω))≤CΩ W1,q+1,L∞/bracketleftBig (1+CP)/ba∇dbl∇ut/ba∇dblL∞(0,T;Lq+1(Ω)) +CΩ 2/ba∇dblut/ba∇dblL∞(0,T;L2(Ω))/bracketrightBig ,(1.14) which will be used to avoid degeneracy of the factor 1 −2kutin the problem (1.9). Employing (1.12) and the estimate /ba∇dblut/ba∇dbl2 L2(0,T;L2(Ω))≤T/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω), (1.15) we can get an L∞estimate on u /ba∇dblu/ba∇dblL∞(0,T;L∞(Ω))≤CΩ W1,q+1,L∞/bracketleftBig (1+CP)|∇u(t)|Lq+1(Ω) +CΩ 1/vextendsingle/vextendsingle/vextendsingle/integraldisplay Ω(u0+/integraldisplayt 0ut(s)ds)dx/vextendsingle/vextendsingle/vextendsingle/bracketrightBig ≤CΩ W1,q+1,L∞/bracketleftBig (1+CP)/ba∇dbl∇u/ba∇dblL∞(0,T;Lq+1(Ω)) +CΩ 1|u0|L1(Ω)+CΩ 2√ T/ba∇dblut/ba∇dblL2(0,T;L2(Ω))/bracketrightBig ≤CΩ W1,q+1,L∞/bracketleftBig (1+CP)/ba∇dbl∇u/ba∇dblL∞(0,T;Lq+1(Ω)) +CΩ 1|u0|L1(Ω)+CΩ 2T/ba∇dblut/ba∇dblL∞(0,T;L2(Ω))/bracketrightBig ,(1.16) which we will apply when investigating (1.8). From (1.12) we can as well obtain |u(t)|L∞(Ω)≤CΩ W1,q+1,L∞/bracketleftBig (1+CP)|∇u0+/integraldisplayt 0∇ut(s)ds|Lq+1(Ω) +CΩ 1|u0|L1(Ω)+CΩ 2/integraldisplayt 0|ut(t)|L2(Ω)ds/bracketrightBig ≤CΩ W1,q+1,L∞/bracketleftBig (1+CP)(|∇u0|Lq+1(Ω)+(tq/integraldisplayt 0|∇ut|q+1 Lq+1(Ω)ds)1/q+1) +CΩ 1|u0|L1(Ω)+CΩ 2/integraldisplayt 0|ut(t)|L2(Ω)ds/bracketrightBig , which leads to the estimate /ba∇dblu/ba∇dblL∞(0,T;L∞(Ω))≤CΩ W1,q+1,L∞/bracketleftBig (1+CP)(|∇u0|Lq+1(Ω) +Tq q+1/ba∇dbl∇ut/ba∇dblLq+1(0,T;Lq+1(Ω))) +CΩ 1|u0|L1(Ω)+CΩ 2T/ba∇dblut/ba∇dblL∞(0,T;L2(Ω))/bracketrightBig ,(1.17) that will be employed when dealing with the possible degeneracy of the coefficient 1−2kuin (1.6) and (1.7). We will also frequently make use of Young’s inequality in the form ab≤εas+C(ε,s)bs s−1(a,b>0, ε>0,1<s<∞), (1.18) withC(ε,s) = (s−1)ss s−1ε−1 1−s. When dealing with the q-Laplace damping term in the equations, the inequality (cf. [11]) /a\}b∇acketle{t|b|q−1b−|a|q−1a,b−a/a\}b∇acket∇i}ht ≥0, a,b∈Rd, (1.19) 56 V. NIKOLI ´C valid for all q, will be of use as well. 2.Westervelt’s equation in the formulation (1.6)and(1.7) We will begin by looking at the problems (1.6) and (1.7) with β=γ= 0: (1−2ku)utt−c2∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig = 2k(ut)2in Ω×(0,T], c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gon Γ×(0,T], αut+c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0},(2.1) Following the approach in [2], we will first consider the equation where n onlinearity appears only in the damping term autt−c2∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig +fut = 0 in Ω ×(0,T] c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gon Γ×(0,T], αut+c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0},(2.2) and prove local well-posedness. Proposition 2.1. LetT >0,c2,b>0,α≥0,δ∈(0,1),q≥1and assume that (i)•a∈L∞(0,T;L∞(Ω)),at∈L∞(0,T;L2(Ω)),0<a≤a(t,x)≤a, •f∈L∞(0,T;L2(Ω)), •g∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), •u0∈H1(Ω),u1∈L2(Ω), with /ba∇dblf−1 2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<min/braceleftBigb(1−δ) 2(CΩ H1,L4)2,a 4T(CΩ H1,L4)2/bracerightBig .(2.3) Then(2.2)has a weak solution u∈˜X:={v:v∈C(0,T;H1(Ω))∩C1(0,T;L2(Ω)) ∧vt∈Lq+1(0,T;W1,q+1(Ω))},(2.4) which is unique and satisfies the energy estimate /bracketleftBiga 4−ˆb(CΩ H1,L4)2T−ǫ0/bracketrightBig /ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+c2 4/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω)) +/bracketleftBigb(1−δ) 2−ˆb(CΩ H1,L4)2/bracketrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) +/bracketleftBigbδ 2−ǫ1/bracketrightBig /ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))(2.5) ≤a 2|u1|2 L2(Ω)+c2 2|∇u0|2 L2(Ω)+1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ))LOCAL EXISTENCE RESULTS 7 +C(ǫ1,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), for some constants 0<ǫ0<a 4−ˆb(CΩ H1,L4)2T,0<ǫ1<bδ 2. (2.6) If, in addition to (i), (ii)•f∈L∞(0,T;H1(Ω)),/ba∇dblf/ba∇dblL∞(0,T;H1(Ω))≤˜b, •g∈L∞(0,T;W−q q+1,q+1 q(Γ)),gt∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), •u1∈W1,q+1(Ω), then u∈X:=C1(0,T;W1,q+1(Ω))∩H2(0,T;L2(Ω)), (2.7) and satisfies the energy estimate µa−τ 2/ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω))+µ[b(1−δ) 4−σ]/ba∇dbl∇ut/ba∇dbl2 L∞(0,T;L2(Ω)) +/bracketleftBiga 4−(CΩ H1,L4)2ˆbT−ǫ0(µ+1)−µ1 2τ(CΩ H1,L4)4˜b2T/bracketrightBig /ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω)) +/bracketleftBigb(1−δ) 2−(CΩ H1,L4)2ˆb−µ(1 2τ(CΩ H1,L4)4˜b2+c2)/bracketrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) +c2 4(1−µc2 σ)/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))+µ[bδ 2(q+1)−η]/ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω)) +[bδ 2−ǫ1(µ+1)]/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) +µα 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(ˆΓ)) ≤C/parenleftBig CΓ(g)+|u1|2 H1(Ω)+|∇u0|2 L2(Ω)+|u1|q+1 W1,q+1(Ω)+|u1|2 L2(ˆΓ)/parenrightBig ,(2.8) for some sufficiently small constants µ,σ,τ,η> 0, some large enough C >0, and CΓ(g) =1/summationdisplay s=0/ba∇dblds dtsg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ))+1/summationdisplay s=0/ba∇dblds dtsg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +/ba∇dblg/ba∇dbl2 L∞(0,T;W−q q+1,q+1 q(Γ))+/ba∇dblg/ba∇dblq+1 q L∞(0,T;W−q q+1,q+1 q(Γ)).(2.9) Proof.The weak form of (2.2) is given as/integraldisplay Ω/braceleftBig auttw+c2∇u·∇w+b/parenleftBig (1−δ)+δ|∇ut|q−1/parenrightBig ∇ut·∇w/bracerightBig dx+α/integraldisplay ˆΓutwdx =−/integraldisplayt 0/integraldisplay Ωfutwdx+/integraldisplay Γgwdx,∀w∈W1,q+1(Ω), (2.10) with initial conditions ( u0,u1). We will use the standardGalerkinmethod (see forinstanceSection 7 .2, [3] forthe case of second-order linear hyperbolic equations and Section 2, [2] for the problem (2.2) with homogeneous Dirichlet boundary data), where we will first construct approximations of the solution, and then by obtaining energy estima tes guarantee weak convergence of these approximations. 1. Smooth approximation of a,f, andg.Let us first introduce sequences (ak)k∈N, (fk)k∈Nand (gk)k∈Nwhich represent smooth in time approximations of a, f, andg:8 V. NIKOLI ´C •(ak)k∈N⊆C∞([0,T]×Ω)∩W1,∞(0,T;L2(Ω)), ak→ainL∞(0,T;L∞(Ω)),ak,t→atinL∞(0,T;L2(Ω)), 0<a≤ak(t,x)≤a, •(fk)k∈N⊆C∞((0,T)×Ω),fk→finL∞(0,T;L2(Ω)), •(gk)k∈N⊆C∞(0,T;W−q q+1,q+1 q(Γ)),gk→ginLq+1 q(0,T;W−q q+1,q+1 q(Γ)), • /ba∇dblfk−1 2ak,t/ba∇dblL∞(0,T;L2(Ω))≤ˆb, and, for fixed k∈N, prove that there exists a solution u(k)of /integraldisplay Ω/braceleftBig aku(k) ttw+c2∇u(k)·∇w+b/parenleftBig (1−δ)+δ|∇u(k) t|q−1/parenrightBig ∇u(k) t·∇w/bracerightBig dx +α/integraldisplay ˆΓu(k) twdx=−/integraldisplay Ωfku(k) twdx+/integraldisplay Γgwdx, ∀w∈W1,q+1(Ω),(2.11) with initial conditions ( u0,u1). (a) Galerkin approximations. We start by proving existence and uniqueness of a solution for a finite-dimensional approximation of (2.11). We cho ose smooth functionswm=wm(x),m∈Nsuch that {wm}m∈Nis an orthonormal basis of L2 ˜ak(Ω), {wm}m∈Nis a basis of W1,q+1(Ω), {wm|ˆΓ}m∈Nis an orthonormal basis of L2(ˆΓ), whereL2 ˜akis the weighted L2-space based on the inner product /a\}b∇acketle{tf,g/a\}b∇acket∇i}htL2 ˜ak(Ω):= /integraltext Ω˜akfgdx, with ˜ak=1 T/integraltextT 0ak(t)dt. Next, we construct a sequence of finite dimensional subspaces VnofL2 ˜ak(Ω)∩ W1,q+1(Ω), Vn= span{w1,w2,...,w n}. Clearly,Vn⊆Vn+1,Vn⊆L2 ˜ak(Ω)∩W1,q+1(Ω) and/uniontext n∈NVn=W1,q+1(Ω). Let (u0,n)n∈N, (u1,n)n∈Nbe sequences such that •u0,n∈Vn,u0,n→u0inH1(Ω), •u1,n∈Vn,u1,n→u1inL2(Ω). We can now consider a sequence of discretized versions of (2.11), /integraldisplay Ω/braceleftBig aku(k) n,ttwn+c2∇u(k) n·∇wn+b/parenleftBig (1−δ)+δ|∇u(k) n,t|q−1/parenrightBig ∇u(k) n,t·∇wn/bracerightBig dx +α/integraldisplay ˆΓu(k) n,twndx=−/integraldisplay Ωfku(k) n,twndx+/integraldisplay Γgkwndx,∀wn∈Vn,(2.12) withu(k) n(t)∈Vnand initial conditions ( u0,n,u1,n). For each n∈N, we face an initial value problem for a second order system of ordinary different ial equations with coefficients and right hand side that are C∞functions of t. According to standard existence theory for ordinary differential equations (c f. [12]), there exists a unique solution u(k) n∈C∞(0,˜T,Vn) of (2.12) for some ˜T≤Tsufficiently small. By employing the uniform energy estimates obtained below, we can co nclude that ˜T=T.LOCAL EXISTENCE RESULTS 9 (b) Lower energy estimate. Testing (2.12) with wn=u(k) n,t(t)∈Vnand inte- grating with respect to time results in 1 2/bracketleftbigg/integraldisplay Ωak/parenleftBig u(k) n,t/parenrightBig2 dx+c2|∇u(k) n|2 L2(Ω)/bracketrightbiggt 0+α/integraldisplayt 0/integraldisplay ˆΓ|u(k) n,t|2dxds +b/integraldisplayt 0/integraldisplay Ω/parenleftBig (1−δ)+δ|∇u(k) n,t|q−1/parenrightBig |∇u(k) n,t|2dxds =−/integraldisplayt 0/integraldisplay Ω(fk−1 2ak,t)/parenleftBig u(k) n,t/parenrightBig2 dxds+/integraldisplayt 0/integraldisplay Γgku(k) n,tdxds ≤/ba∇dblfk−1 2ak,t/ba∇dblL∞(0,T;L2(Ω))/integraldisplayt 0|u(k) n,t|2 L4(Ω)ds+/integraldisplayt 0/integraldisplay Γgku(k) n,tdxds.(2.13) For estimating the boundary integral appearing on the right side, w e will make use of (1.13) to obtain /integraldisplayt 0/integraldisplay Γgku(k) n,tdxds≤/integraldisplayt 0|u(k) n,t(s)| W1−1 q+1,q+1(Γ)|gk(s)| W−q q+1,q+1 q(Γ)ds ≤Ctr 1/integraldisplayt 0|u(k) n,t(s)|W1,q+1(Ω)|gk(s)| W−q q+1,q+1 q(Γ)ds ≤Ctr 1/integraldisplayt 0/bracketleftBig (1+CP)|∇u(k) n,t(s)|Lq+1(Ω) (2.14) +CΩ 2|u(k) n,t(s)|L2(Ω)/bracketrightBig |gk(s)| W−q q+1,q+1 q(Γ)ds ≤ǫ1/ba∇dbl∇u(k) n,t/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+ǫ0/ba∇dblu(k) n,t/ba∇dbl2 L∞(0,T;L2(Ω)) +C(ǫ1,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblgk/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblgk/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)), withǫ0,ǫ1>0. By taking the essential supremum with respect to tin (2.13) and employing the embedding H1(Ω)֒→L4(Ω), as well as the inequality (1.15), we obtain the estimate /bracketleftBiga 4−ˆb(CΩ H1,L4)2T−ǫ0/bracketrightBig /ba∇dblu(k) n,t/ba∇dbl2 L∞(0,T;L2(Ω))+c2 4/ba∇dbl∇u(k) n/ba∇dbl2 L∞(0,T;L2(Ω)) +/bracketleftBigb(1−δ) 2−ˆb(CΩ H1,L4)2/bracketrightBig /ba∇dbl∇u(k) n,t/ba∇dbl2 L2(0,T;L2(Ω))+α 2/ba∇dblu(k) n,t/ba∇dbl2 L2(0,T;L2(ˆΓ)) +/bracketleftBigbδ 2−ǫ1/bracketrightBig /ba∇dbl∇u(k) n,t/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))(2.15) ≤C(ǫ1,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblgk/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ))+a 2|u(k) 1,n|2 L2(Ω) +1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblgk/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ))+c2 2|∇u(k) 0,n|2 L2(Ω). We choose ǫ0,ǫ1small enough 0<ǫ0<a 4−ˆb(CΩ H1,L4)2T,0<ǫ1<bδ 2, (2.16)10 V. NIKOLI ´C so that coefficients appearing in the estimate remain positive. As by a ssumption gk∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), we conclude that the sequence of Galerkin approx- imations/parenleftbig u(k) n/parenrightbig n∈Nis bounded in the Banach space ˜X:={v:v∈C(0,T;H1(Ω))∩C1(0,T;L2(Ω))∧vt∈Lq+1(0,T;W1,q+1(Ω))}. It follows from (2.15) that /parenleftbig u(k) n,t/parenrightbig n∈Nis uniformly bounded in L2(0,T;L2(Ω)), (2.17) /parenleftbig ∇u(k) n,t/parenrightbig n∈Nis uniformly bounded in Lq+1(0,T;Lq+1(Ω)), (2.18) |∇u(k) n,t|q−1∇u(k) n,tis uniformly bounded in Lq+1 q(0,T;Lq+1 q(Ω)),and (2.19) /parenleftbig u(k) n,t|ˆΓ/parenrightbig ∈Nis uniformly bounded in L2(0,T;L2(ˆΓ)), (2.20) which are all reflexive Banach spaces. (c) Convergence of Galerkin approximations. Due to (2.17)-(2.20) there existsaweaklyconvergentsubsequenceof( u(k) n)n∈N, whichwestilldenote( u(k) n)n∈N, and au(k)such that u(k) n,t⇀u(k) tinL2(0,T;L2(Ω)), (2.21) ∇u(k) n,t⇀∇u(k) tinLq+1(0,T;Lq+1(Ω)), (2.22) |∇u(k) n,t|q−1∇u(k) n,t⇀|∇u(k) t|q−1∇u(k) tinLq+1 q(0,T;Lq+1 q(Ω)), (2.23) u(k) n,t|ˆΓ⇀u(k) t|ˆΓinL2(0,T;L2(ˆΓ)). (2.24) Our task next is to prove that the weak limit u(k)solves (2.11). Fix k,m∈Nand letφm∈C∞(0,T,Vm)⊂Lq+1(0,T;W1,q+1(Ω)) withφm(T) = 0. For any n≥m, byVm⊆Vnwe have /integraldisplayT 0/integraldisplay Ω/braceleftBig aku(k) ttφm+c2∇u(k)·∇φm+b/parenleftBig (1−δ)+δ|∇u(k) t|q−1/parenrightBig ∇u(k) t·∇φm +fku(k) tφm/bracerightBig dxds+α/integraldisplayT 0/integraldisplay ˆΓu(k) tφmdxds−/integraldisplayT 0/integraldisplay Γgkφmdxds =−/integraldisplayT 0/integraldisplay Ω[u(k) t−u(k) n,t]/parenleftBig akφm/parenrightBig tdxds−/integraldisplay Ω[u1−u1,n]ak(0)φm(0)dxds +c2/integraldisplayT 0/integraldisplay Ω[∇u(k)−∇u(k) n]·∇φmdxds (2.25) +/integraldisplayT 0/integraldisplay Ω[u(k) t−u(k) n,t]fkφmdxds+b(1−δ)/integraldisplayT 0/integraldisplay Ω[∇u(k) t−∇u(k) n,t]·∇φmdxds +bδ/integraldisplayT 0/integraldisplay Ω[|∇u(k) t|q−1∇u(k) t−|∇u(k) n,t|q−1∇u(k) n,t]·∇φmdxds +α/integraldisplayT 0/integraldisplay ˆΓ[u(k) t−u(k) n,t]φmdxds→0 asn→ ∞, due to (2.21)-(2.24). Since/uniontext m∈NVmis dense in W1,q+1(Ω),u(k)indeed solves (2.11). By testing the problem (2.11) with u(k) tand proceding as in 1.(b) we can conclude that this weak limit satisfies the estimate (2.15) with u(k) nreplaced by u(k).LOCAL EXISTENCE RESULTS 11 2.kkk→∞∞∞.Owing to the previous conclusion, we can find a weakly convergent subsequence of ( u(k)), which we again denote ( u(k)), andu∈˜Xsuch that u(k) t⇀utinL2(0,T;L2(Ω)), (2.26) ∇u(k) t⇀∇utinLq+1(0,T;Lq+1(Ω)), (2.27) |∇u(k) t|q−1∇u(k) n,t⇀|∇ut|q−1∇utinLq+1 q(0,T;Lq+1 q(Ω)),(2.28) u(k) t|ˆΓ⇀ut|ˆΓinL2(0,T;L2(ˆΓ)). (2.29) It remains to show that usatisfies (2.10). For all w∈C∞(0,T;W1,q+1(Ω)) with w(T) = 0 we have /integraldisplayt 0/integraldisplay Ω/braceleftBig auttw+c2∇u·∇w+b/parenleftBig (1−δ)+δ|∇ut|q−1/parenrightBig ∇ut·∇w+futw/bracerightBig dxds +α/integraldisplayt 0/integraldisplay ˆΓutwdxds−/integraldisplayt 0/integraldisplay Γgwdxds =−/integraldisplayt 0/integraldisplay Ω[ut−u(k) t]/parenleftBig aw/parenrightBig tdxds−/integraldisplayt 0/integraldisplay Ωu(k) t/parenleftBig [a−ak]w/parenrightBig tdxds −/integraldisplay Ωu1/parenleftBig [a(0)−ak(0)]w(0)/parenrightBig dx+/integraldisplayt 0/integraldisplay Ωc2[∇u−∇u(k))]·∇wdxds +/integraldisplayt 0/integraldisplay Ωb(1−δ)[∇ut−∇u(k) t]·∇wdxds +/integraldisplayt 0/integraldisplay Ωbδ[|∇ut|q−1∇ut−|∇u(k) t|q−1∇u(k) t]·∇wdxds +/integraldisplayt 0/integraldisplay Ω[ut−u(k) t]fwdxds +/integraldisplayt 0/integraldisplay Ω[f−fk]u(k) twdxds +α/integraldisplayt 0/integraldisplay ˆΓ[ut−u(k) t]wdxds−/integraldisplayt 0/integraldisplay Γ[g−gk]wdxds→0 ask→ ∞, since we demanded that ak→ainL∞(0,T;L∞(Ω)),ak,t→atinL∞(0,T;L2(Ω)), fk→finL∞(0,T;L2(Ω)) andgk→ginLq+1 q(0,T;W−q q+1,q+1 q(Γ)). This relation proves that usolves (2.10). The weak limit then satisfies the estimate (2.5). 3. Uniqueness. To confirm uniqueness, note that the difference ˆ u=u1−u2 between any two weak solutions u1,u2of (2.2) is a weak solution of the problem aˆutt−c2∆ˆu−b(1−δ)∆ˆut−bδdiv/parenleftBig |∇u1 t|q−1∇u1 t−|∇u2 t|q−1∇u2 t/parenrightBig +fˆut= 0, c2∂ˆu ∂n+b(1−δ)∂ˆut ∂n+bδ(|∇u1 t|q−1∂u1 t ∂n−|∇u2 t|q−1∂u2 t ∂n) = 0 on Γ , αˆut+c2∂ˆu ∂n+b(1−δ)∂ˆut ∂n+bδ(|∇u1 t|q−1∂u1 t ∂n−|∇u2 t|q−1∂u2 t ∂n) = 0 on ˆΓ, (ˆu,ˆut)|t=0= (0,0). (2.30) Multiplication of (2.30) by ˆ utyields 1 2/bracketleftbigg/integraldisplay Ωa(ˆut)2dx+c2|∇ˆu|2 L2(Ω)/bracketrightbiggt 0+b(1−δ)/integraldisplayt 0/integraldisplay Ω|∇ˆut|2dxds+α/integraldisplayt 0|ˆut|2 L2(ˆΓ)ds12 V. NIKOLI ´C +/integraldisplayt 0/integraldisplay Ω(f−1 2at)(ˆut)2dxds≤0, since due to the inequality (1.19) we have bδ/integraldisplayt 0/integraldisplay Ω(|∇u1 t|q−1∇u1 t−|∇u2 t|q−1∇u2 t)·∇ˆutdxds≥0.(2.31) From here we conclude that ˆ ut= 0 and∇ˆu= 0 almost everywhere, which results in thesolutionbeingunique uptoanadditiveconstant. Theinitial condit ion ˆu|t=0= 0 provides us with uniqueness. 4. Higher energy estimate. To obtain higher order estimate (2.8), we will test (2.12) with wn=u(k) n,tt(t)∈Vnand then combine the result with the lower order estimate (2.5) we derived previously. Multiplication by u(k) n,tt(t) and integrationwith respect to time produces /integraldisplayt 0/integraldisplay Ωak(u(k) n,tt)2dxds+/bracketleftbiggb(1−δ) 2|∇u(k) n,t|2 L2(Ω)+bδ q+1|∇u(k) n,t|q+1 Lq+1(Ω)/bracketrightbiggt 0 +α 2/bracketleftBig/integraldisplay ˆΓ(u(k) n,t)2dx/bracketrightBigt 0 =c2/integraldisplayt 0|∇u(k) n,t|2 L2(Ω)ds−c2/bracketleftbigg/integraldisplay Ω∇u(k) n·∇u(k) n,tdx/bracketrightbiggt 0+/integraldisplayt 0/integraldisplay Γgku(k) n,ttdxds(2.32) −/integraldisplayt 0/integraldisplay Ωfku(k) n,tu(k) n,ttdxds. To estimate the boundary integral on the right side, we employ (1.13 ) to obtain /integraldisplayt 0/integraldisplay Γgku(k) n,ttdxds ≤Ctr 1|u(k) n,t(t)|W1,q+1(Ω)|gk(t)| W−q q+1,q+1 q(Γ)−/integraldisplay Γgk(0)u(k) n,t(0)dx −/integraldisplayt 0/integraldisplay Γgk,tu(k) n,tdxds ≤η/ba∇dbl∇u(k) n,t/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+ǫ0/ba∇dblu(k) n,t/ba∇dbl2 L∞(0,T;L2(Ω)) +C(η,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblgk/ba∇dblq+1 q L∞(0,T;W−q q+1,q+1 q(Γ)) +1 2ǫ0(Ctr 1CΩ 2)2/ba∇dblgk/ba∇dbl2 L∞(0,T;W−q q+1,q+1 q(Γ))+|u1,n|q+1 W1,q+1(Ω) +C(1,q+1)(Ctr 1|gk(0)| W−q q+1,q+1 q(Γ))q+1 q +C(ǫ1,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblgk,t/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +ǫ1/ba∇dbl∇u(k) n,t/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+1 2ǫ0(Ctr 1CΩ 2)2/ba∇dblgk,t/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)),(2.33) which together with /integraldisplayt 0/integraldisplay Ωfku(k) n,tu(k) n,ttdxds ≤1 2τ(CΩ H1,L4)4/ba∇dblfk/ba∇dbl2 L∞(0,T;H1(Ω))/bracketleftBig T/ba∇dblu(k) n,t/ba∇dbl2 L∞(0,T;L2(Ω))LOCAL EXISTENCE RESULTS 13 +/ba∇dbl∇u(k) n,t/ba∇dbl2 L2(0,T;L2(Ω))/bracketrightBig +τ 2/ba∇dblu(k) n,tt/ba∇dbl2 L2(0,T;L2(Ω)), (2.34) and taking esssup [0,T]in (2.32) leads to the estimate a−τ 2/ba∇dblu(k) n,tt/ba∇dbl2 L2(0,T;L2(Ω))+/parenleftBigb(1−δ) 4−σ/parenrightBig /ba∇dbl∇u(k) n,t/ba∇dbl2 L∞(0,T;L2(Ω)) +/parenleftBigbδ 2(q+1)−η/parenrightBig /ba∇dbl∇u(k) n,t/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+α 4/ba∇dblu(k) n,t/ba∇dbl2 L∞(0,T;L2(ˆΓ)) ≤c2/ba∇dbl∇u(k) n,t/ba∇dbl2 L2(0,T;L2(Ω))+σ|∇u1,n|2 L2(Ω)+c4 4σ(/ba∇dbl∇u(k) n/ba∇dbl2 L∞(0,T;L2(Ω)) +|∇u0,n|2 L2(Ω))+1 2τ(CΩ H1,L4)4˜b2[T/ba∇dblu(k) n,t/ba∇dbl2 L∞(0,T;L2(Ω)) +/ba∇dbl∇u(k) n,t/ba∇dbl2 L2(0,T;L2(Ω))]+η/ba∇dbl∇u(k) n,t/ba∇dblq+1 L∞(0,T;Lq+1(Ω)) +ǫ1/ba∇dbl∇u(k) n,t/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+ǫ0/ba∇dblu(k) n,t/ba∇dbl2 L∞(0,T;L2(Ω)) +1 2ǫ0(Ctr 1CΩ 2)2/parenleftBig /ba∇dblgk/ba∇dbl2 L∞(0,T;W−q q+1,q+1 q(Γ))+/ba∇dblgk,t/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ))/parenrightBig +|u1,n|q+1 W1,q+1(Ω)+C(1,q+1)(Ctr 1|gk(0)| W−q q+1,q+1 q(Γ))q+1 q +b(1−δ) 2|∇u1,n|2 L2(Ω)+α 2|u1,n|2 L2(ˆΓ)+bδ q+1|∇u1,n|q+1 Lq+1(Ω) +(Ctr 1(1+CP))q+1 q/parenleftBig C(ǫ1,q+1)/ba∇dblgk,t/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +C(η,q+1)/ba∇dblgk/ba∇dblq+1 q L∞(0,T;W−q q+1,q+1 q(Γ))/parenrightBig .(2.35) Since there are terms on the right side in (2.35) which cannot be domin ated by the terms on the left hand side, we need to also employ the lower estim ate (2.15). Adding (2.15) and µtimes (2.35) yields (2.8) with ureplaced by u(k) n, provided that we choose 0<τ <a,0<η<bδ 2(q+1),0<σ<b(1−δ) 4, 0<µ<min/braceleftBigb(1−δ) 2−(CΩ H1,L4)2ˆb 1 2τ(CΩ H1,L4)4˜b2+c2,a 4−(CΩ H1,L4)2ˆbT−ǫ0 ǫ0+1 2τ(CΩ H1,L4)4˜b2T,σ c2,bδ 2−ǫ1 ǫ1/bracerightBig ,(2.36) so that the coefficients in (2.8) are positive. Asbyassumption gk∈L∞(0,T;W−q q+1,q+1 q(Γ))andgk,t∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)),/parenleftbig u(k) n/parenrightbig n∈Nis a bounded sequence in X:=C1(0,T;W1,q+1(Ω))∩H2(0,T;L2(Ω)). We further obtain /parenleftbig u(k) n,t/parenrightbig ∈Nis uniformly bounded in L2(0,T;L2(Ω)), (2.37) /parenleftbig ∇u(k) n,t/parenrightbig ∈Nis uniformly bounded in Lq+1(0,T;Lq+1(Ω)), (2.38) |∇u(k) n,t|q−1∇u(k) n,tis uniformly bounded in Lq+1 q(0,T;Lq+1 q(Ω)) and (2.39) /parenleftbig u(k) n,t|ˆΓ/parenrightbig ∈Nis uniformly bounded in L2(0,T;L2(ˆΓ)), (2.40)14 V. NIKOLI ´C which are reflexive Banach spaces. From here, after proceeding as in the step 1.(c) and 2, we can conc lude that (2.10) has a unique solution u∈Xwhich satisfies the estimate (2.8). /square Let us now consider the boundary value problem (2.2) with an added lo wer order linear damping term: autt−c2∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig +βut +fut= 0 in Ω ×(0,T), c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gon Γ×(0,T), αut+c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T), (u,ut) = (u0,u1) onΩ×{t= 0},(2.41) whereβ >0. This is a linearized version of the problem (1.6) with nonlinearity appearing only through the damping term. The additionaly introduce dβ−lower order term will allow us to remove restrictions on final time Tin the estimates (2.5) and (2.8). Indeed, by testing the equation with utand integrating with respect to space and time, we obtain 1 2/bracketleftbigg/integraldisplay Ωa(ut)2dx+c2|∇u|2 L2(Ω)/bracketrightbiggt 0+b/integraldisplayt 0/integraldisplay Ω/parenleftBig (1−δ)+δ|∇ut|q−1/parenrightBig |∇ut|2dxds +β/integraldisplayt 0/integraldisplay Ω|ut|2dxds+α/integraldisplayt 0/integraldisplay ˆΓ|ut|2dxds ≤ˆb(CΩ H1,L4)2/integraldisplayt 0|ut|2 H1(Ω)ds+ǫ1/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) +C(ǫ1,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +ǫ0/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω)+1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)), which leads to the lower order energy estimate (a 4−ǫ0)/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+/parenleftBigb(1−δ) 2−ˆb(CΩ H1,L4)2/parenrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) +/parenleftBigbδ 2−ǫ1/parenrightBig /ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+(β 2−ˆb(CΩ H1,L4)2))/ba∇dblut/ba∇dbl2 L2(0,T;L2(Ω)) +c2 4/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) ≤a 2|u1|2 L2(Ω)+c2 2|∇u0|2 L2(Ω)+1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)) +C(ǫ1,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)),(2.42) provided that /ba∇dblf−1 2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb <min{β 2(CΩ H1,L4)2,b(1−δ) 2(CΩ H1,L4)2}and that 0<ǫ0<a 4, 0<ǫ1<bδ 2. Testing with uttand adding µtimes the obtained estimate to (2.42) results in theLOCAL EXISTENCE RESULTS 15 higher order energy estimate valid for arbitrary time: µa−τ 2/ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω))+µ/parenleftBigb(1−δ) 4−σ/parenrightBig /ba∇dbl∇ut/ba∇dbl2 L∞(0,T;L2(Ω)) +µ(a+µβ 4−ǫ0(µ+1))/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+ˇb/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) +µα 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(ˆΓ))+(bδ 2−µ(ǫ1+1))/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) +/parenleftBigβ 2−(CΩ H1,L4)2ˆb−µ 2τ(CΩ H1,L4)4˜b2/parenrightBig /ba∇dblut/ba∇dbl2 L2(0,T;L2(Ω)) +c2 4/parenleftBig 1−µc2 σ/parenrightBig /ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) +µ(bδ 2(q+1)−η)/ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω)) ≤C/parenleftBig CΓ(g)+|u1|2 H1(Ω)+|∇u0|2 L2(Ω)+|u1|q+1 W1,q+1(Ω)+|u1|2 L2(ˆΓ)/parenrightBig ,(2.43) withˇb=b(1−δ) 2−(CΩ H1,L4)2ˆb−µ(1 2τ(CΩ H1,L4)4˜b2+c2), for some appropriately chosen C >0. Therefore we obtain: Proposition 2.2. Letβ >0and the assumptions (i) in Proposition 2.1hold, with /ba∇dblf−1 2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<min{β 2(CΩ H1,L4)2,b(1−δ) 2(CΩ H1,L4)2}. Then(2.41)has a unique weak solution in ˜X, with˜Xdefined as in (2.4), which satisfies (2.42)for some sufficiently small constants ǫ0,ǫ1>0. If, in addition to (i), the assumptions (ii) in Proposition 2.1are satisfied, then u∈X, withXdefined as in (2.7), andusatisfies the energy estimate (2.43) for some sufficiently small constants ǫ0,ǫ1,µ,σ,τ > 0and some large enough C, independent of T. We continue with considering an equation with an added lower order no nlinear damping term: autt−c2∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig +γ|ut|q−1ut+fut = 0 in Ω ×(0,T], c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gon Γ×(0,T], αut+c2∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0},(2.44) withγ >0, which is motivated by the problem (1.7). Once we multiply (2.44) by utand integrate by parts, we produce 1 2/bracketleftbigg/integraldisplay Ωa(ut)2dx+c2|∇u|2 L2(Ω)/bracketrightbiggt 0+b/integraldisplayt 0/integraldisplay Ω/parenleftBig (1−δ)+δ|∇ut|q−1/parenrightBig |∇ut|2dxds +α/integraldisplayt 0/integraldisplay ˆΓ|ut|2dxds+γ/integraldisplayt 0/integraldisplay Ω|ut|q+1dxds =/integraldisplayt 0/integraldisplay Ω(f−1 2at)(ut)2dxds+/integraldisplayt 0/integraldisplay Γgutdxds.16 V. NIKOLI ´C We will make use of the following inequality /integraldisplayt 0/integraldisplay Γgutdxds≤ǫ0 2/ba∇dblut/ba∇dblq+1 Lq+1(0,T;W1,q+1(Ω)) +C(ǫ0 2,q+1)(Ctr 1/ba∇dblg/ba∇dbl Lq+1 q(0,T,W−q q+1,q+1 q(Γ)))q+1 q,(2.45) and, forq>1, /integraldisplayt 0/integraldisplay Ω(f−1 2at)(ut)2dxds≤/integraldisplayt 0/parenleftBig/integraldisplay Ω|ut|q+1dx/parenrightBig2 q+1/parenleftBig/integraldisplay Ω|f−1 2at|q+1 q−1/parenrightBigq−1 q+1ds =/integraldisplayt 0|ut|2 Lq+1(Ω)|f−1 2at| Lq+1 q−1(Ω)ds ≤ǫ0 2/ba∇dblut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+C(ǫ0 2,q+1 2)/ba∇dblf−1 2at/ba∇dblq+1 q−1 Lq+1 q−1(0,T;Lq+1 q−1(Ω)), to obtain lower order energy estimate a 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+c2 4/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))+b(1−δ) 2/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) +bδ−ǫ0 2/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+(γ 2−ǫ0)/ba∇dblut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) +α 2/ba∇dblut/ba∇dbl2 L2(ˆΓ) ≤C(ǫ0 2,q+1)(Ctr 1/ba∇dblg/ba∇dbl Lq+1 q(0,T;W−q q+1,q+1 q(Γ)))q+1 q+a 2|u1|2 L2(Ω) (2.46) +C(ǫ0 2,q+1 2)/ba∇dblf−1 2at/ba∇dblq+1 q−1 Lq+1 q−1(0,T;Lq+1 q−1(Ω))+c2 2|∇u0|2 L2(Ω), assuming that f,at∈Lq+1 q−1(0,T;Lq+1 q−1(Ω)) and 0 <ǫ0<γ 2. For obtaining higher order estimate, we multiply (2.44) with utt, integrate with respect to space and time and make use of the estimate /integraldisplayt 0/integraldisplay Γguttdxds≤η/ba∇dblut/ba∇dblq+1 L∞(0,T;W1,q+1(Ω))+ǫ0 2/ba∇dblut/ba∇dblq+1 Lq+1(0,T;W1,q+1(Ω)) +C(η,q+1)(Ctr 1/ba∇dblg/ba∇dbl L∞(0,T,W−q q+1,q+1 q(Γ)))q+1 q (2.47) +|u1|q+1 W1,q+1(Ω)+C(1,q+1)(Ctr 1|g(0)| W−q q+1,q+1 q(Γ))q+1 q +C(ǫ0 2,q+1)(Ctr 1/ba∇dblgt/ba∇dbl Lq+1 q(0,T,W−q q+1,q+1 q(Γ)))q+1 q. In order to avoid dependence on time, we approach estimate (2.34) differently this time: by employing the embedding Lq+1(Ω)֒→L4(Ω), valid for q≥3, we obtain /integraldisplayt 0/integraldisplay Ωfututtdxds≤1 2τ(CΩ H1,L4)2(CΩ Lq+1,L4)2/integraldisplayt 0|f(s)|2 H1(Ω)|ut(s)|2 Lq+1(Ω)ds +τ 2/ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω)) ≤ǫ0 2/ba∇dblut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+τ 2/ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω)) +C(ǫ0 2,q+1 2)(1 2τ(CΩ H1,L4CΩ Lq+1,L4)2/ba∇dblf/ba∇dbl2 L2(q+1) q−1(0,T;H1(Ω)))q+1 q−1,LOCAL EXISTENCE RESULTS 17 which, together with (2.46), leads to the higher order estimate µa−τ 2/ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω))+µ/parenleftBigb(1−δ) 4−σ/parenrightBig /ba∇dbl∇ut/ba∇dbl2 L∞(0,T;L2(Ω)) +a 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+c2 4/parenleftBig 1−µc2 σ/parenrightBig /ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω)) +µα 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(ˆΓ))+/parenleftBigbδ 2−ǫ0 2(µ+1)/parenrightBig /ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) +/parenleftBigb(1−δ) 2−µc2/parenrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) +/parenleftBigγ 2−ǫ0(µ+1)/parenrightBig /ba∇dblut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) +µ/parenleftBigbδ 2(q+1)−η/parenrightBig /ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω)) +µ/parenleftBigγ 2(q+1)−η/parenrightBig /ba∇dblut/ba∇dblq+1 L∞(0,T;Lq+1(Ω)) ≤C/parenleftBig1/summationdisplay s=0/ba∇dblds dtsg/ba∇dblq+1 q Lq+1 q(0,T,W−q q+1,q+1 q(Γ))+/ba∇dblg/ba∇dblq+1 q L∞(0,T,W−q q+1,q+1 q(Γ)) +/ba∇dblf−1 2at/ba∇dblq+1 q Lq+1 q(0,T;Lq+1 q(Ω))+/ba∇dblf/ba∇dbl2(q+1) q−1 L2(q+1) q−1(0,T;H1(Ω))+|u1|2 H1(Ω) +|∇u0|2 L2(Ω)+|u1|q+1 W1,q+1(Ω)+|u1|2 L2(ˆΓ)/parenrightBig ,(2.48) assumingf∈L2(q+1) q−1(0,T;H1(Ω) and choosing τ,σ,η,µ> 0 to be sufficiently small. Proposition 2.3. LetT >0,c2,b,γ >0,α≥0,δ∈(0,1),q>1and (i)•a∈L∞(0,T;L∞(Ω)),at∈Lq+1 q−1(0,T;Lq+1 q−1(Ω)),0<a≤a(t,x)≤a, •f∈Lq+1 q−1(0,T;Lq+1 q−1(Ω)), •g∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), •u0∈H1(Ω),u1∈L2(Ω). Then(2.44)has a unique weak solution in ˜X, with˜Xdefined as in (2.4), which satisfies (2.46)for some 0<ǫ0<γ 2. If, in addition to (i), the following assumptions are satisfi ed (ii)•q≥3, •f∈L2(q+1) q−1(0,T;H1(Ω)), •g∈L∞(0,T;W−q q+1,q+1 q(Γ)),gt∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), •u1∈W1,q+1(Ω), then(2.44)has a unique weak solution in X, withXdefined as in (2.4), which satisfies the energyestimate (2.48)for some sufficiently small constants µ,σ,τ,η> 0 and some large enough C >0, independent of T. Remark2.4. Duetotheterms /ba∇dblf−1 2at/ba∇dblq+1 q−1 Lq+1 q−1(0,T;Lq+1 q−1(Ω))and/ba∇dblf/ba∇dbl2(q+1) q L2(q+1) q−1(0,T;H1(Ω)) appearing on the right hand side in the estimate (2.48), we will not be a ble to prove local well-posedness of the problem (1.9) by employing this estimate. Instead, pro- vided the assumptions (ii) in Proposition 2.1 hold, we could proceed with the same estimates as in the proof of that proposition, and for evaluating bo undary integrals18 V. NIKOLI ´C apply (2.45) and (2.47), to obtain the following energy estimate: µa−τ 2/ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω))+µ/parenleftBigb(1−δ) 4−σ/parenrightBig /ba∇dbl∇ut/ba∇dbl2 L∞(0,T;L2(Ω)) +/parenleftBiga 4−(CΩ H1,L4)2ˆbT−µ1 2τ(CΩ H1,L4)4˜b2T/parenrightBig /ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω)) +c2 4/parenleftBig 1−µc2 σ/parenrightBig /ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))+µα 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(ˆΓ)) +ˇb/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+/parenleftBigbδ 2−ǫ0(µ+1)/parenrightBig /ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) +(γ 2−ǫ0(µ+1))/ba∇dblut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) +µ/parenleftBigbδ 2(q+1)−η/parenrightBig /ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω)) +µ/parenleftBigγ 2(q+1)−η/parenrightBig /ba∇dblut/ba∇dblq+1 L∞(0,T;Lq+1(Ω)) ≤C/parenleftBig1/summationdisplay s=0/ba∇dblds dtsg/ba∇dblq+1 q Lq+1 q(0,T,W−q q+1,q+1 q(Γ))+/ba∇dblg/ba∇dblq+1 q L∞(0,T,W−q q+1,q+1 q(Γ)) +|u1|2 L2(ˆΓ)+|u1|2 H1(Ω)+|∇u0|2 L2(Ω)+|u1|q+1 W1,q+1(Ω)/parenrightBig ,(2.49) withˇb=b(1−δ) 2−(CΩ H1,L4)2ˆb−µ(1 2τ(CΩ H1,L4)4˜b2+c2), for some appropriately chosen constantsτ,σ,ǫ0,η,µ>0 and large enough C, independent of T. RelyingonProposition2.1, wecannowprovethelocalwell-posedness forthebound- ary value problem (2.1). Theorem 2.5. Letc2,b >0,α≥0,δ∈(0,1),k∈R,q > d−1,q≥1, g∈L∞(0,T;W−q q+1,q+1 q(Γ)),gt∈Lq+1 q(0,T;H−q q+1,q+1 q(Γ)). For anyT >0there is aκT>0such that for all u0,u1∈W1,q+1(Ω), with CΓ(g)+|u0|2 L1(Ω)+|∇u0|2 Lq+1(Ω)+|u1|2 H1(Ω)+|∇u0|2 L2(Ω)+|u1|q+1 W1,q+1(Ω) +|u1|2 L2(ˆΓ)≤κ2 T (2.50) there exists a unique weak solution u∈ Wof(2.1), where W={v∈X:/ba∇dblvtt/ba∇dblL2(0,T;L2(Ω))≤m ∧/ba∇dblvt/ba∇dblL∞(0,T;H1(Ω))≤m ∧/ba∇dbl∇vt/ba∇dblLq+1(0,T;Lq+1(Ω))≤M ∧(v,vt)|t=0= (u0,u1)},(2.51) with 2|k|CΩ W1,q+1,L∞/bracketleftBig max{1+CP,CΩ 1}κT+(1+CP)Tq q+1M+CΩ 2Tm)/bracketrightBig <1,(2.52) andmandMsufficiently small, where CΓ(g)is defined as in (2.9). Proof.We will carry out the proof by using a fixed point argument. We define an operator T:W →X,v/ma√sto→ Tv=u, whereusolves (2.10) with a= 1−2kv, f=−2kvt. (2.53)LOCAL EXISTENCE RESULTS 19 We will show that assumptions of Proposition 2.1 are satisfied. Since v∈ W, and q>d−1 so we can make use of the embedding W1,q+1(Ω)֒→L∞(Ω), we have by (1.17) |2kv(x,t)| ≤2|k|CΩ W1,q+1,L∞/bracketleftBig max{1+CP,CΩ 1,}κT+(1+CP)Tq q+1M +CΩ 2Tm/bracketrightBig , andat=−2kvt∈L∞(0,T;L2(Ω). It follows that 0 <a= 1−a0<a<a= 1+a0, where a0= 2|k|CΩ W1,q+1,L∞/bracketleftBig max{1+CP,CΩ 1}κT+(1+CP)Tq q+1M+CΩ 2Tm/bracketrightBig . Furthermore, /ba∇dblf−1 2at/ba∇dblL∞(0,T;L2(Ω))=/ba∇dblkvt/ba∇dblL∞(0,T;L2(Ω))≤ |k|m, /ba∇dblf/ba∇dblL∞(0,T;H1(Ω))= 2|k|/ba∇dblvt/ba∇dblL∞(0,T;H1(Ω))≤2|k|m. Hence the higher order energy estimate (2.8) holds and by choosing m,M >0 such that 2|k|CΩ W1,q+1,L∞((1+CP)Tq q+1M+CΩ 2Tm)<1, m<1 |k|min/braceleftBigb(1−δ) 2(CΩ H1,L4)2,a 4T(CΩ H1,L4)2/bracerightBig , and making the bound κTon initial and boundary data small enough κT<1 max{1+CP,CΩ 1}/parenleftBig1 2|k|CΩ W1,q+1,L∞−(1+CP)Tq q+1M−CΩ 2Tm/parenrightBig , κ2 T≤1 Cmin/braceleftBig/parenleftBiga 4−(CΩ H1,L4)2ˆbT−ǫ0(µ+1)−µ1 2τ(CΩ H1,L4)4˜b2T/parenrightBig m2, µa−τ 2m2,µ/parenleftBigb(1−δ) 4−σ/parenrightBig m2,/parenleftBigbδ 2−ǫ1(µ+1)/parenrightBig Mq+1/bracerightBig , we achieve that u∈ W, with constants ǫ0,ǫ1,τ,η,σ,µ chosen as in (2.16) and (2.36) andCas in (2.8). In order to prove contractivity, consider vi∈ W,ui=Tvi∈ W,i= 1,2 and denote ˆu=u1−u2,ˆv=v1−v2. Subtracting the equation (2.2) for u1andu2yields: (1−2kv1)ˆutt−c2∆ˆu−b(1−δ)∆ˆut−bδdiv/parenleftBig |∇u1 t|q−1∇u1 t−|∇u2 t|q−1∇u2 t/parenrightBig , = 2k(ˆvu2 tt+v1 tˆut+ ˆvtu2 t) in Ω, c2∂ˆu ∂n+b(1−δ)∂ˆut ∂n+bδ(|∇u1 t|q−1∂u1 t ∂n−|∇u2 t|q−1∂u2 t ∂n) = 0 on Γ , αˆut+c2∂ˆu ∂n+b(1−δ)∂ˆut ∂n+bδ(|∇u1 t|q−1∂u1 t ∂n−|∇u2 t|q−1∂u2 t ∂n) = 0 on ˆΓ, (ˆu,ˆut)|t=0= (0,0).(2.54) After testing (2.54) with ˆ utand making use of the inequality (2.31), we obtain 1 2/bracketleftBig/integraldisplay Ω(1−2kv1)(ˆut)2dx+c2|∇ˆu|2 L2(Ω)/bracketrightBigt 0+b(1−δ)/integraldisplayt 0|∇ˆut|2 L2(Ω)ds +α/integraldisplayt 0/integraldisplay ˆΓ|ˆut|2dxds20 V. NIKOLI ´C ≤2|k|/integraldisplayt 0/integraldisplay Ω(1 2v1 t(ˆut)2+ ˆvu2 ttˆut+ ˆvtu2 tˆut)dxds, and therefore we have 1 2/bracketleftBig/integraldisplay Ω(1−2kv1)(ˆut)2dx+c2|∇ˆu|2 L2(Ω)/bracketrightBigt 0+b(1−δ)/integraldisplayt 0|∇ˆut|2 L2(Ω)ds +α/integraldisplayt 0/integraldisplay ˆΓ|ˆut|2dxds ≤ |k|(CΩ H1,L4)2/parenleftBig /ba∇dblv1 t/ba∇dblL∞(0,T;L2(Ω))/integraldisplayt 0|ˆut|2 H1(Ω)ds +/ba∇dblu2 tt/ba∇dblL2(0,T;L2(Ω))[/ba∇dblˆv/ba∇dbl2 L∞(0,T;H1(Ω))+/integraldisplayt 0|ˆut|2 H1(Ω)ds] +/ba∇dblu2 t/ba∇dblL∞(0,T;L2(Ω))[/ba∇dblˆvt/ba∇dbl2 L2(0,T;H1(Ω))+/integraldisplayt 0|ˆut|2 H1(Ω)ds]/parenrightBig . Utilizing the fact that v1,v2,u1,u2∈ Wand the inequalities /ba∇dbl∇ˆv/ba∇dbl2 L∞(0,T,L2(Ω))≤ T/ba∇dbl∇ˆvt/ba∇dbl2 L2(0,T,L2(Ω)),/ba∇dblˆv/ba∇dbl2 L∞(0,T,L2(Ω))≤T/ba∇dblˆvt/ba∇dbl2 L2(0,T,L2(Ω))and/ba∇dblˆvt/ba∇dbl2 L2(0,T;L2(Ω))≤ T/ba∇dblˆvt/ba∇dbl2 L∞(0,T;L2(Ω))leads to 1−a0 4/ba∇dblˆut/ba∇dbl2 L∞(0,T;L2(Ω))+c2 4/ba∇dbl∇ˆu/ba∇dbl2 L∞(0,T;L2(Ω))+b(1−δ) 2/ba∇dbl∇ˆut/ba∇dbl2 L2(0,T;L2(Ω)) ≤ |k|(CΩ H1,L4)2m/parenleftBig 3(T/ba∇dblˆut/ba∇dbl2 L∞(0,T;L2(Ω))+/ba∇dbl∇ˆut/ba∇dbl2 L2(0,T;L2(Ω)))+T2/ba∇dblˆvt/ba∇dbl2 L∞(0,T;L2(Ω)) +T/ba∇dbl∇ˆvt/ba∇dbl2 L2(0,T;L2(Ω))+T/ba∇dblˆvt/ba∇dbl2 L∞(0,T;L2(Ω))+/ba∇dbl∇ˆvt/ba∇dbl2 L2(0,T;L2(Ω))/parenrightBig . It follows that /parenleftBig1−a0 4−3T|k|(CΩ H1,L4)2m/parenrightBig /ba∇dblˆut/ba∇dbl2 L∞(0,T;L2(Ω))+c2 4/ba∇dbl∇ˆu/ba∇dbl2 L∞(0,T;L2(Ω)) +/parenleftBigb(1−δ) 2−3|k|(CΩ H1,L4)2m/parenrightBig /ba∇dbl∇ˆut/ba∇dbl2 L2(0,T;L2(Ω)) ≤ |k|(CΩ H1,L4)2m(T+1)max {1,T}/parenleftBig /ba∇dblˆvt/ba∇dbl2 L∞(0,T;L2(Ω))+/ba∇dbl∇ˆvt/ba∇dbl2 L2(0,T;L2(Ω))) and altogether we have min{1−a0 4−3T|k|(CΩ H1,L4)2m,b(1−δ) 2−3|k|(CΩ H1,L4)2m,c2 4}|||u|||2 ≤ |k|(CΩ H1,L4)2m(T+1)max {1,T}|||v|||2,(2.55) where|||u|||2=/ba∇dblˆut/ba∇dbl2 L∞(0,T;L2(Ω))+/ba∇dbl∇ˆut/ba∇dbl2 L2(0,T;L2(Ω))+/ba∇dbl∇ˆu/ba∇dbl2 L∞(0,T;L2(Ω)). We concludefrom(2.55)that Tisacontractionwithrespecttothe norm |||·|||, provided thatmis sufficiently small. This, together with the self-mapping property an dW being closed, provides existence and uniqueness of a solution. /square Relying on Proposition 2.2 we can obtain local well-posedness for the p roblem (1.6)withβ >0. Sinceweneedtoavoiddegeneracyoftheterm1 −2kuandtherefore make use of the estimate (1.17) to get the condition (2.52), we cann ot completely avoid restriction on final time in the fully nonlinear equation. Inspect ing the proof of Theorem 2.5 immediately yields: Theorem 2.6. Letβ >0and the assumptions of Theorem 2.5hold. For any T >0 there is aκT>0such that for all u0,u1∈W1,q+1(Ω), with(2.50), there exists aLOCAL EXISTENCE RESULTS 21 unique weak solution u∈ Wof(2.1), whereWis defined as in (2.51), with(2.52) andmandMsufficiently small. For obtaining well-posedness for the problem (1.7) with γ >0, we cannot rely on estimates in Proposition 2.3 to prove self-mapping of the fixed-po int operator T, instead we make use of (2.49); therefore restrictions on final tim e persist in the nonlinear equation. Analogously to Theorem 2.5 we obtain: Theorem 2.7. Letγ >0and the assumptions of Theorem 2.5hold. For any T >0 there is aκT>0such that for all u0,u1∈W1,q+1(Ω), with 1/summationdisplay s=0/ba∇dblds dtsg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ))+/ba∇dblg/ba∇dblq+1 q L∞(0,T;W−q q+1,q+1 q(Γ))+|u0|2 L1(Ω) +|∇u0|2 Lq+1(Ω)+|u1|2 H1(Ω)+|∇u0|2 L2(Ω)+|u1|q+1 W1,q+1(Ω)+|u1|2 L2(ˆΓ)≤κ2 T,(2.56) there exists a unique weak solution u∈ Wof(2.1), whereWis defined as in (2.51), with(2.52), andmandMsufficiently small. 3.Acoustic-acoustic coupling We will now consider the problem of an acoustic-acoustic coupling whic h can be modeled by the equation with coefficients varying in space (1.10). We w ill make the following assumptions on the coefficients in (1.10): λ,̺,b,δ,k ∈L∞(Ω), ∃λ,λ,̺,̺: 0<λ≤λ(x)≤λ,0<̺≤̺(x)≤̺,in Ω, ∃b,b,δ,δ: 0<b≤b(x)≤b,0<δ≤δ(x)≤δ<1 in Ω.(3.1) Wecanagainfirstinspecttheproblemwithnonlinearitypresentonlyin thedamping term: autt−div(1 ̺(x)∇u)−div/parenleftBig b(x)(((1−δ(x))+δ(x)|∇ut|q−1)∇ut/parenrightBig +fut= 0 in Ω ×(0,T], 1 ̺(x)∂u ∂n+b(x)((1−δ(x))+δ(x)|∇ut|q−1)∂ut ∂n=gon Γ×(0,T], α(x)ut+1 ̺(x)∂u ∂n+b(x)((1−δ(x))+δ(x)|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0}. Analogously to Propositon 2.1 and Theorem 2.5 we obtain Corollary 3.1. Let the assumptions (3.1)and the assumptions (i) in Propositon 2.1be satisfied, with /ba∇dblf−1 2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<min/braceleftBigb(1−δ) 2(CΩ H1,L4)2,a 4T(CΩ H1,L4)2/bracerightBig . Then(3.2)has a weak solution u∈˜X, with˜Xdefined as in (2.4), which satisfies the energy estimate /bracketleftBiga 4−ˆb(CΩ H1,L4)2T−ǫ0/bracketrightBig /ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+c2 4/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω)) +/bracketleftBigb(1−δ) 2−ˆb(CΩ H1,L4)2/bracketrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))22 V. NIKOLI ´C +/bracketleftBigbδ 2−ǫ1/bracketrightBig /ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+α 2/ba∇dblu(k) n,t/ba∇dbl2 L2(0,T;L2(ˆΓ)) ≤a 2|u1|2 L2(Ω)+c2 2|∇u0|2 L2(Ω)+1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)) +C(ǫ1,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), for some sufficiently small constants ǫ0,ǫ1>0. If, additionally, assumptions (ii) of Proposition 2.1hold, thenu∈X, whereXis defined as in (2.7), and satisfies the energy estimate (2.8), whereαis replaced with α,b(1−δ)withb(1−δ),bδwithbδandc2 2(1−µc2 σ)withc2 4−µc4 4σ, for some small enough constants τ,η,σ,µ> 0and some large enough C >0, independent of T. Corollary 3.2. Letg∈L∞(0,T;W−q q+1,q+1 q(Γ)),gt∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), q > d−1,q≥1and assumptions (3.1)be satisfied. For any T >0there is a κT>0such that for all u0,u1∈W1,q+1(Ω), with(2.50), there exists a unique weak solutionu∈ Wof(1.10), whereWis defined as in (2.51), with(2.52)with|k| replaced by |k|L∞(Ω), andmandMsufficiently small. 4.Westervelt’s equation in the formulation (1.8) We begin with the problem (1.8) in the case β= 0: (1−2ku)utt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut = 2k(ut)2in Ω×(0,T], c2∂u ∂n+c2ε|∇u|q−1∂u ∂n+b∂ut ∂n=gon Γ×(0,T], αut+c2∂u ∂n+c2ε|∇u|q−1∂u ∂n+b∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0}.(4.1) We will first study the problem with the nonlinearityappearing only thr ough damp- ing: autt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut+fut= 0 in Ω ×(0,T], c2∂u ∂n+c2ε|∇u|q−1∂u ∂n+b∂ut ∂n=gon Γ×(0,T], αut+c2∂u ∂n+c2ε|∇u|q−1∂u ∂n+b∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0}.(4.2) Proposition 4.1. LetT >0,c2,b,ε>0,α≥0,δ∈(0,1),q≥1and assume that •a∈L∞(0,T;L∞(Ω)),at∈L∞(0,T;L2(Ω)),0<a<a(x,t)<a, •f∈L∞(0,T;L2(Ω)), •g∈L2(0,T;H−1/2(Γ)), •u0∈W1,q+1(Ω),u1∈L2(Ω), with /ba∇dblf−1 2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<min{a 4T(CΩ H1,L4)2,b 2(CΩ H1,L4)2}. Then(4.2)has a weak solution u∈˜X:=C1(0,T;L2(Ω))∩C(0,T;W1,q+1(Ω))∩H1(0,T;H1(Ω)),(4.3)LOCAL EXISTENCE RESULTS 23 which satisfies the energy estimate /bracketleftBiga 4−(Ctr 2)2τ−Tˆb(CΩ H1,L4)2/bracketrightBig /ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+c2 4/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω)) +/bracketleftBigb 2−(Ctr 2)2τ−ˆb(CΩ H1,L4)2/bracketrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) +c2ε 2(q+1)/ba∇dbl∇u/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) ≤1 4τ(/ba∇dblg/ba∇dbl2 L1(0,T;H−1/2(Γ))+/ba∇dblg/ba∇dbl2 L2(0,T;H−1/2(Γ)))+a 2|u1|2 L2(Ω) +c2 2|∇u0|2 L2(Ω)+c2ε q+1|∇u0|q+1 Lq+1(Ω),(4.4) for some constant 0<τ <min/braceleftBigb−2ˆb(CΩ H1,L4)2 2(Ctr 2)2,a−4(CΩ H1,L4)2ˆbT 4(Ctr 2)2/bracerightBig . Proof.The proof follows along the line of the standard Galerkin approximatio n method. Here we will focus on deriving the energy estimate. The weak form of the problem is given as follows: /integraldisplay Ω/braceleftBig auttw+c2(∇u+ε|∇u|q−1∇u)·∇w+b∇ut·∇w/bracerightBig dx+α/integraldisplay ˆΓutwdx =−/integraldisplayt 0/integraldisplay Ωfutwdx+/integraldisplay Γgwdx, ∀w∈W1,q+1(Ω).(4.5) Testing (4.5) with utand integrating with respect to space and time yields /bracketleftBig1 2/integraldisplay Ωa(ut)2dx+c2 2|∇u|2 L2(Ω)+c2ε q+1|∇u|q+1 Lq+1(Ω)/bracketrightBigt 0 +b/integraldisplayt 0|∇ut(s)|2 L2(Ω)ds+α/integraldisplayt 0|ut(s)|2 L2(ˆΓ)ds =/integraldisplayt 0/integraldisplay Ω(f−1 2at)(ut)2dxds+/integraldisplayt 0/integraldisplay Γgutdxds ≤ /ba∇dblf−1 2at/ba∇dblL∞(0,T;L2(Ω))/integraldisplayt 0|ut(s)|2 L4(Ω)ds+/integraldisplayt 0/integraldisplay Γgutdxds.(4.6) By taking esssup [0,T]in (4.6) and making use of the embedding H1(Ω)֒→L4(Ω) and estimating the boundary integral in the following way /integraldisplayt 0/integraldisplay Γgutdxds≤τ(Ctr 2)2/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+1 4τ/ba∇dblg/ba∇dbl2 L2(0,T;H−1/2(Γ)) +τ(Ctr 2)2/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+1 4τ/ba∇dblg/ba∇dbl2 L1(0,T;H−1/2(Γ)), we obtain a 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+b 2/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+c2ε 2(q+1)/ba∇dbl∇u/ba∇dblq+1 L∞(0,T;Lq+1(Ω)) +c2 4/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) ≤ˆb(CΩ H1,L4)2/ba∇dblut/ba∇dbl2 L2(0,T;H1(Ω))+τ(Ctr 2)2(/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)))24 V. NIKOLI ´C +1 4τ(/ba∇dblg/ba∇dbl2 L2(0,T;H−1/2(Γ))+/ba∇dblg/ba∇dbl2 L1(0,T;H−1/2(Γ)))+a 2|u1|2 L2(Ω)+c2 2|∇u0|2 L2(Ω) +c2ε q+1|∇u0|q+1 Lq+1(Ω), which leads to (4.4). /square If we consider an equation with an added linear lower order damping te rm autt−c2div(∇u+ε|∇u|q−1∇u)−b∆ut+βut+fut= 0 in Ω ×(0,T], c2∂u ∂n+c2ε|∇u|q−1∂u ∂n+b∂ut ∂n=gonΓ×(0,T], αut+c2∂u ∂n+c2ε|∇u|q−1∂u ∂n+b∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0},(4.7) we will be able to obtain an energy estimate valid for arbitrary time: Proposition 4.2. Letβ >0and the assumptions in Proposition 4.1hold with /ba∇dblf−1 2at/ba∇dblL∞(0,T;L2(Ω))≤ˆb<1 2(CΩ H1,L4)2min{b,β}. Then(4.7)has a weak solution in ˜X, defined as in (4.3), which satisfies the energy estimate a 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+/bracketleftBigb 2−(Ctr 2)2τ−ˆb(CΩ H1,L4)2/bracketrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) +/bracketleftBigβ 2−(Ctr 2)2τ−ˆb(CΩ H1,L4)2/bracketrightBig /ba∇dblut/ba∇dbl2 L2(0,T;L2(Ω))+c2 4/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω)) +c2ε 2(q+1)/ba∇dbl∇u/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) ≤1 4τ/ba∇dblg/ba∇dbl2 L2(0,T;H−1/2(Γ))+a 2|u1|2 L2(Ω)+c2 2|∇u0|2 L2(Ω)+c2ε q+1|∇u0|q+1 Lq+1(Ω),(4.8) for some constant 0<τ <min/braceleftBigb−2ˆb(CΩ H1,L4)2 2(Ctr 2)2,β−2(CΩ H1,L4)2ˆb 2(Ctr 2)2/bracerightBig . We now proceed to the question of local existence of weak solutions for the problem (4.1). Theorem 4.3. Letc2,b >0,α≥0,δ∈(0,1),k∈R,q > d−1,q≥1, g∈L2(0,T;H−1/2(Γ)). For anyT >0there is aκT>0such that for all u0∈ W1,q+1(Ω),u1∈L2(Ω)with /ba∇dblg/ba∇dbl2 L2(0,T;H−1/2(Γ))+/ba∇dblg/ba∇dbl2 L1(0,T;H−1/2(Γ))+|u1|2 L2(Ω)+|∇u0|2 L2(Ω)+|∇u0|q+1 Lq+1(Ω) +|u0|2 L1(Ω)≤κ2 T there exists a weak solution u∈ Wof(4.1)where W={v∈˜X:/ba∇dblvt/ba∇dblL∞(0,T;L2(Ω))≤m ∧/ba∇dbl∇vt/ba∇dblL2(0,T;L2(Ω))≤m ∧/ba∇dbl∇v/ba∇dblL∞(0,T;Lq+1(Ω))≤M},(4.9) with 2|k|CΩ W1,q+1,L∞[CΩ 1κT+(1+CP)M+CΩ 2Tm]<1, (4.10)LOCAL EXISTENCE RESULTS 25 andmandMare sufficiently small. Proof.We define an operator T:W →˜X,v/ma√sto→ Tv=u, whereusolves (4.2) with a= 1−2kv, f=−2kvt. (4.11) Proposition 4.1 will allow us to prove that Tis a self-mapping. The assumptions of the proposition are satisfied, since for v∈ Wbecause of (1.16) we have 0<a= 1−a0<a(x,t)<a= 1+a0, where a0= 2|k|CΩ W1,q+1,L∞[CΩ 1κT+(1+CP)M+CΩ 2Tm], and by (4.9) /ba∇dblf−1 2at/ba∇dblL∞(0,T;L2(Ω))=/ba∇dblkvt/ba∇dblL∞(0,T;L2(Ω))≤ |k|m. The energy estimate (4.4) holds and we can conclude that for any m,M >0 such that 2|k|CΩ W1,q+1,L∞((1+CP)M+CΩ 2Tm)<1, m≤1 |k|min{a 4(CΩ H1,L4)2,b 2(CΩ H1,L4)2}, and under the assumption on smallness of initial and boundary data κT<1 CΩ 1/parenleftBig1 2|k|CΩ W1,q+1,L∞−(1+CP)M−CΩ 2Tm/parenrightBig , κ2 T≤min{1 C(a 4−Tˆb(CΩ H1,L4)2−(Ctr 2)2τ)m2, 1 C(b 2−ˆb(CΩ H1,L4)2−(Ctr 2)2τ)m2,1 Cc2ε 2(q+1)Mq+1}, whereC= max{1 4τ,a 2,c2 2,c2ε q+1}, operator Tmaps into W. SinceWisclosedandboundedinthedualofaseparableBanachspace, Wisweakly- star compact. Existence of solutions then results from a compact ness argument (see Theorem 6.1, [2]): the sequence of fixed point iterates undefined byun=Tun−1, (1−2kun−1)un tt−c2div(∇un+ǫ|∇un|q−1∇un)−b∆un t+βun t= 2kun−1 tun t, withu0chosen to be compatible with initial and boundary conditions, hasa we akly- starconvergentsubsequencewhosew ∗-limit ¯uliesinW. Thislimitisaweaksolution of the problem since /integraldisplayT 0/integraldisplay Ω/braceleftbig (1−2k¯u)¯uttφ+[c2(∇¯u+ε|∇¯u|q−1∇¯u)+b∇¯ut]·∇φ−2k(¯ut)2φ +βutφ/bracerightbig dxds+α/integraldisplayT 0/integraldisplay ˆΓ¯utφdxds−/integraldisplayT 0/integraldisplay Γgφdxds =/integraldisplayT 0/integraldisplay Ω/braceleftbig −¯ut((1−2k¯u)φ)t+[c2(∇¯u+ε|∇¯u|q−1∇¯u)+b∇¯ut]·∇φ−2k(¯ut)2φ +βutφ/bracerightbig dxds+α/integraldisplayT 0/integraldisplay ˆΓ¯utφdxds−/integraldisplayT 0/integraldisplay Γgφdxds =/integraldisplayT 0/integraldisplay Ω/braceleftBig −(¯u−un)t((1−2k¯u)φ)t+2kun t((¯u−un−1)φ)t −2k(¯ut−un t)¯utφ−2k(¯ut−un−1 t)un tφ+β(¯ut−un t)φ+[c2∇(¯u−un)26 V. NIKOLI ´C +b∇(¯u−un)t]·∇φ+c2ε/integraldisplay1 0|∇(un+σˆu)|q−3[|∇(un+σˆu)|2∇ˆu +(q−1)(∇(un+σˆu)·∇ˆu)∇(un+σˆu)]dσ·∇φ/bracerightBig dxds +α/integraldisplayT 0/integraldisplay ˆΓ(¯u−un)φdxds→0 ask→ ∞, for anyφ∈C∞ 0((0,T)×Ω), where ˆu= ¯u−un. /square Relying on Proposition 4.2, we can also achieve short-time existence o f solutions for the problem (1.8), with β >0. Due to the estimate (1.16) and therefore bound (4.10), the dependency on final time Tcannot be completely avoided. Theorem 4.4. Let the assumptions of Theorem 4.3hold andβ >0. For anyT >0 there is aκT>0such that for all u0∈W1,q+1(Ω),u1∈L2(Ω)with /ba∇dblg/ba∇dbl2 L2(0,T;H−1/2(Γ))+|u1|2 L2(Ω)+|∇u0|2 L2(Ω)+|∇u0|q+1 Lq+1(Ω)+|u0|2 L1(Ω)≤κ2 T, there exists a weak solution u∈ Wof(1.8), whereWis defined as in (4.9), with (4.10), andmandMare sufficiently small. Note that here, as in the case of homogeneous Dirichlet boundary c onditions, the uniqueness remains an open problem due to the presence of q−Laplace damping term which hinders the derivation of higher order energy estimates . For details, the reader is refered to Remark 8, [2]. 5.Westervelt’s equation in the formulation (1.9) We begin with investigations of the problem (1.9) in the case γ= 0: utt−c2 1−2˜kut∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig = 0 in Ω ×(0,T], c2 1−2˜kut∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gon Γ×(0,T], αut+c2 1−2˜kut∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0}.(5.1) We will once again first consider an equation with the nonlinearity only a ppearing through the damping term: utt−a∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig = 0 in Ω ×(0,T], a∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gonΓ×(0,T], αut+a∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0}.(5.2) Proposition 5.1. LetT >0,b>0,α≥0,δ∈(0,1)and assume that •a∈L2(0,T;L∞(Ω)),∇a∈L2(0,T;L2(Ω)), •g∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), •u0∈H1(Ω),u1∈L2(Ω), •q>d−1,q≥1,LOCAL EXISTENCE RESULTS 27 /braceleftBiggˆb:=b(1−δ) 2−T 2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))−(√ T+1 2)/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))>0, ˜b:=1 4−2T(CΩ W1,q+1,L∞CΩ 2)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0,forq>1,(5.3) ˆb:=b 2−(T 2+(CΩ H1,L∞)2)/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω)) −(√ T+1 2)/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))>0, ˜b:=1 4−T(CΩ H1,L∞)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0,forq= 1.(5.4) Then(5.2)has a weak solution u∈˜X:=C1(0,T;L2(Ω))∩W1,q+1(0,T;W1,q+1(Ω))}, (5.5) which, forq>1, satisfies the energy estimate ˆb/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+/parenleftBig ˜b−ǫ0/parenrightBig /ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω) +(bδ 2−ǫ1)/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) ≤/parenleftBig /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBig1 2|∇u0|2 L2(Ω)+1 2|u1|2 L2(Ω) +C(ǫ1 2,q+1 2)T/parenleftBig (CΩ W1,q+1,L∞(1+CP))2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBigq+1 q−1 +C(ǫ1 2,q+1)(CΩ W1,q+1,L∞(1+CP))q+1 q/ba∇dblg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)),(5.6) and forq= 1satisfies /parenleftBig ˆb−ǫ0/parenrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+/parenleftBig ˜b−ǫ0/parenrightBig /ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω)+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) ≤/parenleftBig /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBig1 2|∇u0|2 L2(Ω)+1 2|u1|2 L2(Ω) +1 4ǫ0(Ctr 2)2(/ba∇dblg/ba∇dbl2 L1(0,T;H−1/2(Γ))+/ba∇dblg/ba∇dbl2 L2(0,T;H−1/2(Γ))), for some sufficiently small constants ǫ0,ǫ1>0. Proof.We will focus on acquiring crucial energy estimates. Testing the pro blem withutand integrating with respect to space and time leads to 1 2/bracketleftBig |ut(s)|2 L2(Ω)/bracketrightBigt 0+/integraldisplayt 0/parenleftBig b(1−δ)|∇ut(s)|2 L2(Ω) +bδ|∇ut(s)|q+1 Lq+1(Ω)+α|ut(s)|2 L2(ˆΓ)/parenrightBig ds =/integraldisplayt 0/integraldisplay Ω/parenleftBig −a∇ut·∇u−ut∇a·∇u/parenrightBig dxds+/integraldisplayt 0/integraldisplay Γgutdxds ≤/integraldisplayt 0/parenleftBig |a(s)|L∞(Ω)|∇ut(s)|L2(Ω)+|∇a(s)|L2(Ω)|ut(s)|L∞(Ω)/parenrightBig ·/bracketleftBig |∇u0|L2(Ω)+/radicalBigg s/integraldisplays 0|∇ut(σ)|2 L2(Ω)dσ/bracketrightBig ds+/integraldisplayt 0/integraldisplay Γgutdxds ≤/parenleftBig /ba∇dbla/ba∇dblL2(0,t;L∞(Ω))/ba∇dbl∇ut/ba∇dblL2(0,t;L2(Ω))28 V. NIKOLI ´C +/ba∇dbl∇a/ba∇dblL2(0,t;L2(Ω))/radicalBigg/integraldisplayt 0|ut(s)|2 L∞(Ω)ds/parenrightBig ·/bracketleftBig |∇u0|L2(Ω)+√ t/ba∇dbl∇ut/ba∇dblL2(0,t;L2(Ω))/bracketrightBig +/integraldisplayt 0/integraldisplay Γgutdxds (5.7) ≤ /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))/parenleftBig1 2/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+1 2|∇u0|2 L2(Ω) +√ T/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))/parenrightBig +/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/integraldisplayT 0|ut(s)|2 L∞(Ω)ds +/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenleftBig1 2|∇u0|2 L2(Ω)+1 2T/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))/parenrightBig +ǫ1 2/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+ǫ0/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω)) +C(ǫ1 2,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)), where we have applied (2.14) to estimate the boundary integral on t he right side. We can make use of the embedding W1,q+1(Ω)֒→L∞(Ω) together with the in- equality (1.13) to obtain /integraldisplayT 0|ut(s)|2 L∞(Ω)ds≤(CΩ W1,q+1,L∞)2/integraldisplayT 0|ut(s)|2 W1,q+1(Ω)ds ≤(CΩ W1,q+1,L∞)2/integraldisplayt 0/parenleftBig (1+CP)|∇ut(s)|Lq+1(Ω)+CΩ 2|ut(s)|L2(Ω)/parenrightBig2 ds ≤2(CΩ W1,q+1,L∞)2(1+CP)2/integraldisplayt 0|∇ut(s)|2 Lq+1(Ω)ds +2(CΩ W1,q+1,L∞CΩ 2)2/ba∇dblut/ba∇dbl2 L2(0,T;L2(Ω)), and then from (5.7), for q>1, we further get 1 2/bracketleftBig |ut|2 L2(Ω)/bracketrightBigt 0+/integraldisplayt 0/parenleftBig b(1−δ)|∇ut(s)|2 L2(Ω)+bδ|∇ut(s)|q+1 Lq+1(Ω)ds/parenrightBig +α/integraldisplayt 0|ut(s)|2 L2(ˆΓ)ds ≤ /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))/parenleftBig (√ T+1 2)/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+1 2|∇u0|2 L2(Ω)/parenrightBig +ǫ1/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+ǫ0/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω)) +C(ǫ1 2,q+1 2)T((CΩ W1,q+1,L∞(1+CP))2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω)))q+1 q−1 (5.8) +2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))(CΩ W1,q+1,L∞CΩ 2)2/ba∇dblut/ba∇dbl2 L2(0,T;L2(Ω)) +/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenleftBig1 2T/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+1 2|∇u0|2 L2(Ω)/parenrightBig +C(ǫ1 2,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)),LOCAL EXISTENCE RESULTS 29 for someǫ0,ǫ1>0. By taking esssup [0,T]in (5.8) and making ǫ0andǫ1small enough we gain (5.6). /square Proposition 5.2. LetT >0,b>0,α≥0,δ∈(0,1)and assume that •a(t,x)≥a>0, •a∈L∞(0,T;L∞(Ω)),at∈L2(0,T;L2(Ω)),∇a∈L2(0,T;L4(Ω)), •g∈L∞(0,T;W−q q+1,q+1 q(Γ)),gt∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)), •u0∈W1,4(Ω),u1∈W1,q+1(Ω), •q≥3, with ˜a:=a 4−1 2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))>0, ˜b:=1 4−/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T(CΩ Lq+1,L4CΩ 2)2>0,(5.9) then(5.2)has a weak solution u∈X:=C1(0,T;W1,q+1(Ω))∩H2(0,T;L2(Ω))}, (5.10) which satisfies the energy estimate µ/bracketleftBig1 2−τ/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/bracketrightBig /ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω)) +µb(1−δ) 8/ba∇dbl∇ut/ba∇dbl2 L∞(0,T;L2(Ω))+/bracketleftBig ˜b−ǫ0(µ+1)/bracketrightBig /ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω) +/bracketleftBigb(1−δ) 2−µ/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/bracketrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) +/bracketleftBig µbδ 2(q+1)−η(2µ+1)/bracketrightBig /ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+µα 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(ˆΓ)) +/bracketleftBig ˜a−µ2 b(1−δ)/ba∇dbla/ba∇dbl2 L∞(0,T;L∞(Ω))/bracketrightBig /ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω)) +/bracketleftBigbδ 2−ǫ1(µ+1)/bracketrightBig /ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ))(5.11) ≤C/parenleftBig (T/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1 q−1+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))|∇u0|2 L2(Ω) +(/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))|∇u0|2 L4(Ω) +/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig |∇u0|2 L2(Ω)+|∇u1|L2(Ω)|∇u0|L2(Ω)/parenrightBig +((1 2+T3/4)√ T/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω)))q+1 q−1+(T2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1 q−1 +(T5/2/ba∇dblat/ba∇dblL2(0,T;L2(Ω)))q+1 q−1+/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√ T|∇u0|2 L4(Ω) +|u1|2 H1(Ω)+|u1|q+1 W1,q+1(Ω)+|u1|2 L2(ˆΓ)+CΓ(g)/parenrightBig , for some sufficiently small constants ǫ0,ǫ1,η,µ,τ > 0, some large enough C >0 andCΓ(g)defined as in (2.9). Proof.In order to obtain higher order estimate, we will multiply (5.2) first by ut, proceeding differently than in Proposition 5.1, and then by uttand combine the two obtained estimates. Multiplication by utand integration with respect to space and30 V. NIKOLI ´C time produces 1 2/bracketleftBig |ut(s)|2 L2(Ω)+|√a∇u|2 L2(Ω)/bracketrightBigt 0+/integraldisplayt 0/parenleftBig b(1−δ)|∇ut(s)|2 L2(Ω) +bδ|∇ut(s)|q+1 Lq+1(Ω)/parenrightBig ds+α/integraldisplayt 0|ut(s)|2 L2(ˆΓ)ds =/integraldisplayt 0/integraldisplay Ω/parenleftBig1 2at|∇u|2−ut∇a·∇u/parenrightBig dxds+/integraldisplayt 0/integraldisplay Γgutdxds ≤/integraldisplayt 0/integraldisplay Ω1 2at|∇u|2dxds+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/parenleftBigT 2/ba∇dblut/ba∇dbl2 L∞(0,T;L4(Ω)) (5.12) +1 2/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))/parenrightBig +ǫ1/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) +C(ǫ1,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +ǫ0/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω)+1 4ǫ0(Ctr 1CΩ 2)2/ba∇dblg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)), for someǫ0,ǫ1>0. We will make use of the embedding Lq+1(Ω)֒→L4(Ω) and Young’s inequality (1.18) to estimate 1 2/integraldisplayt 0/integraldisplay Ωat|∇u|2dxds ≤1 2/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L4(Ω))/integraldisplayt 0|at|L2(Ω)ds ≤1 2/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√ T/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L4(Ω)) ≤ /ba∇dblat/ba∇dblL2(0,T;L2(Ω))√ T/bracketleftBig T2/ba∇dbl∇ut/ba∇dbl2 L∞(0,T;L4(Ω))+|∇u0|2 L4(Ω)/bracketrightBig ≤ /ba∇dblat/ba∇dblL2(0,T;L2(Ω))√ T/bracketleftBig T2(CΩ Lq+1,L4)2/ba∇dbl∇ut/ba∇dbl2 L∞(0,T;Lq+1(Ω))+|∇u0|2 L4(Ω)/bracketrightBig ≤η 2/ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√ T|∇u0|2 L4(Ω) +C(η 2,q+1 2)(/ba∇dblat/ba∇dblL2(0,T;L2(Ω))T5/2(CΩ Lq+1,L4)2)q+1 q−1,(5.13) for someη>0 andq>1. We can also obtain /ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T 2/ba∇dblut/ba∇dbl2 L∞(0,T;L4(Ω)) ≤ /ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T 2(CΩ Lq+1,L4)2/ba∇dblut/ba∇dbl2 L∞(0,T;Lq+1(Ω)) ≤ /ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T(CΩ Lq+1,L4)2/bracketleftBig C2 P/ba∇dbl∇ut/ba∇dbl2 L∞(0,T;Lq+1(Ω)) +(CΩ 2)2/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))/bracketrightBig ≤η 2/ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+C(η 2,q+1 2)((CPCΩ Lq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T)q+1 q−1 +/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T(CΩ Lq+1,L4CΩ 2)2/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω)), which together with (5.13) results in the following estimate: (˜b−ǫ0)/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+˜a/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ))LOCAL EXISTENCE RESULTS 31 +b(1−δ) 2/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+(bδ 2−ǫ1)/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) ≤η/ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√ T|∇u0|2 L4(Ω) +C(η 2,q+1 2)((CPCΩ Lq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T)q+1 q−1 (5.14) +C(η 2,q+1 2)(/ba∇dblat/ba∇dblL2(0,T;L2(Ω))T5/2(CΩ Lq+1,L4)2)q+1 q−1 +1 2/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))|∇u0|2 L2(Ω)+1 4ǫ0/ba∇dblg/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)) +1 2|u1|2 L2(Ω)+C(ǫ1,q+1)(Ctr 1(1+CP)/ba∇dblg/ba∇dbl Lq+1 q(0,T;W−q q+1,q+1 q(Γ)))q+1 q. Testing with uttyields /integraldisplayt 0|utt(s)|2 L2(Ω)ds+/bracketleftbiggb(1−δ) 2|∇ut|2 L2(Ω)+bδ q+1|∇ut|q+1 Lq+1(Ω)+α 2|ut|2 L2(ˆΓ)/bracketrightbiggt 0 =/integraldisplayt 0/integraldisplay Ω(−a∇utt·∇u−utt∇a·∇u)dxds+/integraldisplayt 0/integraldisplay Γguttdxds =/integraldisplayt 0/integraldisplay Ω/parenleftbig at∇ut·∇u+a|∇ut|2−utt∇a·∇u/parenrightbig dxds−/bracketleftbigg/integraldisplay Ωa∇ut·∇udx/bracketrightbiggt 0 +/integraldisplayt 0/integraldisplay Γguttdxds ≤/integraldisplayt 0/parenleftbig |at(s)|L2(Ω)|∇ut(s)|L4(Ω)+|∇a(s)|L4(Ω)|utt(s)|L2(Ω)/parenrightbig ·/bracketleftBig |∇u0|L4(Ω)+4/radicalBigg s3/integraldisplays 0|∇ut(σ)|4 L4(Ω)dσ/bracketrightBig ds +/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig |∇ut(t)|L2(Ω)|∇u(t)|L2(Ω)+|∇u1|L2(Ω)|∇u0|L2(Ω) +/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))/parenrightBig +/integraldisplayt 0/integraldisplay Γguttdxds ≤ /ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))/parenleftbigg (1 2+T3 4)/ba∇dbl∇ut/ba∇dbl2 L4(0,T;L4(Ω))+1 2|∇u0|2 L4(Ω)/parenrightbigg +/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))(τ/ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω))+1 2τ|∇u0|2 L4(Ω) +1 2τT3 2/ba∇dbl∇ut/ba∇dbl2 L4(0,T;L4(Ω)))+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig |∇u1|L2(Ω)|∇u0|L2(Ω) +/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))/parenrightBig +2 b(1−δ)/ba∇dbla/ba∇dbl2 L∞(0,T;L∞(Ω))|∇u(t)|2 L2(Ω) +b(1−δ) 8|∇ut(t)|2 L2(Ω)+/integraldisplayt 0/integraldisplay Γguttdxds, for someτ >0. We can make use of Young’s inequality and the inequality (2.33) for the boundary integral together with the inequality /ba∇dbl∇ut/ba∇dblL4(0,T;L4(Ω))≤ T1 4/ba∇dbl∇ut/ba∇dblL∞(0,T;L4(Ω))≤T1 4CΩ Lq+1,L4/ba∇dbl∇ut/ba∇dblL∞(0,T;Lq+1(Ω))to obtain /integraldisplayt 0|utt(s)|2 L2(Ω)ds+/bracketleftbiggb(1−δ) 2|∇ut|2 L2(Ω)+bδ q+1|∇ut|q+1 Lq+1(Ω)+α 2|ut|2 L2(ˆΓ)/bracketrightbiggt 032 V. NIKOLI ´C ≤ /ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))1 2|∇u0|2 L4(Ω)+η 2/ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω)) +C(η 2,q+1 2)((1 2+T3 4)√ T(CΩ Lq+1,L4)2/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω)))q+1 q−1 +/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/parenleftBig τ/ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω))+1 2ǫ|∇u0|2 L4(Ω)/parenrightBig +η 2/ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+C(η 2,q+1 2)(1 2τT2(CΩ Lq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1 q−1 +/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig |∇u1|L2(Ω)|∇u0|L2(Ω)+/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))/parenrightBig +2 b(1−δ)/ba∇dbla/ba∇dbl2 L∞(0,T;L∞(Ω))|∇u(t)|2 L2(Ω)+b(1−δ) 8|∇ut(t)|2 L2(Ω) +η/ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+C(η,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblg/ba∇dblq+1 q L∞(0,T;W−q q+1,q+1 q(Γ)) +ǫ0/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+1 2ǫ0(Ctr 1CΩ 2)2/ba∇dblg/ba∇dbl2 L∞(0,T;W−q q+1,q+1 q(Γ)) +|u1|q+1 W1,q+1(Ω)+C(1,q+1)(Ctr 1|g(0)| W−q q+1,q+1 q(Γ)))q+1 q +C(ǫ1,q+1)(Ctr 1(1+CP))q+1 q/ba∇dblgt/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ)) +ǫ1/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+1 2ǫ0(Ctr 1CΩ 2)2/ba∇dblgt/ba∇dbl2 L1(0,T;W−q q+1,q+1 q(Γ)), which, by taking essential supremum with respect to tand then adding µtimes obtained inequality to (5.14) results in the higher order estimate (5.1 1). /square Let us now consider the problem with the added lower order nonlinear damping term utt−a∆u−bdiv/parenleftBig ((1−δ)+δ|∇ut|q−1)∇ut/parenrightBig +γ|ut|q−1ut= 0 in Ω ×(0,T], a∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n=gonΓ×(0,T], αut+a∂u ∂n+b((1−δ)+δ|∇ut|q−1)∂ut ∂n= 0 onˆΓ×(0,T], (u,ut) = (u0,u1) onΩ×{t= 0},(5.15) whereγ >0. This is a linearized version of (1.9), where nonlinearity appears only through the damping terms. We can utilize the embedding W1,q+1(Ω)֒→ L∞(Ω), Young’s inequality in the form (1.18) and estimate the boundary integral by employing (2.45), to obtain 1 2/bracketleftBig |ut(s)|2 L2(Ω)/bracketrightBigt 0+/integraldisplayt 0/parenleftBig b(1−δ)|∇ut(s)|2 L2(Ω)+bδ|∇ut(s)|q+1 Lq+1(Ω) +γ|ut(s)|q+1 Lq+1(Ω)+α|ut(s)|2 L2(ˆΓ)/parenrightBig ds ≤ /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))/parenleftBig√ T/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+1 2/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) (5.16) +1 2|∇u0|2 L2(Ω)/parenrightBig +ǫ0/integraldisplayT 0|ut(s)|q+1 W1,q+1(Ω)dsLOCAL EXISTENCE RESULTS 33 +C(ǫ0 2,q+1 2)T((CΩ W1,q+1,L∞)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω)))q+1 q−1 +/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenleftBig1 2T/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+1 2|∇u0|2 L2(Ω)/parenrightBig +C(ǫ0 2,q+1 2)(Ctr 1/ba∇dblg/ba∇dbl Lq+1 q(0,T;W−q q+1,q+1 q(Γ)))q+1 q, for someǫ0>0 andq>1,q>d−1. By taking esssup [0,T]in (5.16) we get ˆb/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+1 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) +(bδ 2−ǫ0)/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+(γ 2−ǫ0)/ba∇dblut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) ≤/parenleftBig /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBig1 2|∇u0|2 L2(Ω)+1 2|u1|2 L2(Ω) +C(ǫ0 2,q+1 2)T((CΩ W1,q+1,L∞)2/ba∇dbl∇a/ba∇dblL∞(0,T;L2(Ω)))q+1 q−1 +C(ǫ0 2,q+1 2)(Ctr 1/ba∇dblg/ba∇dbl Lq+1 q(0,T;W−q q+1,q+1 q(Γ)))q+1 q,(5.17) for some 0<ǫ0<1 2min{bδ,γ}andˆb>0 defined as in (5.3). Note that the addition of the lower order damping term allows us to re move the second assumption in (5.3) on smallness of a. In the case of q= 1 (andd= 1),usatisfies /parenleftBig ˆb−ǫ0/parenrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+1 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω) +/parenleftBig ˜b−ǫ0/parenrightBig /ba∇dblut/ba∇dbl2 L2(0,T;L2(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) ≤/parenleftBig /ba∇dbla/ba∇dblL2(0,T;L∞(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))/parenrightBig1 2|∇u0|2 L2(Ω)+1 2|u1|2 L2(Ω) +1 4ǫ0(Ctr 2)2/ba∇dblg/ba∇dbl2 L2(0,T;H−1/2(Γ)),(5.18) where˜b:=γ 2−(CΩ H1,L∞)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0, andˆbis defined as in (5.4). To obtain higher order estimate, we test the problem again by utand integrate with respect to space and time to obtain 1 2/bracketleftBig |ut|2 L2(Ω)+|√a∇u|2 L2(Ω)/bracketrightBigt 0+/integraldisplayt 0/parenleftBig b(1−δ)|∇ut|2 L2(Ω)+bδ|∇ut|q+1 Lq+1(Ω) +γ|ut|q+1 Lq+1(Ω)/parenrightBig ds+α/integraldisplayt 0/integraldisplay ˆΓ|ut|2 L2(ˆΓ)dxds =/integraldisplayt 0/integraldisplay Ω/parenleftBig1 2at|∇u|2−ut∇a·∇u/parenrightBig dxds+/integraldisplayt 0/integraldisplay Γgutdxds (5.19) ≤/integraldisplayt 0/integraldisplay Ω1 2at|∇u|2dxds+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/parenleftBigT 2/ba∇dblut/ba∇dbl2 L∞(0,T;L4(Ω)) +1 2/ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))/parenrightBig +ǫ1/ba∇dblut/ba∇dblq+1 Lq+1(0,T;W1,q+1(Ω))ds +C(ǫ1,q+1 2)(Ctr 1/ba∇dblg/ba∇dbl Lq+1 q(0,T;W−q q+1,q+1 q(Γ)))q+1 q.34 V. NIKOLI ´C Taking esssup [0,T]in (5.19) and making use of (5.13) and /ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T 2/ba∇dblut/ba∇dbl2 L∞(0,T;L4(Ω)) ≤η/ba∇dblut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+C(η,q+1 2)((CΩ Lq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))T)q+1 q−1, leads to the estimate 1 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω))+/parenleftBiga 4−1 2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))/parenrightBig /ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω)) +b(1−δ) 2/ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω))+(bδ 2−ǫ1)/ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) +(γ 2−ǫ1)/ba∇dblut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) ≤η/ba∇dblut/ba∇dblq+1 L∞(0,T;W1,q+1(Ω))+C(η,q+1 2)(/ba∇dblat/ba∇dblL2(0,T;L2(Ω))T5/2(CΩ Lq+1,L4)2)q+1 q−1(5.20) +/ba∇dblat/ba∇dblL2(0,T;L2(Ω))√ T|∇u0|2 L4(Ω)+1 2/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))|∇u0|2 L2(Ω) +C(η,q+1 2)(T(CΩ Lq+1,L4)2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1 q−1+1 2|u1|2 L2(Ω) +C(ǫ1 2,q+1 2)(Ctr 1/ba∇dblg/ba∇dbl Lq+1 q(0,T;W−q q+1,q+1 q(Γ)))q+1 q, for someη>0. Testing with uttand proceeding as in the case of γ= 0, with the use of (2.47) for the estimation of the boundary integral, results in the higher order energy estimate µ/parenleftBig1 2−τ/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/parenrightBig /ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω))+1 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω) +/parenleftBigbδ 2−ǫ1(µ+1)/parenrightBig /ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)) +/parenleftBigb(1−δ) 2−µ/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) +/parenleftBig µbδ 2(q+1)−µ(η+1)/parenrightBig /ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) +/parenleftBig ˜a−µ2 b(1−δ)/ba∇dbla/ba∇dbl2 L∞(0,T;L∞(Ω))/parenrightBig /ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω)) +(γ 2−µ(ǫ1+1))/ba∇dblut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω)+µb(1−δ) 8/ba∇dbl∇ut/ba∇dbl2 L∞(0,T;L2(Ω))(5.21) +/parenleftBig µγ 2(q+1)−η(2µ+1)/parenrightBig /ba∇dblut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+µα 2/ba∇dblut/ba∇dbl2 L∞(0,T;L2(ˆΓ)) ≤C/parenleftBig (T/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1 q−1+/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))|∇u0|2 L2(Ω) +(/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))+/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))|∇u0|2 L4(Ω) +/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/parenleftBig |∇u0|2 L2(Ω)+|∇u1|L2(Ω)|∇u0|L2(Ω)/parenrightBig +((1 2+T3/4)√ T/ba∇dblat/ba∇dblL4/3(0,T;L2(Ω)))q+1 q−1+|u1|2 H1(Ω)+|u1|q+1 W1,q+1(Ω) +(T2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω)))q+1 q−1+(T5/2/ba∇dblat/ba∇dblL2(0,T;L2(Ω)))q+1 q−1+|u1|2 L2(ˆΓ) +1/summationdisplay s=0/ba∇dblds dtsg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ))+/ba∇dblg/ba∇dblq+1 q L∞(0,T;W−q q+1,q+1 q(Γ))/parenrightBigLOCAL EXISTENCE RESULTS 35 with ˜adefined in (5.9), for some sufficiently small constants ǫ1,η,µ,τ > 0 and some large enough C >0. Note that here the second assumption in (5.9) on smallness of awas not needed. Proposition 5.3. LetT >0,b>0,α≥0,δ∈(0,1),γ >0and let the assump- tions in Proposition 5.1hold with b(1−δ) 2−(√ T+1 2)/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))−T 2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0,forq>1, ˆb=b 2−(T 2+(CΩ H1,L∞)2)/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))−(√ T+1 2)/ba∇dbla/ba∇dblL2(0,T;L∞(Ω))>0, ˜b=γ 2−(CΩ H1,L∞)2/ba∇dbl∇a/ba∇dblL2(0,T;L2(Ω))>0,forq= 1. Then(5.15)has a weak solution u∈˜X, with˜Xdefined as in (5.5), which satisfies the energy estimate (5.17)forq>1and estimate (5.18)forq= 1. If the assumptions in Proposition 5.2are satisfied with ˜a=a 4−1 2/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))>0, thenu∈X, withXas in(5.10), and satisfies the energy estimate (5.21). We will now proceed to investigate existence of solutions for (5.1). Theorem5.4. Letc2,b>0,α≥0,δ∈(0,1),˜k∈R,q≥3,g∈L∞(0,T;W−q q+1,q+1 q(Γ)), gt∈Lq+1 q(0,T;W−q q+1,q+1 q(Γ)). There exist κ >0,T >0such that for all u0∈W1,4(Ω),u1∈W1,q+1(Ω), with CΓ(g)+|∇u0|2 L2(Ω)+|u1|2 H1(Ω)+|u1|q+1 W1,q+1(Ω)+|u1|2 L2(ˆΓ)≤κ2 there exists a weak solution u∈ Wof(5.1)where W={v∈X:/ba∇dblvtt/ba∇dblL2(0,T;L2(Ω))≤m ∧/ba∇dblvt/ba∇dblL∞(0,T;H1(Ω))≤m ∧/ba∇dbl∇vt/ba∇dblL∞(0,T;Lq+1(Ω))≤M},(5.22) withmandMsufficiently small, and CΓ(g)is defined as in (2.9). Proof.We define an operator T:W →X,v/ma√sto→ Tv=uwhereusolves (5.2) with a=c2 1−2˜kvt. (5.23) From (1.14), we obtain for v∈ W /ba∇dbl2˜kvt/ba∇dblL∞(0,T;L∞(Ω))≤2|˜k|CΩ W1,q+1,L∞/bracketleftBig (1+CP)M+CΩ 2m/bracketrightBig , and, assuming 2 |˜k|CΩ W1,q+1,L∞/bracketleftBig (1 +CP)M+CΩ 2m/bracketrightBig <1, we can verify hypothesis of Proposition 5.2: a(t,x)≥c2 1+2|˜k|/ba∇dblvt/ba∇dblL∞(0,T;L∞(Ω)) ≥c2 1+2|˜k|CΩ W1,q+1,L∞((1+CP)M+CΩ 2m):=a, /ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))≤c2 1−2|˜k|/ba∇dblvt/ba∇dblL∞(0,T;L∞(Ω))36 V. NIKOLI ´C ≤c2 1−2|˜k|CΩ W1,q+1,L∞/bracketleftBig (1+CP)M+CΩ 2m/bracketrightBig, /ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))=/ba∇dbl2˜kc2 (1−2˜kvt)2∇vt/ba∇dblL2(0,T;L4(Ω)) ≤2|˜k|c2 (1−2|˜k|CΩ W1,q+1,L∞((1+CP)M+CΩ 2m))2CΩ Lq+1,L4√ TM, /ba∇dblat/ba∇dblL4/3(0,T;L2(Ω))=/ba∇dbl2˜kc2 (1−2˜kvt)2vtt/ba∇dblL4/3(0,T;L2(Ω)) ≤2|˜k|c2 (1−2|˜k|CΩ W1,q+1,L∞((1+CP)M+CΩ 2m))24√ Tm, /ba∇dblat/ba∇dblL2(0,T;L2(Ω))≤2|˜k|c2 (1−2|˜k|CΩ W1,q+1,L∞((1+CP)M+CΩ 2m))2m. It follows that assumptions are satisfied provided m,M,κandTare sufficiently small such that 2|˜k|CΩ W1,q+1,L∞/bracketleftBig (1+CP)M+CΩ 2m/bracketrightBig <1, 2|˜k|c2 (1−2|˜k|CΩ W1,q+1,L∞((1+CP)M+CΩ 2m))2CΩ Lq+1,L4√ TM ≤c2 2(1+2|˜k|CΩ W1,q+1,L∞((1+CP)M+CΩ 2m)),and 2|˜k|c2T3/2M(CΩ Lq+1,L4)3(CΩ 2)2 (1−2|˜k|CΩ W1,q+1,L∞((1+CP)M+CΩ 2m))2<1 4. Therefore the energy estimate (5.11) is satisfied and we have µ/bracketleftBig1 2−τ/ba∇dbl∇a/ba∇dblL2(0,T;L4(Ω))/bracketrightBig /ba∇dblutt/ba∇dbl2 L2(0,T;L2(Ω))+µb(1−δ) 8/ba∇dbl∇ut/ba∇dbl2 L∞(0,T;L2(Ω)) +/bracketleftBigbδ 2−ǫ1(µ+1)/bracketrightBig /ba∇dbl∇ut/ba∇dblq+1 Lq+1(0,T;Lq+1(Ω))+/bracketleftBig ˜b−ǫ0(µ+1)/bracketrightBig /ba∇dblut/ba∇dbl2 L∞(0,T;L2(Ω) +/bracketleftBigb(1−δ) 2−µ/ba∇dbla/ba∇dblL∞(0,T;L∞(Ω))/bracketrightBig /ba∇dbl∇ut/ba∇dbl2 L2(0,T;L2(Ω)) +/bracketleftBig ˜a−µ2 b(1−δ)/ba∇dbla/ba∇dbl2 L∞(0,T;L∞(Ω))/bracketrightBig /ba∇dbl∇u/ba∇dbl2 L∞(0,T;L2(Ω))+α 2/ba∇dblut/ba∇dbl2 L2(0,T;L2(ˆΓ)) +/bracketleftBig µbδ 2(q+1)−η(2µ+1)/bracketrightBig /ba∇dbl∇ut/ba∇dblq+1 L∞(0,T;Lq+1(Ω))+µα 4/ba∇dblut/ba∇dbl2 L∞(0,T;L2(ˆΓ)) ≤˜C/parenleftBig (T√ TM)q+1 q−1+|∇u0|2 L2(Ω)+|u1|2 H1(Ω)+(T5/2M)q+1 q−1 +(4√ Tm+√ Tm+√ TM)|∇u0|2 L4(Ω)+|u1|q+1 W1,q+1(Ω)+|u1|2 L2(ˆΓ) +((1 2+T3/4)T3/4m)q+1 q−1+(T5/2m)q+1 q−1+CΓ(g)/parenrightBig , for some large enough ˜C, and hence if Tand the bound κare sufficiently small, and we choose mandMappropriately, Tis a self-mapping. Since Wis closed, we obtain existence of solutions through compactness argument. /squareLOCAL EXISTENCE RESULTS 37 RelyingonProposition5.3, wecanobtainlocalexistenceofsolutionsf ortheproblem (1.9) withγ >0. Theorem 5.5. Let the assumptions of Theorem 5.4hold andγ >0. There exist κ>0,T >0such that for all u0∈W1,4(Ω),u1∈W1,q+1(Ω), with 1/summationdisplay s=0/ba∇dblds dtsg/ba∇dblq+1 q Lq+1 q(0,T;W−q q+1,q+1 q(Γ))+/ba∇dblg/ba∇dblq+1 q L∞(0,T;W−q q+1,q+1 q(Γ))) +|∇u0|2 L2(Ω)+|u1|2 H1(Ω)+|u1|q+1 W1,q+1(Ω)+|u1|2 L2(ˆΓ)≤κ2 there exists a weak solution u∈ Wof(1.9), whereWis defined as in (5.22), and mandMare sufficiently small. Due to the presence of q−Laplace damping term, the derivation of energy es- timates is possible only for multipliers of lower order (see Remark 4, [2]) and the question of uniqueness remains open. Acknowledgments. The author thanks Barbara Kaltenbacher for many fruitful discussions and comments. The financial support by the FWF (Aust rian Science Fund) under grant P24970 is gratefully acknowledged. References [1] A. Bamberger, R. Glowinski and Q.H. Tran, A domain decomp osition method for the acoustic wave equation with discontinuous coefficients and grid chang e,SIAM J. Numer. Anal. ,34 (1997), 603–639. [2] R. Brunnhuber, B. Kaltenbacher and P. Radu, Relaxation o f regularity for the Westervelt equation by nonlinear damping with application in acoustic -acoustic and elastic-acoustic cou- pling,Evol. Eq. Control Theory , to appear [3] L. C. Evans, Partial Differential Equations , American Mathematical Society, Providence, 1998. [4] M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics , Academic Press, New York, 1997. [5] B. Kaltenbacher, Boundary observability and stabiliza tion for Westervelt type wave equations without interior damping, Applied Mathematics and Optimization /62 (2010), 381–410. [6] B. Kaltenbacher and I. Lasiecka, Global existence and ex ponential decay rates for the West- ervelt equation, Discrete and Continuous Dynamical Systems Series S ,2(2009), 503–525. [7] B. Kaltenbacher, I. Lasiecka and S. Veljovi´ c, Well-pos edness and exponential decay for the Westervelt equation with inhomogeneous Dirichlet boundar y data, J. Escher et al (Eds): Progress in Nonlinear Differential Equations and Their Appl ications,60, (2011), 357–387. [8] B.Kaltenbacher and I. Lasiecka, Well-posedness of the W estervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, AIMS Proceedings , (2011). [9] M. Kaltenbacher, Numerical Simulations of Mechatronic Sensors and Actuator s, Springer, Berlin, 2004. [10] G. Leoni, A first course in Sobolev spaces , American Mathematical Society, Providence, 2009. [11] P. Lindqvist, Notes on p-Laplace equation , Lecture notes, University of Jyv¨ askyl¨ a, 2006. [12] G. Teschl, Ordinary Differential Equations and Dynamical Systems , American Mathematical Society, Providence, 2012. [13] P.J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America , 35(1963), 535–537. Insitut f ¨ur Mathematik, Universit ¨at Klagenfurt, Universit ¨atsstraße 65-57, 9020 Klagenfurt am W ¨orthersee, Austria E-mail address :vanja.nikolic@aau.at

2014-08-09

We investigate the Westervelt equation with several versions of nonlinear damping and lower order damping terms and Neumann as well as absorbing boundary conditions. We prove local in time existence of weak solutions under the assumption that the initial and boundary data are sufficiently small. Additionally, we prove local well-posedness in the case of spatially varying $L^{\infty}$ coefficients, a model relevant in high intensity focused ultrasound (HIFU) applications.

Local existence results for the Westervelt equation with nonlinear damping and Neumann as well as absorbing boundary conditions

1408.2160v1

arXiv:1911.02775v2 [cond-mat.mes-hall] 15 Apr 2020Quantum Oscillations of Gilbert Damping in Ferromagnetic/ Graphene Bilayer Systems Yuya Ominato1and Mamoru Matsuo1,2 1Kavli Institute for Theoretical Sciences, University of Ch inese Academy of Sciences, Beijing 100190, China and 2CAS Center for Excellence in Topological Quantum Computati on, University of Chinese Academy of Sciences, Beijing 100190, China (Dated: April 16, 2020) We study the spin dynamics of a ferromagnetic insulator on wh ich graphene is placed. We show that the Gilbert damping is enhanced by the proximity exchan ge coupling at the interface. The modulation of the Gilbert damping constant is proportional to the product of the spin-up and spin-down densities of states of graphene. Consequently, t he Gilbert damping constant in a strong magnetic field oscillates as a function of the external magne tic field that originates from the Landau level structure of graphene. We find that a measurement of the oscillation period enables the strength of the exchange coupling constant to be determined . The results theoretically demonstrate that the ferromagnetic resonance measurements may be used t o detect the spin resolved electronic structure of the adjacent materials, which is critically im portant for future spin device evaluations. Introduction .—Graphene spintronics is an emergent field aiming at exploiting exotic spin-dependent proper- tiesofgrapheneforspintronicsdevices[1]. Althoughpris- tinegrapheneisanon-magneticmaterial,therehavebeen effortstointroducemagnetismintographenetofindspin- dependent phenomena and to exploit its spin degrees of freedom. Placing graphene on a magnetic substrate is a reasonable way, which leads to magnetic proximity ef- fect and lifting of spin degeneracy [2, 3]. Subsequently, magnetization was induced in graphene and spin depen- dent phenomena, such as the anomalous Hall effect [4, 5] and non-local spin transport [6, 7], were observed. In all these experiments, a spin-dependent current was gener- ated by an electric field. There is an alternative way to generate a spin current called spin pumping [8–12]. The proximity exchange coupling describes spin transfer at the magnetic interface and a spin current is injected us- ing ferromagnetic resonance (FMR) from ferromagnetic materials into the adjacent materials. The generation of a spin current is experimentally detectable through both the inverse spin Hall effect and modulation of the FMR, which were experimentally confirmed at magnetic inter- faces between graphene and several magnetic materials [13–18]. The theory of spin transport phenomena at magnetic interfaces has been formulated based on the Schwinger- Keldysh formalism [19], which is applicable to magnetic interfaces composed of a variety of systems, such as a paramagnetic metal and a ferromagnetic insulator (FI) [20–23], a superconductor and FI [24, 25], and two FIs [26, 27]. The modulation of FMR has been investigated in several papers. The modulation of Gilbert damping wasfound to be proportionalto the imaginarypartofthe dynamical spin susceptibility [21, 23–25, 28, 29], which means that one can detect spin excitations and electronic properties of adjacent materials through the FMR mea- surements. This implies that the FMR measurements of FI/graphene bilayer systems allow us to access thespin-dependent properties of graphene in quantum Hall regime [30, 31]. However, the modulation of FMR at the magnetic interface between a FI and graphene has not been investigated and the effect of Landau quantization on the FMR signal is unclear. In this work, we study the modified magnetization dy- namics of a FI adjacent to graphene. Figure 1 (a) shows aschematicofthe system. Microwavesareirradiatedand the precession of localized spins is induced. Figure 1 (b) and (c) shows the electronic structure of graphene on the FI under aperpendicular magneticfield. The spin degen- eracy is lifted by the exchange coupling at the interface and spin-split Landau levels are formed. The densities of states for spin-up and spin-down are shown in the right panel; Landau level broadening is included. We find that the modulation of Gilbert damping is proportionalto the product of the densities of states for spin-up and spin- down, so that the FMR measurements may be used as a probe of the spin-resolved densities of states. Owing to the peak structure of the density of states, the mod- ulation of Gilbert damping exhibits peak structure and an oscillation as a function of Fermi level and magnetic field, which reflects the Landau level structure. One may determine the exchange coupling constant by analyzing the period of the oscillation. Model Hamiltonian .—The totalHamiltonian H(t)con- sists of three terms, H(t) =HFI(t)+HGr+Hex. (1) The first term HFI(t) describes the bulk FI HFI(t) =/summationdisplay k/planckover2pi1ωkb† kbk−h+ ac(t)b† k=0−h− ac(t)bk=0,(2) whereb† kandbkdenote the creation and annihilation operators of magnons with momentum k. We assume a parabolic dispersion /planckover2pi1ωk=Dk2−/planckover2pi1γB, withγ(<0) the electron gyromagnetic ratio. The coupling between the2 MicrowaveB(a) System (b) spin splitting Exchange coupling B01230 -1 -2 -3 DOSE up down(c) spin-split Landau level kx kyE E B123 0-1 -2 -3 0 kx kyE FIG. 1. (Color online) Schematic picture of the FMR measurem ent and the energy spectrum of graphene in a strong perpen- dicular magnetic field. (a) Graphene on a ferromagnetic insu lator substrate. The magnetic field perpendicular to graphe ne is applied and the microwave is irradiated to the FI. (b) The s pin degeneracy is lifted by the exchange coupling. (c) The perpendicular magnetic field leads to the spin-split Landau level structure. The density of states has a peak structure a nd the level broadening originating from disorder is included. microwave and magnons is given by h± ac(t) =/planckover2pi1γhac 2√ 2SNe∓iΩt, (3) wherehacand Ω are the amplitude and frequency of the microwave radiation, respectively, and Sis the magni- tude of the localized spin in the FI. The above Hamilto- nian is derived from a ferromagnetic Heisenberg model using the Holstein-Primakoff transformation and the spin-wave approximation ( Sz k=S−b† kbk,S+ k=√ 2Sbk, S− −k=√ 2Sb† k, whereSkis the Fourier transform of the localized spin in the FI). The second term HGrdescribes the electronic states around the Kpoint in graphene under a perpendicular magnetic field, HGr=/summationdisplay nXsεnc† nXscnXs, (4) wherec† nXsandcnXsdenote the creation and annihi- lation operators of electrons with Landau level index n= 0,±1,±2,···, guiding center X, and spin up s= + and spin down s=−. The eigenenergy is given by εn= sgn(n)√ 2e/planckover2pi1v2/radicalbig |n|B, (5) wherevis the velocity and the sign function is defined as sgn(n) := 1 (n >0) 0 (n= 0) −1 (n <0). (6) In the following, we neglect the Zeeman coupling be- tween the electron spin and the magnetic field because it is much smaller than the Landau-level separation and the exchange coupling introduced below. In graphene, there are two inequivalent valleys labelled KandK′. Inthis paper, we assume that the intervalley scattering is negligible. This assumption is valid for an atomically flat interface,whichisreasonablegiventherecentexperimen- tal setups [4, 17, 18]. Consequently, the valley degree of freedom just doubles the modulation of the Gilbert damping. The third term Hexis the exchange coupling at the interface consisting of two terms Hex=HZ+HT, (7) whereHZdenotes the out-of-plane component of the exchange coupling and leads to the spin splitting in graphene, HZ=−JS/summationdisplay nX/parenleftBig c† nX+cnX+−c† nX−cnX−/parenrightBig ,(8) withJthe exchange coupling constant. The z- component of the localized spin is approximated as ∝angbracketleftSz k∝angbracketright ≈S. The out-of-plane component HZis modeled as a uniform Zeeman-like coupling, although in general, HZcontains the effect of surface roughness, which gives off-diagonal terms. The Hamiltonian HTdenotes the in- plane component of the exchange coupling and describes spin transfer between the FI and graphene, HT=−/summationdisplay nX/summationdisplay n′X′/summationdisplay k/parenleftBig JnX,n′X′,ks+ nX+,n′X′−S− k+h.c./parenrightBig , (9) whereJnX,n′X′,kis the matrix element for the spin trans- fer processes and s+ nX+,n′X′−is the spin-flip operator for the electron spin in graphene. Modulation of Gilbert Damping .—To discuss the Gilbert damping, we calculated the time-dependent sta- tistical averageof the localized spin under the microwave irradiation. The first-order perturbation calculation gives the deviation from the thermal average, δ∝angbracketleftS+ k=0(t)∝angbracketright=−h+ ac(t)GR k=0(Ω). (10)3 The retarded Green’s function is written as GR k(ω) =2S//planckover2pi1 ω−ωk+iαGω−(2S//planckover2pi1)ΣR k(ω),(11) where we have introduced the phenomenological dimen- sionlessdampingparameter αG, calledthe Gilbert damp- ing constant, which originates from the magnon-phonon and magnon-magnon coupling, etc [32–34]. In this pa- per, we focus on the modulation of the Gilbert damping stemming from the spin transfer processes at the inter- face. The self-energy from the spin transfer processes at the interface within second-order perturbation is given by ΣR k(ω) =/summationdisplay nX/summationdisplay n′X′|JnX,n′X′,k|2χR n+,n′−(ω).(12) The spin susceptibility is given by χR n+,n′−(ω) =fn+−fn′− εn+−εn′−+/planckover2pi1ω+i0,(13) wherefns= 1//parenleftbig e(εns−µ)/kBT+1/parenrightbig is the Fermi distribu- tion function and εns=εn−JSsis the spin-split Landau level. From the self-energy expression, one sees that the modulation of the Gilbert damping reflects the property of the spin susceptibility of graphene. The modulation of the Gilbert damping under the microwave irradiation is given by [21, 23–25, 28, 29] δαK G=−2SImΣR k=0(ω) /planckover2pi1ω, (14) where the superscript Ksignifies the contribution from theKvalley. To further the calculation, we assume that the ma- trix element JnX,n′X′,k=0is approximated by a constant J0, including detail properties of the interface, that is, JnX,n′X′,k=0≈J0. Withthisassumption,theself-energy becomes ImΣR k=0(ω) =−|J0|2π/planckover2pi1ω/integraldisplay dε/parenleftbigg −∂f(ε) ∂ε/parenrightbigg D+(ε)D−(ε), (15) whereDs(ε) is the density of states for spin s=± Ds(ε) =A 2πℓ2 B/summationdisplay n1 πΓ (ε−εns)2+Γ2,(16) with magnetic length ℓB=/radicalbig /planckover2pi1/(eB) and area of the interface A. Here, we have introduced a constant Γ de- scribing level broadening arising from surface roughness and impurity scattering. This is the simplest approx- imation to include the disorder effect. The density of states shows peaks at the Landau level, which is promi- nent when its separation exceeds the level broadening. Landau level (JS = 20 meV) δα G [δα 0 10-2 ] 0.6 B [T]1.0μ [meV] 0.4 0.240 20 -20 -400 0.8s=-, n=0 s=+, n=0123 -1 -2 -3 Γ = 1 meV kBT = 1 meV (= 11 K) 6 03 0.6 B [T]1.0μ [meV] 0.4 0.240 20 -20 -400 0.8 FIG. 2. (Color online) Modulation of the Gilbert damping constant δαGand spin-split Landau levels as a function of the Fermi level µand the magnetic field B. The spin splitting JSis set to 20meV. In the left panel, δαGhas peaks at the crossing points of spin-up and spin-down Landau levels. In the right panel, the blue and red curves identify the spin-up and spin-down Landau levels, respectively. Finally, the modulation of the Gilbert damping constant δαGis derived as δαG= 2πgvS|J0|2/integraldisplay dε/parenleftbigg −∂f(ε) ∂ε/parenrightbigg D+(ε)D−(ε),(17) wheregv= 2 denotes the valley degree of freedom. From this expression, one sees that the modulation of the Gilbert damping is proportional to the product of the densities of states for spin-up and spin-down. There- fore, combined with the density of states measurement, for example, a capacitance measurement [35], the FMR measurement is used to detect the spin-resolved densities of states. Figure 2 shows the spin-split Landau levels and the modulation of the Gilbert damping δαGas a function of the Fermi level µand the magnetic field B. We use δα0 as a unit of δαG δα0= 2πgvS|J0|2/parenleftbiggA 2πℓ2 B1 meV/parenrightbigg2 .(18) We note that δα0(∝B2) depends on the magnetic field. Both the level broadening Γ and the thermal broadening kBTare set to 1meV, and JSis set to 20meV [2–4]. δαGreflects the Landau level structure and has peaks at crossing points of spin-up and spin-down Landau levels. The peakpositions aredetermined bysolving εn+=εn′− and the inverse of the magnetic field at the peaks is given by 1 B=2e/planckover2pi1v2 (2JS)2/parenleftBig/radicalbig |n|−/radicalbig |n′|/parenrightBig2 . (19) The peak structurebecomes prominent when the Landau level separation exceeds both level and thermal broaden- ing.4 Γ = 1 meV kBT = 1 meV (= 11 K) 5 meV (= 57 K) 10 meV (= 115 K)kBT = 1 meV (= 11 K) Γ = 1 meV 2 meV 4 meV μ = JS = 20 meV μ = JS = 20 meV (a) (b) Δ(1/B) 3 1/B [1/T]4δα G [δα 0 10 -2 ] 2 18 6 4 2 0 5 3 1/B [1/T]4δα G [δα 0 10 -2 ] 2 18 6 4 2 0 5Δ(1/B) FIG. 3. (Color online) Quantum oscillation of the modulatio n of the Gilbert damping constant δαGas a function of the inverse of the magnetic field 1 /B. The Fermi level µand the magnitude of the spin splitting JSare set to 20meV. (a) Γ = 1meV and δαGis plotted at several temperatures. (b) kBT= 1meV and δαGis plotted for several Γ’s. The period of the oscillation ∆(1 /B) is indicated by double-headed arrows. Figure 3 shows the modulation of the Gilbert damping δαGas a function of the inverse of the magnetic field 1/Bwith the Fermi level set to µ= 20meV, where the spin-down zeroth Landau level resides. δαGshows peak structure and a periodic oscillatorybehavior. The period of the oscillation ∆(1 /B) is derived from Eq. (19) and is written as ∆/parenleftbigg1 B/parenrightbigg =2e/planckover2pi1v2 (2JS)2. (20) The above relation means that the magnitude of the spin splitting JSis detectable by measuring the period of the oscillation ∆(1 /B). For the peak structure to be clear, both leveland thermalbroadeningmust to be sufficiently smaller than the Landau level separation; otherwise, the peak structure smears out. Discussion .—To observe the oscillation of Gilbert damping, at least two conditions must be satisfied. First, the well-separated landau levels have to be realized in the magnetic field where the FMR measurements is fea- sible. Second, the FMR modulation caused by the ad- jacent graphene have to be detectable. The graphene Landau levels are observed in recent experiments at 2T [36], andrecentbroadbandferromagneticresonancespec- trometer enables the generation of microwaves with fre- quencies ≤40GHz and FMR measurements in a mag- netic field ≤2T [37]. The modulation of the FMR linewidth in Permalloy/Graphene [14, 16], yttrium iron garnet/Graphene [17, 18] have been reported by sev- eral experimental groups, although all of them were per- formed at room temperature. Therefore, the above two conditionsareexperimentallyfeasibleandourtheoretical predictions can be tested in an appropriate experimental setup.Conclusion .—We have studied the modulation of the Gilbert damping δαGin a ferromagnetic insulator on which graphene is placed. The exchange coupling at the interface and the perpendicular magnetic field lead to the spin-split Landau levels in graphene. We showed thatδαGis proportional to the product of the densities of states for spin-up and spin-down electrons. Therefore, the spin-resolved densities of states can be detected by measuring δαGand the total density of states. When the Fermi level is fixed at a Landau level, δαGoscillates as a function of the inverse of the magnetic field. The period of the oscillation provides information on the magnitude of the spin splitting. Our main message is that the FMR measurement is a probe of spin-resolved electronic struc- ture. 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2019-11-07

We study the spin dynamics of a ferromagnetic insulator on which graphene is placed. We show that the Gilbert damping is enhanced by the proximity exchange coupling at the interface. The modulation of the Gilbert damping constant is proportional to the product of the spin-up and spin-down densities of states of graphene. Consequently, the Gilbert damping constant in a strong magnetic field oscillates as a function of the external magnetic field that originates from the Landau level structure of graphene. We find that a measurement of the oscillation period enables the strength of the exchange coupling constant to be determined. The results demonstrate in theory that the ferromagnetic resonance measurements may be used to detect the spin resolved electronic structure of the adjacent materials, which is critically important for future spin device evaluations.

Quantum Oscillations of Gilbert Damping in Ferromagnetic/Graphene Bilayer Systems

1911.02775v2

arXiv:0705.1432v3 [cond-mat.mes-hall] 4 Feb 2008Effective temperature and Gilbert damping of a current-driv en localized spin Alvaro S. N´ u˜ nez∗ Departamento de F´ ısica, Facultad de Ciencias Fisicas y Mat ematicas, Universidad de Chile, Casilla 487-3, Codigo postal 837-041 5, Santiago, Chile R.A. Duine† Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands (Dated: October 31, 2018) Starting from a model that consists of a semiclassical spin c oupled to two leads we present a microscopic derivation of the Langevin equation for the dir ection of the spin. For slowly-changing direction it takes on the form of the stochastic Landau-Lifs chitz-Gilbert equation. We give ex- pressions for the Gilbert damping parameter and the strengt h of the fluctuations, including their bias-voltage dependence. At nonzero bias-voltage the fluct uations and damping are not related by the fluctuation-dissipation theorem. We find, however, that in the low-frequency limit it is possible to introduce a voltage-dependent effective temperature tha t characterizes the fluctuations in the direction of the spin, and its transport-steady-state prob ability distribution function. PACS numbers: 72.25.Pn, 72.15.Gd I. INTRODUCTION One of the major challenges in the theoretical descrip- tion of various spintronics phenomena1, such as current- induced magnetization reversal2,3,4,5and domain-wall motion6,7,8,9,10,11,12, is their inherent nonequilibrium character. In addition to the dynamics of the collective degreeoffreedom, themagnetization, thenonequilibrium behavior manifests itself in the quasi-particle degrees of freedomthataredrivenoutofequilibriumbythe nonzero bias voltage. Due to this, the fluctuation-dissipation theorem13,14cannot be applied to the quasi-particles. This, in part, has led to controversysurrounding the the- ory of current-induced domain wall motion15,16. Effective equations of motion for order-parameter dynamics that do obey the equilibrium fluctuation- dissipation theorem often take the form of Langevin equations, or their corresponding Fokker-Planck equations13,14,17. In the context of spintronics the rele- vant equation is the stochastic Landau-Lifschitz-Gilbert equationforthe magnetizationdirection18,19,20,21,22,23,24. In this paper we derive the generalization of this equa- tion to the nonzero-current situation, for a simple microscopic model consisting of a single spin coupled to two leads via an onsite Kondo coupling. This model is intended as a toy-model for a magnetic impurity in a tunnel junction25,26,27. Alternatively, one may think of a nanomagnet consisting of a collection of spins that are locked by strong exchange coupling. The use of this simple model is primarily motivated by the fact that it enables us to obtain analytical results. Because the microscopic starting point for discussing more realistic situations has a similar form, however, we believe that our main results apply quali- tatively to more complicated situations as well. Similar models have been used previously to explicitly study the violation of the fluctuation-dissipation relation28, 1 1.1 1.2 1.3 0 0.1 0.2 0.3 0.4 0.5α/α0 |e|V/µ 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10Teff/T |e|V/(kB T) FIG. 1: Effective temperature as a function of bias voltage. Thedashedlineshows thelarge bias-voltage asymptoticres ult kBTeff≃ |e|V/4 +kBT/2. The inset shows the bias-voltage dependence of the Gilbert damping parameter normalized to the zero-bias result. and the voltage-dependence of the Gilbert damping parameter27. Starting from this model, we derive an effective stochastic equation for the dynamics of the spin direction using the functional-integral description of the Keldysh-Kadanoff-Baymnonequilibrium theory29. (For similar approaches to spin and magnetization dynamics, see also the work by Rebei and Simionato30, Nussinov et al.31and Duine et al.32.) This formalism leads in a natural way to the path-integral formulation of stochastic differential equations33,34. One of the attractive features of this formalism is that dissipation and fluctuations enter the theory separately. This allows us to calculate the strength of the fluctuations even when the fluctuation-dissipation theorem is not valid. We find that the dynamics of the direction of the spin is described by a Langevin equation with a damping ker-2 ,T SµL,T µR FIG. 2: Model system of a spin Sconnected to two tight- binding model half-infinite leads. The chemical potential o f the left lead is µLand different from the chemical potential of the right lead µR. The temperature Tof both leads is for simplicity taken to be equal. nel and a stochastic magnetic field. We give explicit expressions for the damping kernel and the correlation function of the stochastic magnetic field that are valid in the entire frequency domain. In general, they are not related by the fluctuation-dissipation theorem. In the low-frequency limit the Langevin equation takes on the form ofthe stochasticLandau-Lifschitz-Gilbertequation. Moreover, in that limit it is always possible to introduce an effective temperature that characterizes the fluctua- tions and the equilibrium probability distribution for the spin direction. In Fig. 1 we present our main results, namely the bias-voltage dependence of the effective tem- perature and the Gilbert damping parameter. We find that the Gilbert damping constant initially varies lin- early with the bias voltage, in agreement with the re- sult of Katsura et al.27. The voltage-dependence of the Gilbert damping parameter is determined by the den- sity of states evaluated at an energy equal to the sum of the Fermi energy and the bias voltage. The effective temperature is for small bias voltage equal to the actual temperature, whereas for large bias voltage it is inde- pendent of the temperature and proportional to the bias voltage. This bias-dependence of the effective tempera- ture is traced back to shot noise35. Effective temperatures for magnetization dynam- ics have been introduced before on phenomenolog- ical grounds in the context of thermally-assisted current-driven magnetization reversal in magnetic nanopillars36,37,38. A current-dependent effective tem- perature enters in the theoretical description of these systems because the current effectively lowers the energy barrier thermal fluctuations have to overcome. In addi- tion to this effect, the presence of nonzero current alters the magnetization noise due to spin current shot noise35. Covington et al.39interpret their experiment in terms of current-dependent noise although this interpretation is still under debate30. Foroset al.35also predict, using a different model and different methods, a crossover from thermal to shot-noise dominated magnetization noise for increasing bias voltage. Our main result in Fig. 1 is an explicit example of this crossover for a specific model. The remainderofthe paperis organizedasfollows. We start in Sec. II by deriving the general Langevin equation forthe dynamicsofthe magneticimpurity coupledtotwo leads. In Sec. III and IV we discuss the low-frequency limit in the absence and presence of a current, respec- tively. We end in Sec. V with our conclusions.II. DERIVATION OF THE LANGEVIN EQUATION We use a model that consists of a spin Son a site that is coupled via hopping to two semi-infinite leads, as shown in Fig. 2. The full probability distribution for the direction ˆΩ of the spin on the unit sphere is written as a coherent-state path integral over all electron Grassmann field evolutions ψ∗(t) andψ(t), and unit-sphere paths S(t), that evolve from −∞totand back on the so-called Keldysh contour Ct. It is given by29 P[ˆΩ,t] =/integraldisplay S(t)=ˆΩd[S]δ/bracketleftBig |S|2−1/bracketrightBig d[ψ∗]d[ψ] ×exp/braceleftbiggi /planckover2pi1S[ψ∗,ψ,S]/bracerightbigg , (1) where the delta functional enforces the length constraint of the spin. In the above functional integral an inte- gration over boundary conditions at t=−∞, weighted by an appropriate initial density matrix, is implicitly in- cluded in the measure. We have not included boundary conditions on the electron fields, because, as we shall see, the electron correlation functions that enter the theory after integrating out the electrons are in practice conve- niently determined assumingthat the electronsareeither in equilibrium or in the transport steady state. The action S[ψ∗,ψ,S] is the sum of four parts, S[ψ∗,ψ,S] =SL/bracketleftBig/parenleftbig ψL/parenrightbig∗,ψL/bracketrightBig +SR/bracketleftBig/parenleftbig ψR/parenrightbig∗,ψR/bracketrightBig +SC/bracketleftBig/parenleftbig ψ0/parenrightbig∗,ψ0,/parenleftbig ψL/parenrightbig∗,ψL,/parenleftbig ψR/parenrightbig∗,ψR/bracketrightBig +S0/bracketleftBig/parenleftbig ψ0/parenrightbig∗,ψ0,S/bracketrightBig . (2) We describe the leads using one-dimensional non- interacting electron tight-binding models with the action SL/R/bracketleftBig/parenleftBig ψL/R/parenrightBig∗ ,ψL/R/bracketrightBig = /integraldisplay Ctdt′ /summationdisplay j,σ/parenleftBig ψL/R j,σ(t′)/parenrightBig∗ i/planckover2pi1∂ ∂t′ψL/R j,σ(t′) +J/summationdisplay /an}bracketle{tj,j′/an}bracketri}ht;σ/parenleftBig ψL/R j,σ(t′)/parenrightBig∗ ψL/R j′,σ(t′) , (3) where the sum in the second term of this action is over nearest neighbors only and proportional to the nearest-neighbor hopping amplitude Jin the two leads. (Throughout this paper the electron spin indices are de- noted byσ,σ′∈ {↑,↓}, and the site indices by j,j′.) The coupling between system and leads is determined by the action SC[/parenleftbig ψ0/parenrightbig∗,ψ0,/parenleftbig ψL/parenrightbig∗,ψL,/parenleftbig ψR/parenrightbig∗,ψR] =/integraldisplay Ctdt′JC/summationdisplay σ/bracketleftBig/parenleftbig ψL ∂L,σ(t′)/parenrightbig∗ψ0 σ(t′)+/parenleftbig ψ0 σ(t′)/parenrightbig∗ψL ∂L,σ(t′)/bracketrightBig +3 /integraldisplay Ctdt′JC/summationdisplay σ/bracketleftBig/parenleftbig ψR ∂R,σ(t′)/parenrightbig∗ψ0 σ(t′)+/parenleftbig ψ0 σ(t′)/parenrightbig∗ψR ∂R,σ(t′)/bracketrightBig , (4) where∂L and∂R denote the end sites of the semi-infinite left and right lead, and the fields/parenleftbig ψ0(t)/parenrightbig∗andψ0(t) de- scribe the electrons in the single-site system. The hop- ping amplitude between the single-site system and the leads is denoted by JC. Finally, the action for the sys- tem reads S0/bracketleftBig/parenleftbig ψ0/parenrightbig∗,ψ∗,S/bracketrightBig =/integraldisplay Ctdt′ /summationdisplay σ/parenleftbig ψ0 σ(t′)/parenrightbig∗i/planckover2pi1∂ ∂t′ψ0 σ(t′) −/planckover2pi1SA(S(t′))·dS(t′) dt′+h·S(t′) +∆/summationdisplay σ,σ′/parenleftbig ψ0 σ(t′)/parenrightbig∗τσ,σ′·S(t′)ψ0 σ′(t′) .(5) The second term in this action is the usual Berry phase for spin quantization40, withA(S) the vector potential of a magnetic monopole ǫαβγ∂Aγ ∂Sβ=Sα, (6) where a sum over repeated Greek indices α,β,γ∈ {x,y,z}is implied throughout the paper, and ǫαβγis the anti-symmetric Levi-Civita tensor. The third term in the action in Eq. (5) describes the coupling of the spin to an external magnetic field, up to dimensionful prefactors given by h. (Note that hhas the dimensions of energy.) The last term in the action models the s−dexchange coupling of the spin with the spin of the conduction elec- trons in the single-site system and is proportional to the exchange coupling constant ∆ >0. The spin of the con- duction electronsisrepresentedbythe vectorofthe Pauli matrices that is denoted by τ. Next, we proceed to integrate out the electrons using second-order perturbation theory in ∆. This results in an effective action for the spin given by Seff[S] =/integraldisplay Ctdt′/bracketleftbigg S/planckover2pi1A(S(t′))·dS(t′) dt′+h·S(t′) −∆2/integraldisplay Ctdt′′Π(t′,t′′)S(t′)·S(t′′)/bracketrightbigg . (7) This perturbation theory is valid as long as the electron band width is much larger than the exchange interac- tion with the spin, i.e., J,JC≫∆. The Keldysh quasi- particleresponsefunctionisgivenintermsoftheKeldysh Green’s functions by Π(t,t′) =−i /planckover2pi1G(t,t′)G(t′,t), (8) where the Keldysh Green’s function is defined by iG(t,t′) =/angbracketleftBig ψ0 ↑(t)/parenleftbig ψ0 ↑(t′)/parenrightbig∗/angbracketrightBig =/angbracketleftBig ψ0 ↓(t)/parenleftbig ψ0 ↓(t′)/parenrightbig∗/angbracketrightBig .(9)We willgiveexplicit expressionsforthe responsefunction and the Green’s function later on. For now, we will only make use of the fact that a general function A(t,t′) with its argumentson the Keldysh contour is decomposed into its analytic pieces by means of A(t,t′) =θ(t,t′)A>(t,t′)+θ(t′,t)A<(t,t′),(10) whereθ(t,t′) is the Heaviside step function on the Keldysh contour. There can be also a singular piece Aδδ(t,t′), but suchageneraldecompositionisnot needed here. Also needed are the advanced and retarded com- ponents, denoted respectively by the superscript ( −) and (+), and defined by A(±)(t,t′)≡ ±θ(±(t−t′))/bracketleftbig A>(t,t′)−A<(t,t′)/bracketrightbig ,(11) and, finally, the Keldysh component AK(t,t′)≡A>(t,t′)+A<(t,t′), (12) which, as we shall see, determines the strength of the fluctuations. Next we write the forward and backward paths of the spin on the Keldysh contour, denoted respectively by S(t+) andS(t−), as a classical path Ω(t) plus fluctua- tionsδΩ(t), by means of S(t±) =Ω(t)±δΩ(t) 2. (13) Moreover,it turns out to be convenient to write the delta functional, which implements the length constraintofthe spin, as a path integral over a Lagrange multiplier Λ( t) defined on the Keldysh contour. Hence we have for the probability distribution in first instance that P[ˆΩ,t] =/integraldisplay S(t)=ˆΩd[S]d[Λ]exp/braceleftbiggi /planckover2pi1Seff[S]+i /planckover2pi1SΛ[S,Λ]/bracerightbigg , (14) with SΛ[S,Λ] =/integraldisplay Ctdt′Λ(t′)/bracketleftBig |S(t′)|2−1/bracketrightBig .(15) We then also have to split the Lagrange multiplier into classical and fluctuating parts according to Λ(t±) =λ(t)±δλ(t) 2. (16) Note that the coordinate transformations in Eqs. (13) and (16) have a Jacobian of one. Before we proceed, we note that in principle we are required to expand the action up to all orders in δΩ. Also note that for some forward and backward paths S(t+) and S(t−) on the unit sphere the classical path Ωis not necessarily on the unit sphere. In order to circumvent these problems we note that the Berry phase term in Eq. (5) is proportional to the area on the unit sphere enclosed by the forward and backward paths. Hence, in4 the semi-classical limit S→ ∞27,40paths whose forward and backward components differ substantially will be suppressed in the path integral. Therefore, we take this limit from now on which allows us to expand the action in terms of fluctuations δΩ(t) up to quadratic order. We will see that the classical path Ω(t) is now on the unit sphere. We note that this semi-classical approximation is not related to the second-order perturbation theory used to derive the effective action. Splitting the paths in classical and fluctuation parts gives for the probability distribution P[ˆΩ,t] =/integraldisplay Ω(t)=ˆΩd[Ω]d[δΩ]d[λ]d[δλ]exp/braceleftbiggi /planckover2pi1S[Ω,δΩ,λ,δλ]/bracerightbigg , (17) with the action, that is now projected on the real-time axis, S[Ω,δΩ,λ,δλ] =/integraldisplay dt/braceleftbigg /planckover2pi1SǫαβγδΩβ(t)dΩα(t) dtΩγ(t) +δΩα(t)hα+2δΩα(t)Ωα(t)λ(t) +δλ(t)/bracketleftbig |Ω(t)|2−1+|δΩ(t)|2/4/bracketrightbig/bracerightbigg −∆2/integraldisplay dt/integraldisplay dt′/braceleftBig δΩα(t)/bracketleftBig Π(−)(t′,t)+Π(+)(t,t′)/bracketrightBig Ωα(t′)/bracerightBig −∆2 2/integraldisplay dt/integraldisplay dt′/bracketleftbig δΩα(t)ΠK(t,t′)δΩα(t′)/bracketrightbig . (18) From this action we observe that the integration over δλ(t) immediately leads to the constraint |Ω(t)|2= 1−|δΩ(t)|2 4, (19) as expected. Implementing this constraint leads to terms of order O(δΩ3) or higher in the above action which we are allowed to neglect because of the semi-classical limit. From now on we can therefore take the path integration overΩ(t) on the unit sphere. Thephysicalmeaningofthetermslinearandquadratic inδΩ(t) becomes clear after a so-called Hubbard- Stratonovich transformation which amounts to rewrit- ing the action that is quadratic in the fluctuations as a path integral over an auxiliary field η(t). Performing this transformation leads to P[ˆΩ,t] =/integraldisplay Ω(t)=ˆΩd[Ω]d[δΩ]d[η]d[λ] ×exp/braceleftbiggi /planckover2pi1S[Ω,δΩ,λ,η]/bracerightbigg ,(20) where the path integration over Ωis now on the unit sphere. The action that weighs these paths is given by S[Ω,δΩ,λ,η] =/integraldisplay dt/bracketleftbigg /planckover2pi1SǫαβγδΩβ(t)dΩα(t) dtΩγ(t) +δΩα(t)hα+2δΩα(t)Ωα(t)λ(t)+δΩα(t)ηα(t)/bracketrightbigg−∆2/integraldisplay dt/integraldisplay dt′/braceleftBig δΩα(t)/bracketleftBig Π(−)(t′,t)+Π(+)(t,t′)/bracketrightBig Ωα(t′)/bracerightBig +1 2∆2/integraldisplay dt/integraldisplay dt′/bracketleftBig ηα(t)/parenleftbig ΠK/parenrightbig−1(t,t′)ηα(t′)/bracketrightBig .(21) Note that the inverse in the last term is defined as/integraltext dt′′ΠK(t,t′′)/parenleftbig ΠK/parenrightbig−1(t′′,t′) =δ(t−t′). Performing now the path integral over δΩ(t), we ob- serve that the spin direction Ω(t) is constraint to obey the Langevin equation /planckover2pi1SǫαβγdΩβ(t) dtΩγ(t) =hα+2λ(t)Ωα(t) +ηα(t)+/integraldisplay∞ −∞dt′K(t,t′)Ωα(t′),(22) with the so-called damping or friction kernel given by K(t,t′) =−∆2/bracketleftBig Π(−)(t′,t)+Π(+)(t,t′)/bracketrightBig .(23) Note that the Heaviside step functions in Eq. (11) appear precisely such that the Langevin equation is causal. The stochastic magnetic field is seen from Eq. (21) to have the correlations ∝an}bracketle{tηα(t)∝an}bracketri}ht= 0 ; ∝an}bracketle{tηα(t)ηβ(t′)∝an}bracketri}ht=iδαβ/planckover2pi1∆2ΠK(t,t′).(24) Using the fact that Ω(t) is a unit vector within our semi- classical approximation, the Langevin equation for the direction of the spin ˆΩ(t) is written as /planckover2pi1SdˆΩ(t) dt=ˆΩ(t)×/bracketleftbigg h+η(t)+/integraldisplay∞ −∞dt′K(t,t′)ˆΩ(t′)/bracketrightbigg , (25) which has the form of a Landau-Lifschitz equation with a stochastic magnetic field and a damping kernel. In the next sections we will see that for slowly-varying spin direction we get the usual form of the Gilbert damping term. So far, we have not given explicit expressions for the responsefunctionsΠ(±),K(t,t′). Todeterminethesefunc- tions, we assume that the left and right leads are in thermal equilibrium at chemical potentials µLandµR, respectively. Although not necessary for our theoretical approachwe assume, for simplicity, that the temperature Tof the two leads is the same. The Green’s functions for the system are then given by41,42 −iG<(ǫ) =A(ǫ) 2/summationdisplay k∈{L,R}N(ǫ−µk) ; iG>(ǫ) =A(ǫ) 2/summationdisplay k∈{L,R}[1−N(ǫ−µk)] ; G≶,K(t−t′) =/integraldisplaydǫ (2π)e−iǫ(t−t′)//planckover2pi1G≶,K(ǫ),(26) withN(ǫ) ={exp[ǫ/(kBT)]+1}−1the Fermi-Dirac dis- tribution function with kBBoltzmann’s constant, and A(ǫ) =i/bracketleftBig G(+)(ǫ)−G(−)(ǫ)/bracketrightBig , (27)5 the spectral function. Note that Eq. (26) has a particu- larlysimpleformbecausewearedealingwithasingle-site system. The retarded and advanced Green’s functions are determined by /bracketleftBig ǫ±−2/planckover2pi1Σ(±)(ǫ)/bracketrightBig G(±)(ǫ) = 1, (28) withǫ±=ǫ±i0, and the retarded self-energy due to one lead follows, for a one-dimensional tight-binding model, as /planckover2pi1Σ(+)(ǫ) =−J2 C Jeik(ǫ)a, (29) withk(ǫ) = arccos[ −ǫ/(2J)]/athe wave vector in the leads at energy ǫ, andathe lattice constant. The ad- vanced self-energy due to one lead is given by the com- plex conjugate of the retarded one. Before proceeding we give a brief physical description of the above results. (More details can be found in Refs. [41] and [42].) They arise by adiabatically elim- inating (“integrating out”) the leads from the system, assuming that they are in equilibrium at their respective chemical potentials. This procedure reduces the problem to a single-site one, with self-energy corrections for the on-site electron that describe the broadening of the on- site spectral function from a delta function at the (bare) on-site energy to the spectral function in Eq. (27). More- over, the self-energy corrections also describe the non- equilibrium occupation of the single site via Eq. (26) For the transport steady-state we have that Π(±),K(t,t′) depends only on the difference of the time arguments. Using Eq. (8) and Eqs. (10), (11), and (12) we find that the Fourier transforms are given by Π(±)(ǫ)≡/integraldisplay d(t−t′)eiǫ(t−t′)//planckover2pi1Π(±)(t,t′) =/integraldisplaydǫ′ (2π)/integraldisplaydǫ′′ (2π)1 ǫ±+ǫ′−ǫ′′ ×/bracketleftbig G<(ǫ′)G>(ǫ′′)−G>(ǫ′)G<(ǫ′′)/bracketrightbig ,(30) and ΠK(ǫ) =−2πi/integraldisplaydǫ′ (2π)/integraldisplaydǫ′′ (2π)δ(ǫ+ǫ′−ǫ′′) ×/bracketleftbig G>(ǫ′)G<(ǫ′′)+G<(ǫ′)G>(ǫ′′)/bracketrightbig .(31) In the next two sections we determine the spin dynamics in the low-frequency limit, using these expressions to- gether with the expressions for G≶(ǫ). We consider first the equilibrium case. III. EQUILIBRIUM SITUATION In equilibrium the chemical potentials of the two leads are equal so that we have µL=µR≡µ. Combining re- sults from the previous section, we find for the retardedand advanced response functions (the subscript “0” de- notes equilibrium quantities) that Π(±) 0(ǫ) =/integraldisplaydǫ′ (2π)/integraldisplaydǫ′′ (2π)A(ǫ′)A(ǫ′′) ×[N(ǫ′−µ)−N(ǫ′′−µ)] ǫ±+ǫ′−ǫ′′.(32) The Keldysh component of the response function is in equilibrium given by ΠK 0(ǫ) =−2πi/integraldisplaydǫ′ (2π)/integraldisplaydǫ′′ (2π)A(ǫ′)A(ǫ′′)δ(ǫ−ǫ′+ǫ′′) {[1−N(ǫ′−µ)]N(ǫ′′−µ)+N(ǫ′−µ)[1−N(ǫ′′−µ)]}.(33) The imaginary part of the retarded and advanced re- sponse functions are related to the Keldysh component by means of ΠK 0(ǫ) =±2i[2NB(ǫ)+1]Im/bracketleftBig Π(±) 0(ǫ)/bracketrightBig ,(34) withNB(ǫ) ={exp[ǫ/(kBT)]−1}−1the Bose distribu- tion function. This is, in fact, the fluctuation-dissipation theorem which relates the dissipation, determined as we shall see by the imaginary part of the retarded and advanced components of the response function, to the strength of the fluctuations, determined by the Keldysh component. For low energies, corresponding to slow dynamics, we have that Π(±) 0(ǫ)≃Π(±) 0(0)∓i 4πA2(µ)ǫ . (35) With this result the damping term in the Langevin equa- tion in Eq. (25) becomes /integraldisplay∞ −∞dt′K(t,t′)ˆΩ(t′) =−/planckover2pi1∆2A2(µ) 2πdˆΩ(t) dt,(36) where we have not included the energy-independent part ofEq. (35) because it does not contribute to the equation of motion for ˆΩ(t). In the low-energy limit the Keldysh component of the response function is given by ΠK 0(ǫ) =A2(µ) iπkBT . (37) Putting all these results together we find that the dy- namics of the spin direction is, as long as the two leads are in equilibrium at the same temperature and chemical potential,determinedbythestochasticLandau-Lifschitz- Gilbert equation /planckover2pi1SdˆΩ(t) dt=ˆΩ(t)×[h+η(t)]−/planckover2pi1α0ˆΩ×dˆΩ(t) dt,(38) with the equilibrium Gilbert damping parameter α0=∆2A2(µ) 2π. (39)6 Using Eqs. (24), (37), and (39) we find that the strength of the Gaussian stochastic magnetic field is determined by ∝an}bracketle{tηα(t)ηβ(t′)∝an}bracketri}ht= 2α0/planckover2pi1kBTδ(t−t′)δαβ.(40) Note that these delta-function type noise correlations are derived by approximating the time dependence of ΠK(t,t′) by a delta function in the difference of the time variables. This means that the noisy magnetic field η(t) corresponds to a Stratonovich stochastic process13,14,17. The stationary probability distribution function gen- erated by the Langevin equation in Eqs. (38) and (40) is given by the Boltzmann distribution18,19,20,21,22,23,24 P[ˆΩ,t→ ∞]∝exp/braceleftBigg −E(ˆΩ) kBT/bracerightBigg , (41) with E[ˆΩ] =−h·ˆΩ, (42) the energy of the spin in the external field. It turns out that Eq. (41) holds for any effective field h= −∂E[ˆΩ]/∂ˆΩ, and in particular for the case that E[ˆΩ] is quadratic in the components of ˆΩ as is often used to model magnetic anisotropy. It is important to realize that the equilibrium prob- ability distribution has precisely this form because of the fluctuation-dissipation theorem, which ensures that dissipation and fluctuations cooperate to achieve ther- mal equilibrium13,14. Finally, it should be noted that this derivation of the stochastic Landau-Lifschitz-Gilbert equation from a microscopic starting point circumvents concerns regarding the phenomenological form of damp- ing and fluctuation-dissipation theorem, which is subject of considerable debate22,23. IV. NONZERO BIAS VOLTAGE In this section we consider the situation that the chem- ical potential of the left lead is given by µL=µ+|e|V, with|e|V >0 the bias voltage in units of energy, and µ=µRthe chemical potential of the right lead. Using the general expressions given for the response functions derived in Sec. II, it is easy to see that the imaginary part of the retarded and advanced components of the response functions are no longer related to the Keldysh component by means of the fluctuation-dissipation theo- rem in Eq. (34). See also the work by Mitra and Millis28 for a discussion of this point. As in the previous section, we proceed to determine the low-frequency behavior of the response functions. Using Eqs. (26), (27), and (30) we find that the re- tarded and advanced components of the response func- tion are given by Π(±)(ǫ) =∓i 8π/bracketleftbig A2(µ+|e|V)+A2(µ)/bracketrightbig ǫ .(43)In this expression we have omitted the energy- independent part andthe contribution followingfrom the principal-value part of the energy integral because, as we have seen previously, these do not contribute to the final equation of motion for the direction of the spin. Follow- ing the same steps as in the previous section, we find that the damping kernel in the general Langevin equa- tion in Eq. (25) reduces to a Gilbert damping term with a voltage-dependent damping parameter given by α(V) =∆2 4π/bracketleftbig A2(µ+|e|V)+A2(µ)/bracketrightbig ≃α0/bracketleftbigg 1+O/parenleftbigg|e|V µ/parenrightbigg/bracketrightbigg . (44) This result is physically understood by noting that the Gilbert damping is determined by the dissipative part of the response function Π(+)(ǫ). In this simple model, this dissipative part gets contributions from processes that correspond to an electron leaving or entering the system, to or from the leads, respectively. The dissipative part is in general proportional to the density of states at the Fermi energy. Since the Fermi energy of left and right lead is equal to µ+|e|Vandµ, respectively, the Gilbert damping has two respective contributions corresponding to the two terms in Eq. (44). Note that the result that the Gilbert damping param- eter initially varies linearly with the voltage is in agree- ment with the results of Katsura et al.27, although these authorsconsideraslightlydifferentmodel. Inthe insetof Fig. 1 we show the Gilbert damping parameter as a func- tion of voltage. The parameters taken are ∆ /J= 0.1, JC=J,µ/J= 1 andµ/(kBT) = 100. Althoughwe cannolongermakeuseofthefluctuation- dissipation theorem, we are nevertheless able to deter- mine the fluctuations by calculating the low-energy be- havioroftheKeldyshcomponentoftheresponsefunction in the nonzero-voltage situation. It is given by ΠK(ǫ) =−i 2/integraldisplaydǫ′ (2π)A2(ǫ′){[N(µL−ǫ′)+N(µR−ǫ′)] ×[N(ǫ′−µL)+N(ǫ′−µR)]}. (45) We define an effective temperature by means of kBTeff(T,V)≡iΠK(ǫ)∆2 2α(V). (46) This definition is motivated by the fact that, as we mention below, the spin direction obeys the stochas- tic Landau-Lifschitz-Gilbert equation with voltage- dependentdampingandfluctuationscharacterizedbythe above effective temperature43. From the expression for α(V) and ΠK(ǫ) we see that in the limit of zero bias voltage we recover the equilibrium result Teff=T. In the situation that |e|Vis substantially larger than kBT, whichis usuallyapproachedin experiments, wehavethat kBTeff(T,V)≃|e|V 4+kBT 2, (47)7 which in the limit that |e|V≫kBTbecomes indepen- dent of the actual temperature of the leads. In Fig. 1 the effective temperature as a function of bias voltage is shown, using the expression for ΠK(ǫ) given in Eq. (45). The parameters are the same as before, i.e., ∆ /J= 0.1, JC=J,µ/J= 1 andµ/(kBT) = 100. Clearly the ef- fective temperature changes from Teff=Tat zero bias voltagetotheasymptoticexpressioninEq.(47)shownby the dashed line in Fig. 1. The crossover between actual temperatureandvoltageasameasureforthefluctuations is reminiscent of the theory of shot noise in mesoscopic conductors44. This is not surprising, since in the single- site model we use the noise in the equation of motion ul- timately arises because of fluctuations in the number of electronsin thesingle-sitesystem, andisthereforeclosely related to shot noise in the current through the system. Foroset al.35calculate the magnetization noise arising from spin currentshot noisein the limit that |e|V≫kBT and|e|V≪kBT. In these limits our results are similar to theirs. With the above definition of the effective temperature wefind that in the nonzerobiasvoltagesituationthe spin direction obeys the stochastic Landau-Lifschitz-Gilbert equation, identical in form to the equilibrium case in Eqs. (38) and (40), with the Gilbert damping parame- ter and temperature replaced according to α0→α(V) ; T→Teff(T,V). (48) Moreover, the transport-steady-state probability distri- bution for the direction of the spinis a Boltzmann distri- bution with the effective temperature characterizing the fluctuations. V. DISCUSSION AND CONCLUSIONS We have presented a microscopic derivation of the stochastic Landau-Lifschitz-Gilbert equation for a semi- classical single spin under bias. We found that the Gilbert damping parameter is voltage dependent and to lowest order acquires a correction linear in the bias volt- age, in agreement with a previous study for a slightly different model27. In addition, we have calculated the strength of the fluctuations directly without using the fluctuation-dissipation theorem and found that, in the low-frequency regime, the fluctuations are characterized by a voltage and temperature dependent effective tem- perature. To arrive at these results we have performed a low frequency expansion of the various correlation functions that enter the theory. Such an approximation is valid as long as the dynamics is much slower than the times set by the other energy scales in the system such as temper- ature and the Fermi energy. Moreover, in order for the leads to remain in equilibrium as the spin changes direc- tion, the processes in the leads that lead to equilibrationhave to be much faster than the precession period of the magnetizationspin. Both these criteria are satisfied in experiments with magnetic materials. In principle how- ever, the full Langevinequationderivedin Sec. II alsode- scribes dynamics beyond this low-frequency approxima- tion. The introduction of the effective temperature relies on the low-frequency approximation though, and for ar- bitrary frequencies such a temperature can no longer be uniquely defined28. An effective temperature for magnetization dynam- ics has been introduced before on phenomenological grounds36,37,38. Interestingly, the phenomenological ex- pression of Urazhdin et al.36, found by experimentally studying thermal activation of current-driven magneti- zation reversal in magnetic trilayers, has the same form as our expression for the effective temperature in the large bias-voltage limit [Eq. (47)] that we derived micro- scopically. Zhang and Li37, and Apalkov and Visscher38, have, on phenomenological grounds, also introduced an effective temperature to study thermally-assisted spin- transfer-torque-induced magnetization switching. In their formulation, however, the effective temperature is proportional to the real temperature because the current effectively modifies the energy barrier for magnetization reversal. Foroset al.35consider spin current shot noise in the largebias-voltagelimit andfind forsufficiently largevolt- age that the magnetization noise is dominated by shot noise. Moreover, they also consider the low bias-voltage limit and predict a crossover for thermal to shot-noise dominated magnetization fluctuations. Our main result in Fig. 1 provides an explicit example of this crossover for a simple model system obtained by methods that are easily generalized to more complicated models. In the experiments of Krivorotov et al.45the temperature de- pendence of the dwell time of parallel and anti-parallel states of a current-driven spin valve was measured. At low temperatures kBT/lessorsimilar|e|Vthe dwell times are no longer well-described by a constant temperature, which could be a signature of the crossover from thermal noise to spin current shot noise. However, Krivorotov et al. interpret this effect as due to ohmic heating, which is not taken into account in the model presented in this paper, nor in the work by Foros et al.35. Moreover, in realistic materials phonons provide an additional heat bath for the magnetization, with an effective tempera- ture that may depend in a completely different manner on the bias voltage than the electron heat-bath effec- tive temperature. Nonetheless, we believe that spin cur- rent shot noise may be observable in future experiments and that it may become important for applications as technological progress enables further miniaturization of magnetic materials. Moreover, the formalism presented hereisanimportantstepinunderstandingmagnetization noise from a microscopic viewpoint as its generalization to more complicated models is in principle straightfor- ward. Possible interesting generalizations include mak- ing one of the leads ferromagnetic (see also Ref. [46]).8 Since spin transfer torques will occur on the single spin as a spin-polarized current from the lead interacts with the single-spin system, the resulting model would be a toy model for microscopically studying the attenuation of spin transfer torques and current-driven magnetiza- tion reversal by shot noise. Another simple and use- ful generalization would be enlarging the system to in- clude more than one spin. The formalism presented here would allow for a straightforward microscopic calcula- tion of Gilbert damping and adiabatic and nonadiabatic spin transfer torques which are currently attracting a lot of interest in the context of current-driven domain wall motion6,7,8,9,10,11,12. The application of our theory in thepresentpaperis, in additiontoitsintrinsicphysicalinter- est, chosen mainly because of the feasibility of analytical results. Theapplicationsmentionedabovearemorecom- plicated and analytical results may be no longer obtain- able. 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Phys. 10, 013017 (2008).

2007-05-10

Starting from a model that consists of a semiclassical spin coupled to two leads we present a microscopic derivation of the Langevin equation for the direction of the spin. For slowly-changing direction it takes on the form of the stochastic Landau-Lifschitz-Gilbert equation. We give expressions for the Gilbert damping parameter and the strength of the fluctuations, including their bias-voltage dependence. At nonzero bias-voltage the fluctuations and damping are not related by the fluctuation-dissipation theorem. We find, however, that in the low-frequency limit it is possible to introduce a voltage-dependent effective temperature that characterizes the fluctuations in the direction of the spin, and its transport-steady-state probability distribution function.

Effective temperature and Gilbert damping of a current-driven localized spin

0705.1432v3

Ultrafast Demagnetization through Femtosecond Generation of Non-thermal Magnons Markus Weißenhofer1, 2,∗and Peter M. Oppeneer1 1Department of Physics and Astronomy, Uppsala University, P. O. Box 516, S-751 20 Uppsala, Sweden 2Department of Physics, Freie Universit¨ at Berlin, Arnimallee 14, D-14195 Berlin, Germany (Dated: January 17, 2024) Ultrafast laser excitation of ferromagnetic metals gives rise to correlated, highly non-equilibrium dynamics of electrons, spins and lattice, which are, however, poorly described by the widely-used three-temperature model (3TM). Here, we develop a fully ab-initio parameterized out-of-equilibrium theory based on a quantum kinetic approach–termed (N+2) temperature model –that describes magnon occupation dynamics due to electron-magnon scattering. We apply this model to per- form quantitative simulations on the ultrafast, laser-induced generation of magnons in iron and demonstrate that on these timescales the magnon distribution is non-thermal: predominantly high- energy magnons are created, while the magnon occupation close to the center of the Brillouin zone even decreases, due to a repopulation towards higher energy states via a so-far-overlooked scattering term. We demonstrate that the simple relation between magnetization and temperature computed at equilibrium does not hold in the ultrafast regime and that the 3TM greatly overestimates the demagnetization. The ensuing Gilbert damping becomes strongly magnon wavevector dependent and requires a description beyond the conventional Landau-Lifshitz-Gilbert spin dynamics. Our ab- initio -parameterized calculations show that ultrafast generation of non-thermal magnons provides a sizable demagnetization within 200fs in excellent comparison with experimentally observed laser- induced demagnetizations. Our investigation emphasizes the importance of non-thermal magnon excitations for the ultrafast demagnetization process. I. INTRODUCTION The discovery that magnetic order can be manipu- lated on sub-picosecond timescales by femtosecond laser pulses [1–3] has fueled the emergence of intensive exper- imental and theoretical research efforts in the field of ul- trafast magnetization dynamics. What makes this field particularly interesting, apart from its technological po- tential in future memory and spintronic devices [4, 5], is that many well-established physical paradigms cannot be simply transferred from the equilibrium to the ultrafast regime, due to its highly non-equilibrium nature. Relat- edly, albeit more than 25 years of intense research, the underlying mechanisms of ultrafast demagnetization are still heavily debated [6–8]: while some works [9–14] lean towards longitudinal excitations – i.e., the reduction of the magnetic moment carried by each atom due to the de- crease of exchange splitting – others [15–19] hint at trans- verse spin excitations – a reduction of the average magne- tization due to the mutual tilting of the moments carried by different atoms – as the main contribution. Non-local contributions due to superdiffusive spin currents [20, 21] are relevant in certain situations [22–25]. However, it has become evident that they are most likely not the only mechanism of ultrafast demagnetization [26, 27]. Theoretical models describing ultrafast magnetization dynamics typically rely on a separation of electronic, phononic and – if magnetization dynamics are to be con- sidered separately – spin degrees of freedom. Beaurepaire et al. [1] introduced the three-temperature model (3TM) to explain the flow of the energy transferred by the laser ∗markus.weissenhofer@fu-berlin.deby assuming that each subsystem is internally in thermal equilibrium and the system can hence be described by three temperatures (for electrons, phonons and spins), to- gether with the respective distributions (Fermi-Dirac and Bose-Einstein). However, it was pointed out in numer- ous investigations that the distributions are non-thermal on ultrafast timescales [28–37]. Also, the 3TM discards completely the transfer of angular momentum due to de- magnetization, which, according to recent experiments [38, 39], appears to be primarily to the lattice. Transverse demagnetization is often studied using atomistic spin dynamics simulations based on the stochastic Landau-Lifshitz-Gilbert (LLG) equation to- gether with an extended Heisenberg model [40–42], which can successfully reproduce experimentally measured de- magnetization curves [43, 44]. The stochastic LLG is a Langevin-type equation with a coupling to a heat bath with given temperature via a single parameter, the Gilbert damping parameter. This parameter includes all possible contributions – Fermi surface breathing, crystal defects, coupling to phonons, s−dcoupling, etc. [45–52] – to damping and while it can in principle be obtained from ab initio calculations, in practice it is typically taken from experimental measurements of ferromagnetic resonance (FMR) [53]. On the one hand, this ensures the versatility of atomistic spin dynamics simulations, but on the other hand, it obscures the details of the underlying micro- scopic energy and angular momentum transfer processes - which are crucial for understanding the fundamentals of ultrafast demagnetization. For this reason, steps have been taken in recent years to explicitly consider the cou- pling of spins to phonons [54–62] and electrons [63–65]. Also, due to the classical nature of the commonly used stochastic LLG, the equilibrium magnon occupations cal-arXiv:2309.14167v3 [cond-mat.mtrl-sci] 15 Jan 20242 culated by it follow Rayleigh-Jeans rather than Bose- Einstein statistics, henceforth leading to the wrong tem- perature scaling of the magnetization [66, 67]. Implemen- tation of quantum statistics in the spin-dynamics simu- lations can however provide the correct low-temperature scaling of the magnetization [68, 69]. In this work, we investigate the laser-induced gener- ation of magnons, the low energy transverse excitations of the spin system, due to electron-magnon scattering. We develop a quantum kinetic approach, which will be termed (N+2)-temperature model [(N+2)TM], to per- form quantitative simulations of the time evolution of the non-thermal magnon dynamics in bcc iron. Being based on ab initio parameters and considering also non- thermal magnon distributions, our work goes well beyond what has been done in Refs. [63, 64, 70] and the conven- tional 3TM. In addition, we show that the 3TM and its relevant parameters can be obtained from our (N+2)TM and, with that, from ab initio calculations. Importantly, using ab initio calculated input parameters, our quantum kinetic theory predicts a sizable and ultrafast demagne- tization of iron within 200 fs, in excellent agreement with experiments [15]. II. OUT-OF-EQUILIBRIUM MAGNON DYNAMICS MODEL To describe the time evolution of the ultrafast non- thermal magnon occupation dynamics, we assume that their creation and annihilation is dominated by electron- magnon scattering processes. In this work, we use the sp−dmodel [71, 72] to describe such processes. The basic idea of both s−dmodel and sp−dmodel is the separation of electrons in localized ( dband) electrons and itinerant ( sband or sandpbands) electrons. The mag- netic moments of the delectrons make up the Heisenberg- type [73] magnetic moments of constant length, the small energy excitations of which are the magnons. The itin- erant electrons are described within a Stoner-type model [74]. While an unambiguous identification of spandd electrons as localized and itinerant is strictly speaking not possible, it has nonetheless been established in liter- ature that these models provide a suitable framework for the description of electron-spin interaction in many phe- nomena relevant for spintronics, e.g. magnetic relaxation [75–77], ultrafast demagnetization [63–65, 70, 78–80] and spin torques [81]. We assume local exchange between the itinerant and localized spins, as given by the Hamiltonian ˆHem∼PN i=1ˆsitin·ˆSloc i, with Nbeing the number of atoms, andˆsitinand ˆSloc ithe spin operators for itinerant ( sp) electrons and localized ( d) electrons at atom i. In sec- ond quantization and second order in magnon variables(details in Method Section V A), the Hamiltonian reads ˆHem≈ −∆X kν ˆc† kν↑ˆckν↑−ˆc† kν↓ˆckν↓ −∆r 2 SNX kνν′,q ˆc† k+qν↑ˆckν′↓ˆb† −q+ ˆc† k+qν↓ˆckν′↑ˆbq +∆ SNX kνν′,qq′ ˆc† k−q+q′ν↑ˆckν′↑−ˆc† k−q+q′ν↓ˆckν′↓ ˆb† qˆbq′. (1) Here, ∆ is the sp−dexchange parameter, Sis the ab- solute value of the localized spins, kandqare vectors in reciprocal space, ˆ c(†) kνσis the fermionic electron annihila- tion (creation) operator for the itinerant electrons – with νbeing the band index and σ∈ {↑,↓}– and ˆb(†) qis the bosonic magnon annihilation (creation) operator. The first term in Equation (1) describes the spin-splitting of the itinerant electrons due to the exchange with the lo- calized magnetic moments, the second one the excitation (annihilation) of a magnon due to a spin flip process and the third one the spin-conserving scattering of a magnon and an electron from one state to another. It is worth noting that the second term leads to a transfer of both energy and angular momentum (i.e., spin) – since it can change the total number of magnons – while the third term can only transfer energy. For this reason, this term was discarded earlier works [63–65], however, our quanti- tative analysis reveals that the energy transferred by this term can exceed the energy transferred by the term first order in magnon operators. We complete our Hamiltonian H=ˆHe+ˆHm+ˆHem by considering ˆHe=P kνσεkνσˆc† kνσˆckνσand ˆHm=P qℏωqˆb† qˆbq, with εkνσ=εkν−2∆δσ↑being the mode and spin dependent electron energies that are calcu- lated from first-principles calculations and ℏωqbeing the magnon energies. Note that we have absorbed the term zero-th order in magnon variables in Equation (1) in the otherwise spin-independent ˆHe. Next, we use the Hamiltonian introduced above to construct a quantum kinetic approach for the descrip- tion of the out-of-equilibrium dynamics of electrons and magnons. We define the rates of energy exchange be- tween both subsystems as ˙Em=X qℏωq˙nq (2) ˙Ee=X kνσεkνσ˙fkνσ=−X qℏωq˙nq. (3) where the dot represents temporal derivative and with the electron ( fkνσ) and magnon ( nq) occupation num- bers. The equivalence in Equation (3) results from the conservation of total energy. The time derivatives of the occupation numbers can be calculated by applying Fermi’s golden rule to the scattering Hamiltonian (1).3 To simplify the calculations, we further assume a ther- mal electron distribution and can hence introduce a sin- gle electronic temperature Tethat relates to the occu- pation of electronic states via the Fermi-Dirac distribu- tion. This allows us to apply and also extend (by in- cluding terms second order in the bosonic operators) the ideas laid out in Allen’s seminal work on electron-phonon interaction [82] to electron-magnon scattering, yielding ˙nq= nBE(ωq, Te)−nq γq+P q′ (nq+ 1)nq′nBE(ωq− ωq′, Te)+(q↔q′) Γqq′, with nBE(ωq, Te) = [eℏωq kBTe−1]−1 being the Bose-Einstein distribution evaluated at the electron temperature. The scattering rates are given by γq=4π∆2 SNωqI↑↓(Te)X kνν′δ(εF−εk−qν↑)δ(εF−εkν′↓), (4) Γqq′=2π∆2 S2N2(ωq−ωq′)X σIσσ(Te) ×X kνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ),(5) with εFbeing the Fermi energy. The functions Iσσ′(Te) have the property lim Te→0Iσσ′(Te) = 1 and account for the smearing of the Fermi-Dirac distribution at high electron temperatures, similar to what has been derived for electron-phonon scattering [35]. The expression for Iσσ′(Te) and details of the derivation of Equations (4)– (5) are in the Method Section V A. Note that a com- parison with linear spin-wave theory in the framework of the Landau-Lifshitz-Gilbert equation [83] reveals that γq/ωq=αqcan be viewed as a mode-dependent Gilbert damping parameter. Due to the assumption that the electron occupation numbers follow the Fermi-Dirac distribution at all times, the change in electron energy is determined by the change in Te, i.e., ˙Ee=P kνσεkνσ(∂fkνσ/∂T e)˙Te= Ce˙Te, with the electronic heat capacity Ce=P kνσεkνσ(∂fkνσ/∂T e). By additionally considering the absorption of a laser pulse with power P(t) by the electrons and a coupling of the electrons to a phonon heat bath as in the 2TM, we finally obtain our out-of- equilibrium magnon dynamics model: ˙nq=h nBE(ωq, Te)−nqi γq +X q′h (nq+ 1)nq′nBE(ωq−ωq′, Te) + (q↔q′)i Γqq′, (6) ˙Te=1 Ceh −X qℏωq˙nq+Gep(Tp−Te) +P(t)i , (7) ˙Tp=−Gep Cp(Tp−Te). (8) Here, Tp,Cpand Gepare the phonon temperature and heat capacity and electron-phonon coupling con-stant, respectively. Note that we do not consider di- rect magnon-phonon coupling, which has been shown to be a reasonable approximation for 3 dferromagnets [43, 44]. We would like to point out that the non- thermal magnon occupations nqcan be translated to mode-specific temperatures via the Bose-Einstein distri- bution, Tq:=ℏωq/(kBln(n−1 q+ 1)). Based on this – and in distinction from the 3TM – we term the framework provided by Equations (6)-(8) the (N+2)-temperature model ((N+2)TM). Below, we reveal by solving these coupled equations numerically that they provide a vi- able framework to describe laser-induced ultrafast mag- netization dynamics and the generation of non-thermal magnons, going beyond the well-established 3TM. Before doing so, we want to shortly discuss the relation between the (N+2)TM introduced here and the 3TM. Al- beit their phenomenological nature, the 2TM ( TeandTp) and the 3TM ( Te,TpandTm) have been successfully ap- plied to explain a plethora of phenomena [84], perhaps most prominently by Beaurepaire et al. to describe the ultrafast demagnetization of Ni [1]. Allen [82] and Man- chon et al. [78] demonstrated that the 2TM and the 3TM can be derived from a microscopic out-of-equilibrium ap- proach similar to the one used here. By assuming instan- taneous relaxation of the magnon occupation numbers to the Bose-Einstein distribution with a single magnon temperature Tm, our (N+2)TM reduces to the 3TM (in absence of magnon-phonon coupling), Cm˙Tm=Gem(Te−Tm), Ce˙Te=Gem(Tm−Te) +Gep(Tp−Te) +P(t), Cp˙Tp=Gep(Te−Tp),(9) with the magnon heat capacity Cm=P qCq=P qℏωq(∂nq/∂T m) and the electron-magnon coupling constant Gem=X qCqh γq+X q′kBTm ℏωqΓqq′i . (10) Details of the derivation are found in Method Section V B. The above expression goes beyond what was derived in Ref. [78] by including terms second order in magnon variables and allows us to determine the electron-magnon coupling fully based on ab initio parameters. We would like to point out that it can be extended further by going to higher order in the magnon variables. III. RESULTS A. Magnon lifetimes and Gilbert damping We apply the (N+2)TM model defined by Equations (6)-(8) to bcc iron. To obtain a full solution of the out- of-equilibrium dynamics, it is required to calculate ma- terial specific quantities. First, we estimate ∆ ≈0.75 eV from the band structure and with that we compute the4 Γ H N Γ P H020406080100120magnon frequency (THz) 100101102103 lifetime (ps) Figure 1. Magnon dispersion of bcc iron with lifetimes γ−1 q given as color code, shown along high-symmetry lines of the BZ. The lifetimes are due to the first-order contribution to the electron-magnon scattering. quantities γq, Γqq′andIσσ′(Te), both using the full- potential linear augmented plane wave code ELK [85] (details can be found in the Method Section V C). For bcc iron it turns out that Iσσ′(Te) only scales weakly with temperature and hence we use the low tempera- ture limit Iσσ′(Te) = 1 hereinafter. The parameters gov- erning the magnon energies ℏωq=S(2d+P jJij[1− exp(−iq·(rj−ri))]) were taken from earlier works: the exchange constants Jijare from first-principles calcula- tions [86] and the magneto-crystalline anisotropy energy d= 6.97µeV per atom is from experiments [87]. Further, we used the saturation moment µs= 2.2µBand spin S= 2.2/2. Based on these parameters and the formulas derived above, we get Cm= 5.720×104Jm−3K−1and Gem= 6.796×1017Wm−3K−1atTm= 300 K. Notably, the term first order in magnon variables leads to a con- tribution to Gemthat is one order of magnitude smaller than the second-order term. We further use the room- temperature values Ce= 1.013×105Jm−3K−1,Cp= 3.177×106Jm−3K−1andGep= 1.051×1018Wm−3K−1 that were obtained in Refs. [35, 37] from first-principles calculations. The influence of a temperature dependent electronic heat capacity Ceon the demagnetization is dis- cussed in the Supporting Information. Both ωqand the inverse of γq, i.e., the lifetime of magnons due to the contribution to electron-magnon scattering linear in the magnon variables, are shown in Figure 1 along high-symmetry lines of the Brillouin zone (BZ). It can readily be observed that the lifetimes of high-frequency magnons are drastically reduced as com- pared to the low energy ones. The q-dependent lifetimes give rise to mode-specific Gilbert damping αq(=ωq/γq). Our finding of mode-dependent Gilbert damping is con- sistent with experiments [88] and also with a recent field- theory derivation [89]. The computed αqvalues, shown in Method Section V C, range between 1 .5×10−3and 1.08×10−2. These values are close to the experimentally obtained ones (via FMR measurements) for Fe ranging from 1 .9×10−3to 7.2×10−3[90–95], however with a 0 1 2 3300400500600700temperature T (K)(a) 0 1 2 3 timet(ps)−1.0−0.50.0∆M/M 0(%) (b)Te Tp /angbracketleftTq/angbracketrightT3TM e T3TM p T3TM m 0.0 0.2 0.4 0.6 0.8 1.0 laser fluence (mJ /cm2)0510demagnetization (%) (c)Figure 2. Laser-induced ultrafast non-equilibrium dynamics of iron calculated from an ab initio parameterized model. (a) Temporal evolution of electron temperature Te, phonon temperature Tpand average magnon temperature ⟨Tq⟩= 1/NP qTqobtained by the (N+2)TM (solid lines). The blue shaded region indicates the temperature range within which all magnon temperatures are contained. Dashed lines show the results of the 3TM solved with ab initio calcu- lated input parameters. (b) Relative change of total mag- netization of the localized magnetic moments ∆ M/M 0=P q(ninit q−nq)/(NS−P qninit q), with ninit q=nBE(ωq,300 K) being the occupation number before the laser pulse. (c) De- magnetization max( |∆M/M 0|) versus laser fluence computed for a ferromagnetic layer with a thickness of 20 nm. The dot- ted line serves as a guide to the eye. somewhat larger variation with qas compared to what was reported in Ref. [83]. We note that the q-dependent Gilbert damping goes beyond the conventional LLG description which assumes one single damping parameter for all spin dynamics. Moreover, a further distinction between the current the- ory and the LLG framework is that, in the latter, there is a single damping term that governs both the energy and angular momentum transfer [96], whereas the cur- rent theory has two terms [see Equation (1)], one that transfers energy and angular momentum and one that transfers only energy. As shown in the following, this 2nd term is found to be important for non-thermal magnon5 generation. B. Ultrafast dynamics Based on the above-given parameters, we calculate the coupled out-of-equilibrium magnon, electron, and phonon dynamics induced by a Gaussian laser pulse P(t) = A/p 2πζ2exp[−(t/ζ)2/2] with A= 9.619× 107Jm−3andζ= 60 fs for N= 203magnon modes. Note that this value of Atranslates to an absorbed fluence of 0.19 mJ /cm2for a ferromagnetic layer with thickness of 20 nm, which is a typical thickness in ultrafast demagne- tization experiments [1]. Figure 2(a) depicts the time evolution of electron, phonon and average magnon temperature – together with the temperature range of all magnon temperatures – cal- culated using the (N+2)TM. The electron temperature reaches a maximum of 685 K at around 52 fs after the maximum of the laser pulse (located at t= 0) and con- verges to the phonon temperature in less than 1 .5 ps. The maximum of the average magnon temperature of 520 K is reached only slightly after the electronic one at around 136 fs, followed by a convergence to the electronic and phononic temperature to a final temperature of around 329 K at 3 ps, in agreement with what can be estimated from the energy supplied by the laser pulse and the in- dividual heat capacities via ∆ T=A/(Cm+Ce+Cp) = 28.8 K. Notably, the magnon temperatures still cover a range of around 50 K at this point in time. Our results clearly demonstrate the shortcomings of the conventional 3TM (shown as dotted lines): While the initial increase of temperatures is comparable to the (N+2)TM, magnon thermalization happens much faster in the 3TM. In Figure 2(b), we show the laser-induced change in magnetization (associated with the localized magnetic moments) due to the creation of additional magnons. We observe ultrafast transversal demagnetization of around one percent in less than 300 fs, demonstrating that the timescales obtained by our ab initio based calculations are in reasonable agreement with experimental measure- ments (see, e.g., [15, 97–99]). Notably, the minimum of the magnetization and the maximum in the average magnon temperature computed by the (N+2)TM are at different points in time. Also, the drop in the (localized) magnetization is much less pronounced than expected from the increase in average temperature: in thermal equilibrium, a temperature increase from 300 K to above 500 K approximately leads a demagnetization of 20% for iron [100]. These observations clearly demonstrate the shortcomings of the 3TM – where a thermal magnon dis- tribution at all times is assumed – and underline the im- portance of treating the full, non-thermal magnon distri- bution in the ultrafast regime. Figure 2(c) depicts the maximum of the demagnetiza- tion versus laser fluence for an iron layer of 20 nm. We find a nonlinear dependence, which is a result of the non- linearity of our (N+2)TM, and a substantial demagneti- 0 2 4 6 8 10 timet(ps)0.850.900.951.00M/M 0 (N+2)TM experimentFigure 3. Comparison between experiment and the (N+2)TM theory for ultrafast demagnetization in iron. The experimen- tal data (symbols) are those of Carpene et al. [15] and the solid lines are calculated from the ab initio parameterized (N+2)TM. zation of around ten percent at 0 .95 mJcm−2. We note that for high fluences, higher-order magnon-magnon scat- tering terms that are not included in the current model could start to play a role. The obtained amount of demagnetization and the mag- netization decay time (below 200 fs) for this fluence are comparable with experiments, which suggests that ultra- fast magnon excitation [15–17] provides a viable mech- anism for ultrafast laser-induced demagnetization. It is also consistent with time-resolved extreme ultraviolet magneto-optical and photoemission investigations that detected magnon excitations during ultrafast demagneti- zation of elemental ferromagnetic samples [18, 101]. For a more precise examination of the predictions of the (N+2)TM, we compare the calculated time- dependent demagnetization with experimental data for Fe in Figure 3. The experimental data were measured by Carpene et al. [15] on a 7-nm thin film, using the time-resolved magneto-optical Kerr effect for two differ- ent pump laser fluences of 1 .5 mJcm−2and 3 mJcm−2. In the calculations we used an absorbed laser fluence that is about five times lower, as the exact value of the absorbed fluence in experiments is difficult to estimate (due to influence of optical losses, sample reflection, etc). Specifically, in the simulations we used absorbed laser energies of 433 Jcm−3and 693 Jcm−3in a 7-nm Fe film. Figure 3 exemplifies that not only the amount of demagnetization but also the full time dependence of the demagnetization predicted by the (N+2)TM is in remarkable agreement with experiments. C. Non-thermal magnon dynamics Next, we analyze the non-thermal magnon dynam- ics in more detail in Figure 4. There, we show the magnon temperatures versus frequency (a) and along high-symmetry lines of the BZ (b) at different points in time. The laser pulse primarily heats up high energy magnons, while the temperature of low energy magnons6 300500 −250 fs 300500 0 fs 300500temperature (K) 250 fs 300500 500 fs 0 20 40 60 80 100 120 magnon frequency (THz)300500 1000 fs Γ H N Γ P H−200 fs−100 fs0 fs100 fs200 fs300 fs400 fs500 fs600 fs700 fs800 fs900 fs1000 fs 300350400450500550600 temperature (K)(a) (b) Figure 4. Magnon temperatures of iron during ultrafast laser excitation at different points in time (w.r.t. the maximum of the laser pulse) calculated from the ab initio parameterized (N+2)TM. (a) Magnon temperatures (dots) versus frequency. The solid line indicates the electron temperature. (b) Magnon dispersion and their temperatures, depicted by the color code, shown along high-symmetry lines of the BZ. barely changes and even decreases slightly in the vicinity of the Γ point (the temperatures drop by up to around 2.5 K). This surprising observation is caused by a redistribution of magnons from this region to other parts of the BZ due to the term second order in the magnon operators in Equation (1); the effective second order scattering rate γ(2) q:=P q′Γqq′is negative for low magnon frequencies (more details can be found in the Method Section V C). It is also observed that although the magnon temperatures reached after the laser pulse are generally higher at higher frequencies, however, there is not necessarily a monotonous increase of temperature with frequency at all times: e.g., at 100 fs after the laser pulse [Figure 4(b)], the temperatures at the points H, N, and P is higher than in between these points. Notably, the position of the maximum magnon temperature in the BZ also varies with time. D. Discussion Different physical mechanisms have been proposed for ultrafast demagnetization in elemental 3 dferromagnets [9, 11, 15, 18]. The preeminent mechanisms are Elliott- Yafet (EY) electron-phonon spin-flip scattering [11, 13] and ultrafast magnon generation [15]. In the former, a Stoner-type picture is used to model the longitudinal re- duction of the atomic moment due to electron-phonon spin-flip scattering, whereas the latter is based on length- conserving transverse spin-wave excitations. Experimen- tal indications of electron-phonon scattering [38, 39] aswell as of electron-magnon scattering have been reported [18, 101]. The strength of the different demagnetization channels is an important issue in the on-going discussion on the dominant origin of ultrafast demagnetization [7]. Ab ini- tiocalculated quantities such as the EY spin-flip prob- ability are essential to achieve reliable estimates [102– 104]. Griepe and Atxitia [14] recently employed the microscopic 3TM [11] and obtained quantitative agree- ment with measured demagnetizations for the elemental 3dferromagnets. They compared the fitted EY spin- flip probability αsfwith ab initio calculated values [104] and found these to be in good agreement, in support of an electron-phonon mechanism of ultrafast demagnetiza- tion. A drawback of their employed approach is how- ever that only magnetization reducing spin flips are in- cluded. EY spin flips that increase the magnetization are also possible and, including these would lead to a signifi- cantly smaller demagnetization amplitude [104]. This in turn would question again what amount of demagneti- zation is precisely due to EY electron-phonon spin flip scattering. Conversely, in our non-thermal magnon ap- proach we employ ab initio calculated quantities without fit parameter. We find that the ab initio predicted ultra- fast demagnetization agrees accurately with experiments, which provides a strong support for the prominence of the non-thermal magnon channel to the ultrafast demagne- tization process.7 IV. CONCLUSIONS We have developed an ab initio parameterized quan- tum kinetic approach to study the laser-induced gen- eration of magnons due to electron-magnon scattering, which we applied to iron. Our results clearly demon- strate that on ultrafast timescales the magnon distribu- tion is non-thermal and that henceforth the simple re- lation between magnetization and temperature via the M(T) curves computed at equilibrium does not hold: since predominantly high-energy magnons are excited the energy transferred from the laser-excited electrons cre- ates relatively few magnons and hence the demagneti- zation (proportional to the total number of magnons) is much less pronounced than expected from the increaseof the average magnon temperature. Notably, the num- ber of magnons actually decreases near the center of the Brillouin zone, which is due to the scattering from low to high energy magnons by a previously neglected scattering term that can transfer energy but not angular momen- tum. This term, which is not included in LLG simula- tions, is a crucial quantity for out-of-equilibrium magnon dynamics. Ourab initio -based calculations of the induced demag- netization in iron furthermore provide strong evidence that non-thermal magnons are excited fast and lead to a sizable demagnetization within 200 fs. The result- ing time-dependent demagnetization agrees remarkably well with experiments, which establishes the relevance of magnon excitations for the process of ultrafast optically induced demagnetization. V. METHOD A. Derivation of electron-magnon scattering rates In this Method Section we derive the (N+2)TM for the description of non-thermal magnons from a microscopic Hamiltonian for electron-magnon scattering. We start with a local sp−dmodel Hamiltonian, ˆHem=−Jsp−dX iδ(r−ri)ˆsitin·Sloc i, (11) with Jsp−dbeing the sp−dvolume interaction energy, ˆsitin=ˆσbeing the spin operators of itinerant ( sandp) electrons and Sloc ibeing the localized ( d) spins located at ri. For now, we treat the latter as classical vectors. The expectation value for a given spin wave function Ψ(r) is given by ⟨ˆHem⟩=−Jsp−dX iZ Ψ†(r)δ(r−ri)ˆsitin·Sloc iΨ(r)dr (12) =−Jsp−dX iZ δ(r−ri)Ψ∗ ↑(r),Ψ∗ ↓(r)
 ˆσxSx i+ ˆσySy i+ ˆσzSz i  Ψ↑(r) Ψ↓(r) dr (13) =−Jsp−dX iZ δ(r−ri)( Ψ∗ ↑(r)Ψ↓(r)S− i+ Ψ∗ ↓(r)Ψ↑(r)S+ i+ (Ψ∗ ↑(r)Ψ↑(r)−Ψ∗ ↓(r)Ψ↓(r))Sz i) dr.(14) Here, we have introduced S± i=Sx i±iSy i. Next, we perform a plane wave expansion of the wave functions (for a single band of itinerant electrons), Ψσ(r) =1√ VX keik·rckσ, (15) and a Holstein-Primakoff transformation of the localized spins, S+ i=p 2S−b∗ ibibi, S− i=b∗ ip 2S−b∗ ibi, Sz i=S−b∗ ibi, (16) together with introducing the Fourier transform of the magnon amplitudes b∗ i=1√ NX qe−iq·rib∗ q, b i=1√ NX qeiq·ribq. (17)8 Insertion of (15)–(17) into (14) and keeping terms up to second order in magnon variables, we get ⟨ˆHem⟩=−Jsp−d VX iX kk′(r 2S NX qe−i(k−k′+q)·ric∗ k↑ck′↓b∗ q+r 2S NX qe−i(k−k′−q)·ric∗ k↓ck′↑bq +Se−i(k−k′)·ri(c∗ k↑ck′↑−c∗ k↓ck′↓)−1 NX qq′e−i(k−k′+q−q′)·ri(c∗ k↑ck′↑−c∗ k↓ck′↓)b∗ qbq′) (18) =−Jsp−dSN VX k(c∗ k↑ck↑−c∗ k↓ck↓)−Jsp−dSN VX kqr 2 SN c∗ k+q↑ck↓b∗ −q+c∗ k+q↓ck↑bq +Jsp−d VX kqq′ c∗ k−q+q′↑ck↑−c∗ k−q+q′↓ck↓ b∗ qbq′.(19) For multiple itinerant bands and in second quantization we obtain ˆHem=−∆X kν(ˆc† kν↑ˆckν↑−ˆc† kν↓ˆckν↓)−∆r 2 SNX kνν′,q ˆc† k+qν↑ˆckν′↓ˆb† −q+ ˆc† k+qν↓ˆckν′↑ˆbq +∆ SNX kνν′,qq′ ˆc† k−q+q′ν↑ˆckν′↑−ˆc† k−q+q′ν↓ˆckν′↓ ˆb† qˆbq′.(20) where we have introduced ∆ =Jsp−dSN V. Note that due to the plane wave ansatz we have implicitly assumed that the itinerant electrons are completely delocalized and interband scattering (from νtoν′̸=ν) fully contributes to the electron-magnon scattering. Next, we use Fermi’s golden rule to get the change of the magnon occupation number nq=⟨ˆb† qˆbq⟩. Fermi’s golden rule computes the probability W(i→f) for a small perturbation term in the Hamiltonian, ˆH′(in our specific case, ˆHem) via W(i→f) =2π ℏ|⟨f|ˆH′|i⟩|2δ(Ef−Ei), (21) where |i⟩and|f⟩denote the initial and final state, respectively. We start with the term first order in the magnon variables, ˙n(1) q=W(nq→nq+ 1)−W(nq→nq−1) =2π ℏ2∆2 SNX kνν′ (1−fk−qν↑)fkν′↓−(fk−qν↑−fkν′↓)nq δ(εkν′↓−εk−qν↑−ℏωq),(22) with fkνσ=⟨ˆc† kνσˆckνσ⟩andεkνσandℏωqbeing the eigenenergies of electrons and magnons, respectively. Hereinafter, we make the assumption that due to the fast equilibration processes for electrons, they always follow the Fermi-Dirac distribution, fFD(εkνσ, Te) = [e(εkνσ−εF)/kBTe+ 1]−1,with a single electron temperature Te. Before we continue we need the following relation, fFD(εkν′↓, Te)(1−fFD(εk−qν↑, Te))δ(εkν′↓−εk−qν↑−ℏωq) = (fFD(εk−qν↑, Te)−fFD(εkν′↓, Te))nBE(ωq, Te)δ(εkν′↓−εk−qν↑−ℏωq)(23) with nBE(ωq, Te) = [eℏωq kBTe−1]−1being the Bose-Einstein distribution evaluated at the electron temperature. Now we can simplify Equation (22), yielding ˙n(1) q≈2π ℏ2∆2 SNX kνν′ nBE(ωq, Te)−nq (fFD(εk−qν↑, Te)−fFD(εkν′↓, Te))δ(εkν′↓−εk−qν↑−ℏωq) = nBE(ωq, Te)−nq γq. (24)9 With γqbeing the linewidth – i.e., the inverse lifetime – of the magnon due to the first order contribution to electron- magnon scattering. Following the ideas laid out by Allen [82] and Maldonado et al. [35], it can be computed as γq=2π ℏ2∆2 SNX kνν′[fFD(εk−qν↑, Te)−fFD(εkν′↓, Te)]δ(εkν′↓−εk−qν↑−ℏωq) (25) =2π ℏ2∆2 SNX kνν′Z dε δ(ε−εk−qν↑)Z dε′δ(ε′−εkν′↓)[fFD(ε, Te)−fFD(ε′, Te)]δ(ε′−ε−ℏωq) (26) ≈2π ℏ2∆2 SNX kνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z dεZ dε′[fFD(ε, Te)−fFD(ε′, Te)]δ(ε′−ε−ℏωq)g↑(ε)g↓(ε′) g↑(εF)g↓(εF) (27) ≈2π ℏ2∆2 SNℏωqX kνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z dε(−1)∂fFD(ε, Te) ∂εg↑(ε)g↓(ε+ℏωq) g↑(εF)g↓(εF)(28) ≈2π ℏ2∆2 SNℏωqX kνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)Z dε(−1)∂fFD(ε, Te) ∂εg↑(ε)g↓(ε) g↑(εF)g↓(εF)(29) =4π∆2 NSωqX kνν′δ(εF−εk−qν↑)δ(εF−εkν′↓)I↑↓(Te) (30) with εFbeing the Fermi energy, the spin-dependent density of states is gσ(ε) =P kνδ(ε−εkνσ) and the thermal correction factor given by Iσσ′(Te) =Z dε(−1)∂fFD(ε, Te) ∂εgσ(ε)g′ σ(ε) gσ(εF)g′σ(εF). (31) It is obvious that lim Te→0Iσσ′(Te) = 1. Note that we have used that the energy scale of magnons is much smaller than the one of electrons, i.e., that ℏωq≪ε, ε′. The contribution of the term second order in magnon variables to the occupation number can be calculated analogous and reads ˙n(2) q=2π ℏ∆ SN2X kνν′σ,q′n (nq+ 1)nq′ (1−fFD(εk−q+q′νσ, Te))fFD(εkν′σ, Te)δ(ℏωq−ℏωq′+εk−q+q′νσ−εkν′σ) − q↔q′o (32) =2π ℏ∆ SN2X kνν′σ,q′n (nq+ 1)nq′nBE(ωq−ωq′, Te) fFD(εk−q+q′νσ, Te)−fFD(εkν′σ, Te) × δ(ℏωq−ℏωq′+εk−q+q′νσ−εkν′σ)− q↔q′o(33) ≈2π ℏ∆ SN2X kνν′σ,q′n (nq+ 1)nq′nBE(ωq−ωq′, Te)(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te)− q↔q′o (34) =2π ℏ∆ SN2X q′n (nq+ 1)nq′nBE(ωq−ωq′, Te) + q↔q′
oX kνν′σ(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te) (35) =2π ℏ∆ SN2X q′n (nq+ 1)nq′nBE(ωq−ωq′, Te) + q↔q′
oX kνν′σ(ℏωq−ℏωq′)δ(εF−εk−q+q′νσ)δ(εF−εkν′σ)Iσσ(Te) (36) =X q′n (nq+ 1)nq′nBE(ωq−ωq′, Te) + q↔q′
o Γqq′(Te)(37) with Γqq′(Te) =2π ℏ∆ SN2 (ℏωq−ℏωq′)X σIσσ(Te)X kνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ). (38)10 B. Derivation of the three temperature model In what follows, it is demonstrated that the three temperature model (3TM) can be obtained from the (N+2)- temperature model derived in the main text, ˙nq=h nBE(ωq, Te)−nqi γq+X q′h (nq+ 1)nq′nBE(ωq−ωq′, Te) + (q↔q′)i Γqq′,(39) ˙Te=1 Ceh −X qℏωq˙nq+Gep(Tp−Te) +P(t)i , (40) ˙Tp=−Gep Cp(Tp−Te), (41) by assuming instantaneous relaxation of the magnon occupation numbers to the Bose-Einstein distribution with a single magnon temperature Tm, i.e., nq=nBE(ωq, Tm). For the sake of readability we rewrite nBE(ωq, Tm) =nq(Tm). We start with the first order scattering term: ˙n(1) q= [nq(Te)−nq(Tm)]γq≈(Te−Tm)∂nq(T) ∂T T=Tmγq(Te) = (Te−Tm)Cqγq ℏωq. (42) Here we have introduced the mode-dependent magnon heat capacity Cq=ℏωq∂nq(Tm) ∂T. In order to calculate the scattering term second order in the magnon variables, we first introduce the following relation nq′(Tm) + 1
nq(Tm) = nq′(Tm)−nq(Tm) nq−q′(Tm). (43) Now we calculate ˙n(2) q=X q′ (nq(Tm) + 1) nq′(Tm)nq−q′(Te) + (q↔q′) Γqq′ (44) =X q′ nq′−q(Tm)nq−q′(Te)−(q↔q′) × nq(Tm)−nq′(Tm)
Γqq′ (45) =X q′1 2 cothℏ(ωq′−ωq) 2kBTe −cothℏ(ωq′−ωq) 2kBTm nq(Tm)−nq′(Tm)
Γqq′ (46) ≈X q′nq(Tm)−nq′(Tm) ℏ(ωq′−ωq)kB(Te−Tm)Γqq′ (47) ≈X q′∂nq(Tm) ∂(ℏωq)kB(Tm−Te)Γqq′ (48) =X q′∂nq(T) ∂T T=TmkBTm ℏωq(Te−Tm)Γqq′ (49) =X q′CqkBTm (ℏωq)2(Te−Tm)Γqq′. (50) Using the expressions for ˙ n(1) qand ˙n(2) q, the change in total energy of the magnons can then be calculated as ∂Em ∂t=∂Em ∂Tm∂Tm ∂t=X qℏωq∂nq(T) ∂T|T=Tm | {z } Cm∂Tm ∂t= (Te−Tm)X qCq γq+X q′kBTm ℏωqΓqq′ . | {z } Gem(51) With that, the (N+2)TM transforms into the 3TM (in the absence of magnon-phonon coupling), which is given by Cm˙Tm=Gem(Te−Tm), Ce˙Te=Gem(Tm−Te) +Gep(Tp−Te) +P(t), Cp˙Tp=Gep(Te−Tp).(52)11 C.Ab initio calculations To obtain a full solution of the (N+2)TM, it is necessary to compute the material specific quantities ∆, γq, Γqq′ andIσσ(Te). For this purpose, we use the full-potential linear augmented plane wave code ELK [85]. As a first step, we determine the coupling parameter ∆ of the sp−dmodel, which sets the general scale of the electron-magnon scattering. As shown in the main text, the first term (zeroth order in magnon variables) in the electron-magnon scattering Hamiltonian reads ˆH(0) em=−∆P kν(ˆc† kν↑ˆckν↑−ˆc† kν↓ˆckν↓), with ν∈ {s, p}. Based on this, ∆ can be estimated from the projected density of states (DOS), since it is one half of the spin-dependent energy splitting of the s- and p-bands. In general, this splitting may vary for different electronic states. This is not accounted for in the model used here, where instead a single parameter is used to model the spin splitting. We find, however, that for bcc iron this is justified, since the shift in both s- and p-bands around the Fermi energy – the relevant region for electron-magnon scattering – between spin up and down states is approximately constant with a value of ∆≈0.75 eV, see left panel of Figure 5. Now we calculate the first and second order scattering rates using the formulas derived above, γq=4π∆2 SNωqI↑↓(Te)X kνν′δ(εF−εk−qν↑)δ(εF−εkν′↓), (53) Γqq′=2π∆2 S2N2(ωq−ωq′)X σIσσ(Te)X kνν′δ(εF−εk−q+q′νσ)δ(εF−εkν′σ). (54) The calculation of both quantities requires a spin-dependent summation over the Fermi surface, analogous to what was done in Ref. [103] for the evaluation of the spin-dependent Eliashberg function for electron-phonon scattering. As in Ref. [103] we use a Gaussian broadening of the Dirac delta distributions by 0 .03 eV. Also, since we only include the contribution of s- and p-states (indicated by ν, ν′) to the scattering, we have to project the Kohn-Sham states (indicated by n, n′) onto the spherical harmonics Ym lvia δ(εF−εkνσ)δ(εF−εk′ν′σ′) =X nn′Pnν kσPn′ν′ k′σ′δ(εF−εknσ)δ(εF−εk′n′σ′), (55) with Pnν kσbeing projector functions. The functions Iσσ′(Te) describe corrections to the scattering rate at high electron temperatures and are given by Iσσ′(Te) =Z dε(−1)∂fFD(ε, Te) ∂εgσ(ε)g′ σ(ε) gσ(εF)g′σ(εF), (56) with gσ(ε) =P kνδ(ε−εkνσ) =P kνP nPnν kσδ(ε−εknσ) being the cumulative DOS of both s- and p-states. We find that they increase monotonously with the electron temperature (see right panel of Figure 5). However, even for temperature up to 2000 K, the Iσσ′(Te) functions are below two. Hence, we concluded that the approximation Iσσ′= 1 is reasonable for the laser fluences – heating the electrons up to around 700 K – considered in the main text. Figure 6 depicts the numerically calculated scattering rates using Iσσ′= 1 and ∆ = 0 .75 eV as obtained above. In the left panel, we show the scattering rate γqthat is first order in the magnon variables through color code on the magnon dispersion. It is strictly positive and tends to increase with magnon frequency. The right panel shows the effective scattering rate γ(2) q=P q′Γqq′due to the scattering term second order in magnon variables. Notably, this quantity is negative for low frequencies and positive for high frequencies, indicating that it leads to a depopulation of magnons at low energies due a scattering from low to high energies (the total magnon number is kept constant). In general, the values of the effective second order scattering rate are comparable to the one first order in magnon variables. They are, however, distributed differently: e.g., for magnons close to the Γ point the second order scattering rate is by far the dominating one. This is the reason why, as demonstrated in the main text, a laser pulse can in fact lead to a cooling of low energy magnons, i.e., to a decrease of their occupation numbers. Lastly, we show in Figure 7 the ab initio computed mode-dependent Gilbert damping, αq=ωq/γq. Interestingly, the Gilbert damping αqis large ( ∼0.01) at the BZ center and at the high-symmetry points H, N and P at the BZ edge. There is also a noticeable directional anisotropy in the Gilbert damping for modes along Γ −H and Γ −P. We emphasize that the Gilbert damping is here due to the electron-magnon scattering term that is first order in the magnon variables. Other scattering mechanisms as phonon-magnon scattering could contribute further to the mode-specific Gilbert damping.12 −10−5 0 5 10 energyε−εF(eV)−0.04−0.020.000.020.040.06projected DOS (eV−1)s p 500 1000 1500 2000 electron temperature Te(K)1.01.21.41.61.8thermal correction factorI↑↓ I↑↑ I↓↓ Figure 5. Left: Projected spin-polarized DOS for bcc iron. Spin-minority density is shown by positive values, spin-majority density by negative values. The exchange splitting is 2∆ ≈1.5 eV in a large interval around the Fermi energy and for both s- andp-states. Right : Thermal correction factors Iσσ′versus electron temperature Tecalculated from the projected DOS. Γ H N Γ P H020406080100120magnon frequency (THz) 12345 scattering rate γq(THz) Γ H N Γ P H020406080100120magnon frequency (THz) −4−3−2−101 scattering rate γ(2) q(THz) Figure 6. Magnon dispersion of bcc iron along high-symmetry lines of the Brillouin zone. The color coding describes ( left) the scattering rates γqdue to the electron-magnon scattering term first order in magnon variables γqand ( right) the effective scattering rate γ(2) q=P q′Γqq′due to the term second order in magnon variables, calculated with Iσσ′= 1 and ∆ = 0 .75 eV. Γ H N Γ P H020406080100120magnon frequency (THz) 0.0020.0040.0060.0080.010 Gilbert damping αq Figure 7. Calculated mode-specific Gilbert damping αq=ωq/γq, depicted by the color code on the magnon dispersion of bcc iron. The mode-specific Gilbert damping αqis due to the electron-magnon scattering term first order in magnon variables.13 ACKNOWLEDGMENTS The authors thank K. Carva for valuable discussions. This work has been supported by the Swedish Re- search Council (VR), the German Research Foundation (Deutsche Forschungsgemeinschaft) through CRC/TRR 227 “Ultrafast Spin Dynamics” (project MF, project-ID: 328545488), and the K. and A. Wallenberg Foundation (Grant No. 2022.0079). Part of the calculations were en- abled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS) at NSC Link¨ oping partially funded by the Swedish Re- search Council through grant agreement No. 2022-06725.CONFLICT OF INTEREST The authors declare no conflict of interest. DATA AVAILABILITY STATEMENT Data available on request from the authors. KEYWORDS Ultrafast magnetism, electron-magnon coupling, non- thermal magnons [1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. 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2023-09-25

Ultrafast laser excitation of ferromagnetic metals gives rise to correlated, highly non-equilibrium dynamics of electrons, spins and lattice, which are, however, poorly described by the widely-used three-temperature model (3TM). Here, we develop a fully ab-initio parameterized out-of-equilibrium theory based on a quantum kinetic approach--termed (N+2) temperature model--that describes magnon occupation dynamics due to electron-magnon scattering. We apply this model to perform quantitative simulations on the ultrafast, laser-induced generation of magnons in iron and demonstrate that on these timescales the magnon distribution is non-thermal: predominantly high-energy magnons are created, while the magnon occupation close to the center of the Brillouin zone even decreases, due to a repopulation towards higher energy states via a so-far-overlooked scattering term. We demonstrate that the simple relation between magnetization and temperature computed at equilibrium does not hold in the ultrafast regime and that the 3TM greatly overestimates the demagnetization. The ensuing Gilbert damping becomes strongly magnon wavevector dependent and requires a description beyond the conventional Landau-Lifshitz-Gilbert spin dynamics. Our ab-initio-parameterized calculations show that ultrafast generation of non-thermal magnons provides a sizable demagnetization within 200fs in excellent comparison with experimentally observed laser-induced demagnetizations. Our investigation emphasizes the importance of non-thermal magnon excitations for the ultrafast demagnetization process.

Ultrafast Demagnetization through Femtosecond Generation of Non-thermal Magnons

2309.14167v3

arXiv:1102.4551v1 [cond-mat.mtrl-sci] 22 Feb 2011APS/123-QED Ab-initio calculation of the Gilbert damping parameter via linear response formalism H. Ebert, S. Mankovsky, and D. K¨ odderitzsch University of Munich, Department of Chemistry, Butenandtstrasse 5-13, D-81377 Munich, Germany P. J. Kelly Faculty of Science and Technology and MESA+ Institute for Na notechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands (Dated: October 14, 2018) A Kubo-Greenwood-like equation for the Gilbert damping par ameterαis presented that is based on the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn- Rostoker (KKR) band structure method in combination with Co herent Potential Approximation (CPA) alloy theory allows it to be applied to a wide range of si tuations. This is demonstrated with results obtained for the bcc alloy system Fe xCo1−xas well as for a series of alloys of permalloy with 5d transition metals. To account for the thermal displaceme nts of atoms as a scattering mechanism, an alloy-analogy model is introduced. The corresponding ca lculations for Ni correctly describe the rapid change of αwhen small amounts of substitutional Cu are introduced. PACS numbers: Valid PACS appear here I. INTRODUCTION The magnetization dynamics that is relevant for the performance of any type of magnetic device is in gen- eral governed by damping. In most cases the magneti- zation dynamics can be modeled successfully by means of the Landau-Lifshitz-Gilbert (LLG) equation [1] that accounts for damping in a phenomenological way. The possibility to calculate the corresponding damping pa- rameter from first principles would open the perspective of optimizing materials for devices and has therefore mo- tivated extensive theoretical work in the past. This led among others to Kambersky’s breathing Fermi surface (BFS) [2] and torque-correlation model (TCM) [3], that in principle provide a firm basis for numerical investi- gations based on electronic structure calculations [4, 5]. The spin-orbit coupling that is seen as a key factor in transferring energy from the magnetization to the elec- tronic degrees of freedom is explicitly included in these models. Most ab-initioresults havebeen obtained for the BFS model though the torque-correlation model makes fewer approximations [4, 6]. In particular, it in principle describes the physical processes responsible for Gilbert damping over a wide range of temperatures as well as chemical(alloy)disorder. However,inpractice,likemany other models it depends on a relaxation time parame- terτthat describes the rate of transfer due to the vari- ous types of possible scattering mechanisms. This weak point could be removed recently by Brataas et al. [7] who described the Gilbert damping by means of scatter- ing theory. This development supplied the formal basis for the first parameter-free investigations on disordered alloys for which the dominant scattering mechanism is potential scattering caused by chemical disorder [8]. As pointed out by Brataas et al. [7], their approach is completelyequivalenttoaformulationintermsofthelin- earresponseorKuboformalism. Thelatterrouteistakenin this communication that presents a Kubo-Greenwood- like expression for the Gilbert damping parameter. Ap- plication of the scheme to disordered alloys demonstrates that this approach is indeed fully equivalent to the scat- tering theory formulation of Brataas et al. [7]. In addi- tion a scheme is introduced to deal with the temperature dependence of the Gilbert damping parameter. Following Brataas et al. [7], the starting point of our scheme is the Landau-Lifshitz-Gilbert (LLG) equation for the time derivative of the magnetization /vectorM: 1 γd/vectorM dτ=−/vectorM×/vectorHeff+/vectorM×/bracketleftBigg˜G(/vectorM) γ2M2sd/vectorM dτ/bracketrightBigg ,(1) whereMsis the saturation magnetization, γthe gyro- magnetic ratio and ˜Gthe Gilbert damping tensor. Ac- cordingly, the time derivative of the magnetic energy is given by: ˙Emag=/vectorHeff·d/vectorM dτ=1 γ2˙/vector m[˜G(/vector m)˙/vector m] (2) in terms of the normalized magnetization /vector m=/vectorM/Ms. Ontheotherhandtheenergydissipationoftheelectronic system ˙Edis=/angbracketleftBig dˆH dτ/angbracketrightBig is determined by the underlying Hamiltonian ˆH(τ). Expanding the normalized magne- tization/vector m(τ), that determines the time dependence of ˆH(τ) about its equilibrium value, /vector m(τ) =/vector m0+/vector u(τ), one has: ˆH=ˆH0(/vector m0)+/summationdisplay µ/vector uµ∂ ∂/vector uµˆH(/vector m0). (3) Using the linear response formalism, ˙Ediscan be written2 as [7]: ˙Edis=−π/planckover2pi1/summationdisplay ii′/summationdisplay µν˙uµ˙uν/angbracketleftBigg ψi|∂ˆH ∂uµ|ψi′/angbracketrightBigg/angbracketleftBigg ψi′|∂ˆH ∂uν|ψi/angbracketrightBigg ×δ(EF−Ei)δ(EF−Ei′),(4) whereEFis the Fermi energy and the sums run over all eigenstates αof the system. Identifying ˙Emag=˙Edis, one gets an explicit expression for the Gilbert damping tensor˜Gor equivalently for the damping parameter α= ˜G/(γMs): αµν=−π/planckover2pi1γ Ms/summationdisplay ii′/angbracketleftBigg ψi|∂ˆH ∂uµ|ψi′/angbracketrightBigg/angbracketleftBigg ψi′|∂ˆH ∂uν|ψi/angbracketrightBigg ×δ(EF−Ei)δ(EF−Ei′).(5) An efficient way to deal with Eq. (5) is achieved by ex- pressing the sum over the eigenstates by means of the retarded single-particle Green’s function Im G+(EF) = −π/summationtext α|ψα/angbracketright/angbracketleftψα|δ(EF−Eα). This leads for the parame- terαto a Kubo-Greenwood-like equation: αµν=−/planckover2pi1γ πMsTrace/angbracketleftBigg ∂ˆH ∂uµImG+(EF)∂ˆH ∂uνImG+(EF)/angbracketrightBigg c(6) with/angbracketleft.../angbracketrightcindicating a configurational average in case of a disordered system (see below). Identifying ∂ˆH/∂uµ with the magnetic torque Tµthis expression obviously gives the parameter αin terms of a torque-torque corre- lation function. However, in contrast to the conventional TCM the electronic structure is not represented in terms of Bloch states but using the retarded electronic Green function giving the present approach much more flexibil- ity. As it corresponds one-to-one to the standard Kubo- Greenwood equation for the electrical conductivity, the techniques developed to calculate conductivities can be straightforwardly adopted to evaluate Eq. (6). The most reliable way to account for spin-orbit cou- pling as the source of Gilbert damping is to evaluate Eq. (6) using a fully relativistic Hamiltonian within the framework of local spin density formalism (LSDA) [9]: ˆH=c/vector α/vector p+βmc2+V(/vector r)+β/vector σ/vector mB(/vector r).(7) Hereαiandβare the standard Dirac matrices and /vector pis the relativistic momentum operator [10]. The functions VandBare the spin-averagedand spin-dependent parts respectively of the LSDA potential. Eq. (7) implies for the magnetic torque Tµoccurring in Eq. (6) the expres- sion: Tµ=∂ ∂uµˆH=βBσµ. (8) The Green’s function G+in Eq. (5) can be obtained in a very efficient way by using the spin-polarized relativisticversion of multiple scattering theory [9] that allows us to treat magnetic solids: G+(/vector rn,/vector rm′,E) =/summationdisplay ΛΛ′Zn Λ(/vector rn,E)τnm ΛΛ′(E)Zm× Λ′(/vector rm′,E) −/summationdisplay ΛZn Λ(/vector r<,E)Jn× Λ′(/vector r>,E)δnm.(9) Here coordinates /vector rnreferring to the center of cell n have been used with |/vector r<|=min(|/vector rn|,|/vector rn′|) and|/vector r>|= max(|/vector rn|,|/vector rn′|). The four component wave functions Zn Λ(/vector r,E) (Jn Λ(/vector r,E)) are regular (irregular) solutions to the single-site Dirac equation for site nandτnm ΛΛ′(E) is the so-called scattering path operator that transfers an electronic wave coming in at site minto a wave going out from site nwith all possible intermediate scattering events accounted for coherently. Using matrix notation, this leads to the following ex- pression for the damping parameter: αµµ=g πµtot/summationdisplay nTrace/angbracketleftbig T0µ˜τ0nTnµ˜τn0/angbracketrightbig c(10) with the g-factor 2(1 + µorb/µspin) in terms of the spin and orbital moments, µspinandµorb, respectively, the total magnetic moment µtot=µspin+µorb, and ˜τ0n ΛΛ′= 1 2i(τ0n ΛΛ′−τ0n Λ′Λ) and the energy argument EFomitted. Thematrix elementsofthe torqueoperator Tnµareiden- tical to those occurring in the context of exchange cou- pling [11] and can be expressed in terms of the spin- dependent part Bof the electronic potential with matrix elements: Tnµ Λ′Λ=/integraldisplay d3rZn× Λ′(/vector r) [βσµBxc(/vector r)]Zn Λ(/vector r).(11) As indicated above, the expressions in Eqs. (6) – (11) can be applied straightforwardly to disordered alloys. In this case the brackets /angbracketleft.../angbracketrightcindicate the necessary configurational average. This can be done by describ- ing in a first step the underlying electronic structure (forT= 0 K) on the basis of the Coherent Potential Approximation (CPA) alloy theory. In the next step the configurational average in Eq. (6) is taken follow- ing the scheme worked out by Butler [12] when dealing with the electrical conducting at T= 0 K or residual resistivity respectively, of disordered alloys. This im- plies in particular that so-called vertex corrections of the type/angbracketleftTµImG+TνImG+/angbracketrightc− /angbracketleftTµImG+/angbracketrightc/angbracketleftTνImG+/angbracketrightcthat account for scattering-in processes in the language of the Boltzmann transport formalism are properly accounted for. Thermal vibrations as a source of electron scattering can in principle be accounted for by a generalization of Eqs. (6) – (11) to finite temperatures and by includ- ing the electron-phonon self-energy Σ el−phwhen calcu- lating the Greens function G+. Here we restrict our- selves to elastic scattering processes by using a quasi- static representation of the thermal displacements of the3 0 0.1 0.2 0.3 0.4 0.50.6 0.7 concentration xCo0123456α(x) x10-3Expt Theory (CPA), bcc Theory (NL CPA)Fe-Co n(EF) n(EF) (sts./Ry) 102030405060 0 FIG. 1: Gilbert damping parameter for bcc Fe xCo1−xas a function of Co concentration: full circles - the present res ults within CPA, empty circles - within non-local CPA (NL CPA), and full diamonds - experimental data by Oogane [14]. atoms from their equilibrium positions. We introduce an alloy-analogy model to average over a discrete set of displacements that is chosen to reproduce the ther- mal root mean square average displacement/radicalbig /angbracketleftu2/angbracketrightTfor a given temperature T. This was chosen according to /angbracketleftu2/angbracketrightT=1 43h2 π2mkΘD[Φ(ΘD/T) ΘD/T+1 4] with Φ(Θ D/T) the De- bye function, hthe Planck constant, kthe Boltzmann constant and Θ Dthe Debye temperature [13]. Ignoring the zero temperature term 1 /4 and assuming a frozen potential for the atoms, the situation can be dealt with in full analogy to the treatment of disordered alloys de- scribed above. The approach described above has been applied to the ferromagnetic 3d-transition metal alloy systems bcc FexCo1−x, fcc Fe xNi1−xand fcc Co xNi1−x. Fig. 1 shows as an example results for bcc Fe xCo1−xforx≤0.7. The calculated damping parameter α(x) forT= 0 K is found in very good agreement with the results based on the scatteringtheoryapproach[8]demonstratingnumerically the equivalence of the two approaches. An indispensable requirement to achieve this agreement is to include the vertex corrections mentioned above. In fact, ignoring them leads in some cases to completely unphysical re- sults. To check the reliability of the standard CPA, that implies a single-site approximation when performing the configurationalaverage,weperformedcalculationsonthe basis of the non-local CPA [15]. In this case four atom cluster have been used leading - apart from the very di- lute case - practically to the same results as the CPA. As found before for fcc Fe xNi1−x[8] the theoretical results forαreproduce the concentration dependence of the ex- perimental data quite well but are found too low (see below). As suggested by Eq. (10) the variation of α(x) with concentration xmay reflect to some extent the vari- ation of the average magnetic moment µtotof alloy. As the moments as well as the spin-orbit coupling strength of Fe and Co don’t differ too much, the variation of α(x) should be determined in the concentrated regime primar- ily by the electronic structure at the Fermi energy EF.As Fig. 1 shows, there is indeed a close correlation of the density of states n(EF) that may be seen as a measure for the available relaxation channels. While the scattering and linear response approach are completely equivalent when dealing with bulk alloys the latter allows us to perform the necessary configuration averagingin a much more efficient way. This allows us to study with moderate effort the influence of varying the alloy composition on the damping parameter α. Corre- sponding work has been done in particular using permal- loy as a starting material and adding transition metals (TM) [16] orrareearthmetals [17]. Fig. 2(top) showsre- sultsobtainedbysubstitutingFeandNiatomsinpermal- loy by 5d TMs. As found by experiment [16] αincreases 00.05 0.1 0.15x01234 α x10-2 Ta W Re OsIr PtAu02468α x10-2 Ta W Re OsIr PtAu-0.300.30.60.9mspin5d (µB)WOs Ir Pt AuRe Ta n5d(EF) 5d spin moment 061218 n5d(EF) (sts./Ry) FIG. 2: Top: Change of the Gilbert damping parameter ∆ α w.r.t. permalloy(Py)forvariousPy/5dTM systemsasafunc- tion of 5d TM concentration; Middle: Gilbert damping pa- rameter αfor Py/5d TM systems with 10 % 5d TM content in comparison with experiment [16]; Bottom: spin magnetic moment m5d spinand density of states n(EF) at the Fermi en- ergy of the 5 dcomponent in Py/5d TM systems with 10 % 5d TM content. in all cases nearly linearly with the 5d TM content. The total damping for 10 % 5d TM content shown in the middle panel of Fig. 2 varies roughly parabolically over the 5d TM series. In contrast to the Fe xCo1−xalloys considered above, there is now an S-like variation of the momentsµ5d spinover the series (Fig. 2, bottom), char- acteristic of 5d impurities in the pure hosts Fe and Ni [18, 19]. In spite of this behaviour of µ5d spinthe variation4 00.050.10.15α(T)expt: pure Ni theory: pure Ni 00.050.10.15α(T)expt: Ni + 0.17 wt.%Cu theory: Ni + 0.2 at.%Cu 0 100 200 300 400 500 Temperature (K)00.050.10.15α(T)expt: Ni + 5 wt.%Cu theory: Ni + 5 at.%Cu FIG. 3: Temperature variation of Gilbert damping of pure Ni and Ni with Cu impurities: present theoretical results vs experiment [20] ofα(x) seems again to be correlated with the density of statesn5d(EF) (Fig. 2 bottom). Again the trend of the experimental data is well reproduced by the theoretical ones that are however somewhat too low. One of the possible reasons for the discrepancy of the theoretical and experimental results shown in Figs. 1 and 2 might be the neglect of the influence of finite temper- atures. This can be incorporated as indicated above by accounting for the thermal displacement of the atoms in a quasi-static way and performing a configurational av- erage over the displacements using the CPA. This leads even for pure systems to a scattering mechanism and this waytoafinite valuefor α. Correspondingresultsforpure Ni are given in Fig. 3 that show in full accordance with experiment a rapid decrease of αwith increasing tem- perature until a regime with a weak variation of αwith Tis reached. This behavior is commonly interpreted as a transition from conductivity-like to resistivity-like be- haviour reflecting the dominance of intra- and inter-band transition, respectively [4], that is related to the increase of the broadening of electron energy bands caused by the increase of scattering events with temperature. Adding only less than 1 at. % Cu to Ni, the conductivity-like behavior at low temperatures is strongly reduced whilethe high temperature behavior is hardly changed. A fur- ther increase of the Cu content leads to the impurity- scattering processes responsible for the band broaden- ing dominating α. This effect completely suppresses the conductivity-likebehavior in the low-temperatureregime because of the increase of scattering events due to chem- ical disorder. Again this is fully in line with the experi- mental data, providing a straightforward explanation for their peculiar variation with temperature and composi- tion. FromtheresultsobtainedforNionemayconcludethat thermal lattice displacements are only partly responsible for the finding that the damping parameters obtained for Py doped with the 5 dTM series, and Fe xCo1−xare somewhatlowcomparedwith experiment. This indicates that additional relaxation mechanisms like magnon scat- tering contribute. Again, these can be included at least in a quasi-static way by adopting the point of view of a disordered local moment picture. This implies scatter- ing due to random temperature-dependent fluctuations of the spin moments that can also be dealt with using the CPA. Insummary, aformulationforthe Gilbert dampingpa- rameterαin terms of a torque-torque-correlation func- tion was derived that led to a Kubo-Greenwood-like equation. The scheme was implemented using the fully relativistic KKR band structure method in combination with the CPA alloy theory. This allows us to account for various types of scattering mechanisms in a parameter- free way. Corresponding applications to disordered tran- sition metal alloys led to very good agreement with re- sults based on the scattering theory approach of Brataas et al. demonstrating the equivalence of both approaches. The flexibility and numerical efficiency of the present scheme was demonstrated by a study on a series of permalloy-5dTMsystems. Toinvestigatetheinfluenceof finite temperatures on α, a so-called alloy-analogymodel was introduced that deals with the thermal displacement of atoms in a quasi-static manner. Applications to pure Ni gave results in very good agreement with experiment and in particular reproduced the dramatic change of α when Cu is added to Ni. Acknowledgments The authors would like to thank the DFG for finan- cial support within the SFB 689 “Spinph¨ anomene in re- duzierten Dimensionen” and within project Eb154/23for financialsupport. PJKacknowledgessupportbyEUFP7 ICT Grant No. 251759 MACALO. [1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443 (2004).[2] V. Kambersky, Can. J. Phys. 48, 2906 (1970). [3] V. Kambersky, Czech. J. Phys. 26, 1366 (1976), URL5 http://dx.doi.org/10.1007/BF01587621 . [4] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. 99, 027204 (2007), URL http://link.aps.org/doi/10.1103/PhysRevLett.99.0272 04. [5] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427 (2006). [6] V. Kambersky, Phys. Rev. B 76, 134416 (2007). [7] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 101, 037207 (2008), URL http://link.aps.org/doi/10.1103/PhysRevLett.101.037 207. [8] A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 105, 236601 (2010), URL http://link.aps.org/doi/10.1103/PhysRevLett.105.236 601. [9] H. Ebert, in Electronic Structure and Physical Properties of Solids , edited by H. Dreyss´ e (Springer, Berlin, 2000), vol. 535 of Lecture Notes in Physics , p. 191. [10] M. E. Rose, Relativistic Electron Theory (Wiley, New York, 1961). [11] H. Ebert and S. Mankovsky, Phys. Rev. B 79, 045209 (2009), URL http://link.aps.org/doi/10.1103/PhysRevB.79.045209 . [12] W. H. Butler, Phys. Rev. B 31, 3260 (1985), URL http://link.aps.org/doi/10.1103/PhysRevB.31.3260 .[13] E. M. Gololobov, E. L. Mager, Z. V. Mezhevich, and L. K. Pan, phys. stat. sol. (b) 119, K139 (1983). [14] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and T. Miyazaki, Jap. J. Appl. Phys. 45, 3889 (2006). [15] D. K¨ odderitzsch, H. Ebert, D. A. Rowlands, and A. Ernst, New Journal of Physics 9, 81 (2007), URL http://dx.doi.org/10.1088/1367-2630/9/4/081 . [16] J. O. Rantschler, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F. Egelhoff, B. B. Maranville, D. Pu- lugurtha, A. P. Chen, and L. M. Connors, J. Appl. Phys. 101, 033911 (2007). [17] G. Woltersdorf, M. Kiessling, G. Meyer, J.-U. Thiele, a nd C. H. Back, Phys. Rev. Lett. 102, 257602 (2009), URL http://link.aps.org/doi/10.1103/PhysRevLett.102.257 602. [18] B. Drittler, N. Stefanou, S. Bl¨ ugel, R. Zeller, and P. H. Dederichs, Phys. Rev. B 40, 8203 (1989), URL http://link.aps.org/doi/10.1103/PhysRevB.40.8203 . [19] N. Stefanou, A. Oswald, R. Zeller, and P. H. Dederichs, Phys. Rev. B 35, 6911 (1987), URL http://link.aps.org/doi/10.1103/PhysRevB.35.6911 . [20] S. M. Bhagat andP. Lubitz, Phys. Rev.B 10, 179 (1974).

2011-02-22

A Kubo-Greenwood-like equation for the Gilbert damping parameter $\alpha$ is presented that is based on the linear response formalism. Its implementation using the fully relativistic Korringa-Kohn-Rostoker (KKR) band structure method in combination with Coherent Potential Approximation (CPA) alloy theory allows it to be applied to a wide range of situations. This is demonstrated with results obtained for the bcc alloy system Fe$_x$Co$_{1-x}$ as well as for a series of alloys of permalloy with 5d transition metals. To account for the thermal displacements of atoms as a scattering mechanism, an alloy-analogy model is introduced. The corresponding calculations for Ni correctly describe the rapid change of $\alpha$ when small amounts of substitutional Cu are introduced.

Ab-initio calculation of the Gilbert damping parameter via linear response formalism

1102.4551v1

arXiv:0811.2235v2 [cond-mat.mes-hall] 27 Jun 2009Intrinsic Coupling between Current and Domain Wall Motion i n (Ga,Mn)As Kjetil Magne Dørheim Hals, Anh Kiet Nguyen, and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway We consider current-induceddomain wall motion and, therec iprocal process, movingdomain wall- induced current. The associated Onsager coefficients are exp ressed in terms of scattering matrices. Uncommonly, in (Ga,Mn)As, the effective Gilbert damping coe fficientαwand the effective out-of- plane spin transfer torqueparameter βware dominated byspin-orbit interaction incombination wit h scattering off the domain wall, and not scattering off extrins ic impurities. Numerical calculations giveαw∼0.01 andβw∼1 in dirty (Ga,Mn)As. The extraordinary large βwparameter allows experimental detection of current or voltage induced by dom ain wall motion in (Ga,Mn)As. The principle of giant magneto resistance is used to detect magnetic information. Large currents in mag- netic nanostructures can manipulate the magnetization via spin transfer torques [1]. A deeper knowledge of the coupled out-of-equilibrium quasi-particle and magnetiza- tion dynamics is needed to precisely control and utilize current-induced spin transfer torques. Themagnetizationrelaxestowardsits equilibriumcon- figuration by releasing magnetic moments and energy into reservoirs. This friction process is usually described by the Gilbert damping constant αin the Landau- Lifshitz-Gilbert (LLG) equation. Spins traversingamag- netic domain wall exert an in-plane and an out-of-plane torqueonthe wall[2]. In dirtysystems, when the domain wall is wider than the mean-free-path, the out-of-plane torque, often denoted the non-adiabatic torque, is pa- rameterized by the so-called β-factor [2]. The Gilbert damping coefficient α, the in-plane spin-transfer torque, and the out-of-plane torque coefficient βdetermine how the magnetization is influenced by an applied current, e.g. the current-induced Walker domain wall drift veloc- ity is proportional to β/α[2, 3, 4]. Scattering off impu- rities are important for αandβ[2, 3, 4]. Additionally, domain wall scattering can contribute to αandβ. In ballistic(Ga,Mn)As, intrinsic spin-orbit coupling causes significant hole reflection at the domain wall, even in the adiabatic limit when the wall is much thicker than the Fermi wavelength [5]. This grossly increases the out-of- plane spin-transfer torque, and consequently the current- driven domain wall mobility. So far, there are no inves- tigations on the effect of these domain wall induced hole reflections on the effective Gilbert damping constant α. Experimental(Ga,Mn)As samples aredirty sothat the effectofdisorderontheeffectiveGilbertdampingandthe out-of-plane spin transfer torque should be taken into account. We find surprisingly that, in systems with a largeintrinsic spin-orbitcoupling, domainwallscattering contributesdominantlyto αandβeveninthedirtylimit. Intrinsic current-domain wall motion coupling is robust against impurity scattering. Current-induced domain-wall motion has been seen in many experiments [3]. The reciprocaleffect, domain-wall motion induced current, is currentlytheoreticallyinvesti-gated[6, 7], andseenexperimentally[8]. Aprecessingdo- main wall induces a charge current in ferromagnetic met- als [6] similar to spin-pumping in layered ferromagnet- normal metal systems [9]. For rigid domain wall motion, the induced chargecurrentis proportionalto β/α[7]. We find that βandβ/αin (Ga,Mn)As are so large that the current, or equivalently, the voltage induced by a moving domain wall is experimentally measurable. Onsager’s reciprocity relations dictate that response coefficients of domain wall motion induced current and current induced domain wall motion are related. In dirty systems, these relations have been discussed in Ref. [7]. Ref.[7]alsousedthescatteringtheoryofadiabaticpump- ing to evaluate the non-adiabatic spin-transfer torque in ballistic systems without intrinsic spin-orbit interaction. We first extend the pumping approach to (Ga,Mn)As with strong intrinsic spin-orbit interaction, and second, also evaluate the Onsager coefficient as a function of sample disorder. In determining all Onsager coefficients, magnetization friction must be evaluated on the same footing. To this end, we generalize the energy pump- ing scattering theory of Gilbert damping [10] to domain wall motion. Our numerical calculation demonstrates, for the first time, that domain wall scattering is typi- cally more important than impurity scattering for the effective domain wall motion friction in systems with a strongintrinsicspin-orbitinteraction. OurnovelOnsager scattering approach can also be used to compute the ef- fective rigid domain wall motion αandβparameters in realistic materials like Fe, Ni, Co, and alloysthereof from first-principles. Let us discuss in more detail the Onsager reciprocity relations in our system. The magnetic field is a thermo- dynamic force for the magnetization since it can move domainwalls. The electric field is athermodynamic force for the charges as it induces currents. In systems where charge carriers also carry spin, the magnetic and charge systems are coupled. Through this coupling, the elec- tric field can move a domain wall and, vice versa, the magnetic field can induce a current. This phenomenon, where the thermodynamic force of one system can induce a flux in another system is well-known in thermodynam- ics [11]: Assume a system described by the quantities2 {qi},Xidenotes the thermodynamic force, and Jithe flux associated with the quantity qi. In linear response, Ji=/summationtext jLijXj, whereLijare the Onsager coefficients. Onsager’s reciprocity principle dictates Lij=ǫiǫjLji, whereǫi= 1 (ǫi=−1) ifqiis even (odd) under time- reversal [11]. Fluxes and forces are not uniquely defined, but the Onsager reciprocity relations are valid when the entropy generation is ˙S=/summationtext iJiXi[11]. We first derive expressions for the Onsager coefficients and determine the Onsager reciprocity relations between a charge current and a moving domain wall in terms of the scattering matrix. Subsequently, we derive the relation between the Onsager coefficients and the effec- tive Gilbert damping parameter αwand the out-of-plane torque parameter βwfor domain wall motion. Finally, we numerically compute αwandβwfor (Ga,Mn)As. We start the derivation of the Onsager coefficients in terms of the scattering matrix by assuming the following free energy functional for the magnetic system F[M] =Ms/integraldisplay dr/parenleftbiggJ 2[(∇θ)2+sin2(θ)(∇φ)2]+ K⊥ 2sin2(θ)sin2(φ)−Kz 2cos2(θ)−Hextcos(θ)/parenrightbigg ,(1) whereMs,JandHextare the saturation magnetization, spin-stiffness and external magnetic field, respectively, andKzandK⊥are magnetic anisotropy constants. The local magnetization angles θandφare defined with re- spect to the z- andx-axis, respectively. The system con- tains a Bloch wall rotating in the (transverse) x-zplane, cos(θ) = tanh([ y−rw]/λw), sin(θ) = 1/cosh([y−rw]/λw), whererwis the position of the wall, and λwis the wall width. We assume the external magnetic field is lower than the Walker threshold, so that the wall rigidly moves (˙φ= 0) with a constant drift velocity. In this case rw andφcompletely characterize the magnetic system, and λw=/radicalBig J/(Kz+K⊥sin2(φ)) [4]. The current is along they-axis. Theheatdissipatedperunittimefromachargecurrent Jcis˙Q=Jc(VL−VR), where VL(VR) is the voltage in the left (right) reservoir. Using the relation dS=dQ/T, this implies an entropy generation ˙S=Jc(VL−VR)/T. Thus,Xc≡(VL−VR)/Tis the thermodynamic force inducing the flux Jc. We assume the magnetic system to be at constant temperature, which means that the heat transported out of the magnetic system as the do- main wall moves equals the loss of free energy. This implies an entropy generation ˙S=˙Q/T=−˙F/T= (−∂F[rw,φ]/T∂rw) ˙rw=XwJw, where we have defined the force Xw≡ −∂F[rw,φ]/T∂rwand flux Jw≡˙rw. Us- ing Eq. (1), wefind Xw=−2AMsHext/T, whereAis the conductor’s cross-section. Fluxes are related to forces by Jw=LwwXw+LwcXc (2) Jc=LccXc+LcwXw, (3)whereLcc=GTandGis the conductance. Lww(Lwc) determine the induced domain wall velocity by an exter- nal magnetic field (a current). The induced current by a moving domain wall caused by an external magnetic field Hextis controlled by Lcw. Both charge and rware even under time-reversal so that Lcw=Lwc[12]. The current induced by a moving domain wall is para- metric pumping in terms of the scattering matrix [9]: Jc,α=e˙rw 2π/summationdisplay β=1,2ℑm/braceleftbigg Tr/bracketleftbigg∂Sαβ ∂rwS† αβ/bracketrightbigg/bracerightbigg ,(4) whereSαβis the scattering matrix between transverse modes in lead βto transverse modes in lead α. The system has two leads ( α,β∈ {1,2}). The trace is over all propagating modes at the Fermi energy EF. From Eqs. (2) and (3) we find Jc=Lcw˙rw/Lww. We considertransportwellbelowthe criticaltransition temperature in (Ga,Mn)As, which is relatively low, and assume the energy loss in the magnetic system is trans- ferred into the leads by holes. Generalizing Ref. [10] to domain wall motion, this energy-flux is related to the scattering matrix: JE=¯h 4πTr/braceleftbiggdS dtdS† dt/bracerightbigg =¯h˙r2 w 4πTr/braceleftbigg∂S ∂rw∂S† ∂rw/bracerightbigg .(5) For a domain wall moved by an external magnetic field, we then find that XwJw=J2 w/Lww=JE/T. In sum- mary, the Onsager coefficients in Eq. (2) and Eq. (3) are Lww=/parenleftbigg¯h 4πTr/braceleftbigg∂S ∂rw∂S† ∂rw/bracerightbigg/parenrightbigg−1 , (6) Lcw=2e ¯h/summationtext β=1,2ℑm/braceleftBig Tr/bracketleftBig ∂Sαβ ∂rwS† αβ/bracketrightBig/bracerightBig Tr/braceleftBig ∂S ∂rw∂S† ∂rw/bracerightBig ,(7) Lcc=e2 hTr/braceleftbig t†t/bracerightbig , (8) wheretis the transmission coefficient in the scatter- ing matrix. We have omitted the temperature factor in the coefficients (6), (7), and (8) since it cancels with the temperature factor in the forces, i.e.we transform L→L/TandX→TX. The Onsager coefficient ex- pressions in terms of the scattering matrix are valid irre- spectiveofimpuritydisorderandspin-orbitinteractionin the band structures or during scattering events, and can treat transport both in ballistic and diffusive regimes. Let us compare the global Onsager cofficients (6), (7), and (8) with the local Onsager coefficients in the dirty limit to gainadditionalunderstanding. In the dirtylimit, all Onsager cofficients become local and the magnetiza- tion dynamics can be described by the following phe- nomenological local LLG equation [2, 3]: ˙m=−γm×Heff+αm×˙m −(1−βm×)(vs·∇)m, (9)3 wheremis the magnetization direction, Heffis the effective magnetic field, γis the gyromagnetic ratio, vs=−¯hPj/(eS0),S0=Ms/γ,Msthe magnetization, αthe Gilbert damping constant, Pthe spin-polarization along−mof the charge carriers [13], and βis the out- of-plane spin-transfer torque parameter. Substituting a Walker ansatz into Eq. (9) gives below the Walker threshold [4]: α˙rw/λw=−γHext−¯hβPj/(eS0λw). In dirty, local, systems this equation determines the rela- tion between the flux Jwand the forces XwandXc asLww=λw/(2AS0α) andLwc=−¯hβPG/(eαS0A), where we have used j=σ(VL−VR)/L, andG=σA/L. Here,Lis the length of the conductor, ethe electron charge, and σthe conductivity. This motivates defining the following dimensionless global coefficients: αw≡λw 2AS0Lww, βw≡ −λwe 2¯hPGLwc Lww. αwis the effective Gilbert damping coefficient and βwis the effective out-of-plane torque on the domain wall. We will in the following investigate αwandβwfor (Ga,Mn)As by calculating the scattering matrix expres- sions in Eq. (6) and Eq. (7). We use the following Hamil- tonian to model quantum transport of itinerant holes: H=HL+h(r)·J+V(r). (10) Here,HLis the 4×4 Luttinger Hamiltonian (parame- terized by γ1andγ2) for zincblende semiconductors in the spherical approximation, while h·Jdescribes the exchange interaction between the itinerant holes and the local magnetic moment of the Mn dopants. We introduce Anderson impurities as V(r) =/summationtext iViδr,Ri, whereRiis the position of impurity i,Viits impurity strength, and δthe Kroneckerdelta. More details about the model and the numericalmethod used can be found in Refs. [14, 15]. We consider a discrete conductor with transverse di- mensions Lx= 23nm,Lz= 17nmand length Ly= 400nm. The lattice constant is 1 nm, much less than the typical Fermi wavelength λF∼8nm. The Fermi energy EF= 82meVis measured from the bottom of the lowest subband. |h|= 0.5×10−20Jandγ1= 7. The typical mean-free path for the systems studied ranges from the diffusive to the ballistic regime l∼23nm→ ∞, and we are in the metallic regime kFl≫1. The domain wall length is λw= 40nm. The spin-density S0from the local magneticmoments is S0= 10¯hx/a3 GaAs,aGaAsthe lattice constant for GaAs, and x= 0.05 the doping level[14]. Fig. 1a shows the computed effective Gilbert damping coefficient αwversusλw/lfor (Ga,Mn)As containing one Bloch wall. Note the relatively high αw∼5×10−3in the ballistic limit. Additional impurities, in combination with the spin-orbit coupling, assist in releasing energy and angular momentum into the reservoirs and increase αw. However, as shown in Fig. 1a, impurities contribute only about 20% to αweven when the domain wall is two0 0.5 1 1.50246x 10−3 00.5 11.5 22.5024x 10−3 0 0.5 1 1.502468 00.5 11.5 22.50123λw/l λw/lαw βw αw βwγ2 γ2(a) (b) FIG. 1: (a): Effective Gilbert damping αwas function of λw/l, whereλwis the domain wall length and lis the mean free path when γ2= 2.5. Here,λwis kept fixed, and lis varied. Inset: αwas a function of spin-orbit coupling γ2for a clean system, l=∞.(b):βwas a function of λw/l, whereλw is the domain wall length and lis the mean free path when γ2= 2.5. Here,λwis kept fixed, and lis varied. Inset: βwas a function of spin-orbit coupling γ2for a clean system, l=∞. In all plots, line is guide to the eye. times longer than the mean free path. Due to the strong spin-orbit coupling, ballistic domain walls have a large intrinsic resistance [5] that survives the adiabatic limit. When itinerant holes scatter off the domain wall their momentum changes and through the spin-orbit coupling their spin also changes. This is the dominate process for releasing energy and magnetization into the reservoirs. The saturated value αw∼6×10−3is of the same order as the estimates in Ref. [16] for bulk (Ga,Mn)As. The inset in Fig. 1a shows the domain-wall contribution to αwversus the spin-orbit coupling for a clean system with no impurities. αwmonotonicallydecreasesfor decreasing γ2and vanish for γ2→0. Since, λw/λF∼5, itinerant holes will, without spin-orbit coupling, traverse the do-4 main wall adiabatically. Fig. 1b shows βwversusλw/l.βwdecreases with increasing disorder strength. This somewhat counter intuitive result stem from the fact that domain walls in systems with spin-orbit coupling have a large intrin- sic domain wall resistance [5] which originates from the anisotropy in the distribution of conducting channels [5]. The reflected spins do not follow the magnetization of the domain wall, and thereby cause a large out-of-plane torque [2]. This causes the large βwin the ballistic limit. Scalar, rotational symmetric impurities tend to reduce the anisotropy in the conducting channels, and thereby reduce the intrinsic domain wall resistance and conse- quently reduce βw. Deeper into the diffusive regime, β saturates. Here, the domain wall resistance and βware kept at high levels due to the increase in the spin-flip rate caused by impurity scattering. The saturated value isβ∼1. For even dirtier systems than a reasonable computing time allows, we expect a further increase in βw. In comparison, simple microscopic theories for fer- romagnetic metals where one disregards the spin-orbit coupling in the band structure predict β∼0.001−0.01 [2, 3, 4]. Similar to the Gilbert damping, in ballistic sys- temsβwincreaseswith spin-orbit coupling because ofthe increased domain wall scattering [5], see Fig. 1b inset. βwcan be measured experimentally by the induced current or voltage from a domain wall moved by an ex- ternal magnetic field as a function of the domain wall velocity [7]. From the Onsager relations we have that Jc=LcwXw. UsingXw=Jw/Lww, the induced current and voltage are [7]: Jc=−2β¯hPG eλw˙rw⇒V=−2βw¯hP eλw˙rw.(11) An estimate of the maximum velocity of a domain wall moved by an external magnetic field below the Walker treshold is ˙ rw∼10m/s[17]. With λw= 40nmand P= 0.66 this indicates an experimentally measurable voltageV∼0.2µV. Inconclusion, wehavederivedOnsagercoefficientsand reciprocity relations between current and domain wall motion in terms of scattering matrices. In (Ga,Mn)As, we find the effective Gilbert damping constant αw∼0.01 and out-of-plane spin transfer torque parameter βw∼1. In contrast to ferromagnetic metals, the main contribu- tions to αwandβwin (Ga,Mn)As are intrinsic, and in- duced by scattering off the domain wall, while impurity scattering is less important. The large βwparameter im- plies a measurable moving domain wall induced voltage. This work was supported in part by the Re-search Council of Norway, Grants Nos. 158518/143 and 158547/431, computing time through the Notur project and EC Contract IST-033749 ”DynaMax”. [1] J.C. Slonczewski,J. Magn. Magn. Mater. 159, L1 (1996); L. Berger,Phys. Rev. B 54, 9353 (1996). [2] G. Tatara, H. Kohno,Phys. Rev. Lett. 92,086601 (2004); S. Zhang, Z. Li,Phys. Rev. Lett. 93, 127204 (2004); S. E. Barnes, S. Maekawa,Phys. Rev. Lett. 95, 107204 (2005); A. Thiaville, Y.Nakatani, J. Miltat, Y.Suzuki,Europhys. Lett.69, 990 (2005); Y. Tserkovnyak, H. J. Skadsem, A. Brataas, G. E. W. Bauer, Phys. Rev. B 74, 144405 (2006); A. K. Nguyen, H. J. Skadsem, A. Brataas, Phys. Rev. Lett. 98, 146602 (2007). [3] For reviews see e.g.D.C. Ralph, M.D. Stiles, J. Magn. Magn. Mater., 3201190 (2008). [4] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, J. Magn. Magn. Mater., 3201282 (2008). [5] A. K. Nguyen, R. V. Shchelushkin and A. Brataas, Phys. Rev. Lett. 97, 136603 (2006); R. Oszwaldowski, J. A. Majewski and T. Dietl,Phys. Rev. B 74, 153310 (2006). [6] G. E. Volovik,J. Phys.C: Sol. State Phys. 20, L83(1987); S.E. BarnesandS.Maekawa,Phys.Rev.Lett. 98, 246601 (2007); J.I. Ohe, A. Takeuchi, and G. Tatara, Phys. Rev. Lett.99, 266603 (2007); S. A.Yang, D. Xiao, andQ. Niu, cond-mat/0709.1117. [7] R. A. Duine, Phys. Rev. B 77, 014409 (2008); arXiv:0809.2201v1; Y. Tserkovnyak and M. Mecklen- burg,Phys. Rev. B 77, 134407 (2008). [8] S. A. Yang et al.,Phys. Rev. Lett. 102, 067201 (2009). [9] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). [10] A. Brataas, Y. Tserkovnyak and G.E.W Bauer, Phys. Rev. Lett. 101, 037207 (2008). [11] S. R. de Groot, Thermodynamics of Irreversible Pro- cesses(North-Holland, Amsterdam, 1952). [12] The magnetic texture is inverted in this equation, i.e. Lcw[m] =Lwc[−m]. [13] We define the spin-polarization P=/angbracketleftJz/angbracketright=P j(ψ† jJzψj)/N(m=−ˆ z), where the sum is over N propagating modes, ψjare the corresponding spinor- valued wavefunctions, and Jzthe dimensionless angular momentum operator. [14] T. Jungwirth, J. Sinova, J. Maˇ sek, J. Kuˇ cera, and A. H. MacDonald, Rev. Mod. Phys. 78, 809 (2006). [15] A. K. Nguyen, and A. Brataas, Phys. Rev. Lett. 101, 016801 (2008). [16] J. Sinova et al., Phys. Rev. B 69, 085209 (2004); Y. Tserkovnyak, G. A. Fiete and B. I. Halperin,Appl. Phys. Lett.84, 5234 (2004); I. Garate and A. MacDonald, arXiv:0808.3923v1. [17] A. Dourlat, V. Jeudy, A. Lemaitre and C. Gourdon, Phys. Rev. B 78, 161303(R) (2008).

2008-11-13

We consider current-induced domain wall motion and, the reciprocal process, moving domain wall-induced current. The associated Onsager coefficients are expressed in terms of scattering matrices. Uncommonly, in (Ga,Mn)As, the effective Gilbert damping coefficient $\alpha_w$ and the effective out-of-plane spin transfer torque parameter $\beta_w$ are dominated by spin-orbit interaction in combination with scattering off the domain wall, and not scattering off extrinsic impurities. Numerical calculations give $\alpha_w \sim 0.01$ and $\beta_w \sim 1$ in dirty (Ga,Mn)As. The extraordinary large $\beta_w$ parameter allows experimental detection of current or voltage induced by domain wall motion in (Ga,Mn)As.

Intrinsic Coupling between Current and Domain Wall Motion in (Ga,Mn)As

0811.2235v2

1 Investigation of Ultrafast Demagnetization and Gilbert Damping and their Correlation in Different Ferromagnetic Thin Films Grown Under Identical Conditions1 Suchetana Mukhopadhyay, Sudip Majumder, Surya Narayan Panda, Anjan Barman* Department of Condensed Matter and Materials Physics, S.N. Bose National Center for Basic Sciences, Block -JD, Sector III, Salt Lake, Kolkata 700106, India Email: abarman@bose.res.in Abstract Following the demonstration of laser -induc ed ultrafast demagnetization in ferromagnetic nickel, several theoretical and phenomenological propositions have sought to uncover its underlying physics. In this work we revisit the three temperature model (3TM) and the microscopic three temperature model (M3TM) to perform a comparative analysis of ultrafast demagnetization in 20-nm-thick cobalt, nickel and permalloy thin films measured using an all-optical pump - probe technique. In addition to the ultrafast dynamics at the femtosecond timescales, the nano second magnetization precession and damping are recorded at various pump excitation fluences revealing a fluence -dependent enhancement in both the demagnetization times and the damping factors. We confirm that the Curie temperature to magnetic moment ratio of a given system acts as a figure of merit for the demagnetization time, while the demagnetization times and damping factors show an apparent sensitivity to the density of states at the Fermi level for a given system. Further, from numerical simulations of the ultrafast demagnetization based on both the 3TM and the M3TM, we extract the reservoir coupling parameters that best reproduce the experimental data and estimate the value of the spin flip scattering probability for each system. We discuss how the f luence -dependence of inter-reservoir coupling parameters so extracted may reflect a role played by nonthermal electrons in the magnetization dynamics at low laser fluences. 1. Introduction Optical excitation of a magnetic material with a short and intense laser pulse sets in motion a chain of microscopic events that result in a macroscopic, measurable quenching of the net magnetization. Since the processing speed of magnetic storage devices is limited by the maximum speeds at which the magnetization can be manipulated, the possibility of controlling magnetization at sub-picosecond timescales by employing ultrashort laser pulses holds tremendous potential for applications in spin-based memory and storage devices with ultrafast processing speeds including laser -induced opto-magnetism and all-optical switching [1–4] as well as THz spintronic devices [5–7]. Meanwhile, the relaxation timescales of the laser - induced magnetization precession in ferromagnetic thin films associated with the magnetic damping set fundamen tal limits on magnetization switching and data transfer rates in spintronic devices. Picosecond laser -induced magnetization quenching in ferromagnetic gadolinium was reported by Vaterlaus et al. in 1991, who also reported for the first time a character istic timescale of 100±80 ps [8] associated with the magnetization loss. In this early work, the timescale was identified with the timescale of the spin-lattice interactions mediating the disruption of magnetic ordering due to laser heating of the lattice, though later works contradicted this interpretation [9, 10]. In fact, by the mid- 1990s, it had begun to be recognized that finer time resolution was necessary to probe this phenomenon to uncover a fuller picture of the associated relaxation processes occu rring at sub-picosecond timescales. In 1996, Beaurepaire et al. demonstrated in a seminal work that faster demagnetization occurring at sub - picosecond timescales could be triggered by femtosecond laser pulses in a nickel thin film [11]. The ultrafast demag netization phenomenon went on to be demonstrated in a wide array of ferromagnetic systems [12–19], triggering a flurry of research that continues till date. A few years after the pioneering experiments, Koopmans et al. characterized for the first time the full magneto -optical response to femtosecond laser pulses in a ferromagnet by time-resolved measurements of Kerr ellipticity and rotation [13]. However, the observation of nonmagnetic contributions to the Kerr rotation signal naturally led some to question whether the ultrafast quenching in response to the laser excitation was indeed a magnetic phenomenon [13, 14]. In 2003, Rhie et al. used time -resolved photoemission 1First submitted in March 2022 2 spectroscopy to probe the collapse of the 3d exchange s plitting in nickel as an unambiguous signature of a photoinduced demagnetization occurring over 300±70 fs [15]. This was soon followed by the first reports of an estimate of the characteristic timescale of femtosecond laser -induced demagnetization derived from quantum -mechanical principles [16]. One of the major reasons behind this sustained interest is the need to achieve a complete understanding of the microscopic mechanisms that underlie the ultrafast demagnetization phenomenon that have so far remained elusive. Much inquiry has been focused on how the laser -induced loss of magnetic order is compensated for via the transfer of angular momentum of the spin system at the associated timescales. Over the years, several mechanisms have been proposed for explai ning the angular momentum conservation associated with the ultrafast demagnetization, which may be broadly categorized in two distinct classes. One of these argues in favor of a dominant contribution to the demagnetization arising from nonlocal transport p rocesses driven by the laser pulse, such as superdiffusive transport of spin - polarized hot electrons [20, 21] or heat currents [22]. The other narrative relies on local spin -flip scattering processes occurring by the collision of excited electrons with imp urities, phonons, and magnons [17, 23, 24]. In this category, electron -lattice and electron -spin relaxation mechanisms are accepted as the major demagnetization channels in the picoseconds following the laser excitation [25], with several works attributing a pivotal role to the material -specific spin -orbit interaction [23, 25 –27]. On the other hand, it is possible to model the demagnetization by adopting a thermal description, considering the laser excitation as a “heat” source that excites electrons close to the Fermi level instantaneously to very high energies, whose eventual relaxation processes drive the magnetization loss. The laser excitation energy controlled by the applied laser pump fluence has a direct influence on the maximum electron temperatures reached and thus plays a pivotal role in the ultrafast magnetization dynamics that follow as a result. In this work, employing an all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) technique, we investigate the ultrafast spin dynamics along w ith the nanosecond magnetization precession and damping at different laser pump fluences in three ferromagnetic thin films: 20 -nm-thick cobalt, nickel and permalloy grown under uniform deposition conditions. Cobalt and nickel are elementary 3d ferromagneti c transition metals well studied in the literature in both their elementary forms and as constituting elements in multilayered structures where their strong spin -orbit coupling mediates interfacial effects such as spin pumping. Permalloy, an alloy comprise d of approximately 80% nickel and 20% iron, is a prototype material for spintronic applications due to several desirable qualities such as low coercivity, large permeability and in particular, low Gilbert damping, which aids in minimizing the maximum power consumption of devices and allows spin wave propagation on length scales of the order of device size. The simultaneous investigation of the fluence -dependent modulation of ultrafast demagnetization and damping in all the above ferromagnetic thin films is motivated by the primary objectives of exploring the dominant microscopic processes underlying the demagnetization in these systems, correlating the observed magnetization dynamics to the material properties of each system, and in the process exploring their tunability with laser pump fluence. We set out to achieve this in two ways: (a) by modelling the ultrafast demagnetization at various pump fluences using two theoretical models sharing a common link and (b) by correlating the laser -induced changes in ultrafast demagnetization to the changes in the Gilbert damping factor which characterizes the magnetization precession. TR -MOKE is a well established local and non - invasive all -optical measurement technique tunable to an extremely fine time resolution limited only by the laser pulse width and is thus well suited to our purpose. For the detailed analysis of ultrafast demagnetization in our samples, we extract the values of demagnetization time and fast relaxation time for each sample using a p henomenological fitting function and record its systematic variation with the pump fluence. We subsequently model the demagnetization data using two well known theoretical models: the phenomenological three -temperature model (3TM) and the microscopic three temperature model (M3TM), and thereby extract values for the microscopic parameters relevant for the demagnetization process and calculate the temporal evolution of the simulated temperatures of the electron, spin and lattice systems within the first few picoseconds of the laser excitation. Both these models assume a thermal picture to explain the initial electron temperature rise and subsequent relaxation but differ significantly in their approach with regards to the treatment of the magnetization. A syst ematic study implementing both models to analyze TR -MOKE data recorded under uniform experimental conditions on identically prepared samples is absent in the literature and will prove instructive. Further, a thorough investigation and characterization of t he ultrafast demagnetization, magnetic damping and their intercorrelation in three different systems deposited under identical deposition conditions has not been carried out before. For the purposes of a comparative analysis, it is vital to study samples g rown under the same deposition conditions. The conductivity of the substrate can also directly influence demagnetization time by promoting or hindering ultrafast spin transport processes [20, 28] while the 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 shown to be quite sensitive to extrinsic factors such as the laser pulse duration and the spectral ba ndwidth [29], it is useful to measure the demagnetization times for various thin films from experiments performed in the same experimental setup under near-identical experimental conditions. 2. Materials and Methods 2.1 Sample fabrication 20 nm -thick cobalt, nickel and permalloy thin films were deposited by electron beam evaporation under an average base pressure of 1 ×10−7 Torr and at a very low deposition rate of 0.2 Å/s chosen to achieve uniform deposition. Each film was deposited on an insulating 8 mm × 8 mm silicon substrate coated with 285-nm-thick SiO 2. Subsequently, the films were capped in-situ with a 5 -nm-thick protective gold layer (base pressure ~1 × 10−7 Torr, deposition rate 0.1 Å/s) to prevent surface oxidation of the ferromagnetic layer and protect it from possible degradation due to high power laser exposure during the optical pump -probe measurements [ 30, 31]. The thickness of the capping layer was kept more than three times smaller than the optical penetration depth of 400 nm light in gold. After the deposition of the capping layer, the surface topography of the samples was investigated by atomic force microscopy (AFM) using a Tap190Al -G tapping mode AFM probe as shown in Figure 1 ( a). The average surface roughness Ra of the cobalt, nickel and permalloy films were obtained as 0.91 nm, 0.36 nm and 0.41 nm respectively. Thus, reasonably low surface roughn ess in the sub-nm range was obtained which is comparable for all the samples. In addition, the static magneto -optical Kerr effect was used to study the magnetic hysteresis of the deposited samples to confirm their ferromagnetic nature (Figure 1(b)). 2.2 All-optical measurement of ultrafast spin dynamics Measurement of laser -induced magnetization dynamics is carried out using a TR -MOKE technique in two -color optical pump -probe arrangement using a 400 nm pump beam and 800 nm probe beam having a pump -probe cr oss-correlation width of ~100 fs [30] shown schematically in Figure 1( c). A typical TR -MOKE trace is shown in Figure 1( d), comprising ultrafast demagnetization (Regime I), fast remagnetization (Regime II) and damped magnetization precession (Regime III).Th is technique enables the direct observation of the spin dynamics in the femto - and picosecond time domain. Femtosecond pulses are generated by an amplified laser system (Libra, Coherent Inc.) employing a chirped -pulse regenerative amplifier and a Ti:sapphi re laser oscillator (Coherent Inc.) pumped by a neodymium -doped yttrium lithium fluoride (Nd -YLF) laser. The second harmonic of the amplified laser output (wavelength = 400 nm, repetition rate = 1 kHz, pulse width >40 fs), generated through a lithium tribo rate (LBO) nonlinear crystal, is used for laser excitation of the ferromagnetic thin films. The time-delayed fundamental beam (wavelength = 800 nm, repetition rate = 1 kHz, pulse width ~40 fs) is used to probe the ensuing magnetization dynamics. In our setup, different wavelengths are employed for the pump and the probe pulse to eliminate the possibility of state blocking effects arising from the use of identical wavelengths for pumping and probing [32]. A computer -controlled variable delay generator offers precise control of the delay time between pump and probe. Before commencing measurements on any sample, the zero delay was carefully estimated by maximizing the transient reflectivity signal of a bare silicon substrate placed adjacent to the sample on the same sample holder. TR-MOKE experiments are performed with a non -collinear pump -probe geometry. The pump beam, focused to a spot size of ~300 µm, is incident obliquely on the sample while the probe beam, with a spot size of ~100 µm, is incident normal to the sample surface and aligned to the center of the pump spot. The pump -probe spatial overlap on the sample was carefully maintained. The choice of a relatively smaller spot size of the probe beam as compared to the pump beam facilitates optical alignment and ensures that the probe beam detects the local magnetization changes from a part of the sample uniformly irradiated by the pump. Before reflection on the samples, the probe beam is polarized orthogonally to the linearly polarized pump beam. After reflec tion, the Kerr -rotated probe beam is split and one part is fed directly into a Si -photodiode to measure the time -resolved reflectivity signal. The other part is fed into an identical photodiode after passing through a Glan -Thompson polarizer adjusted to a small angle from extinction to minimize optical artifacts in the Kerr rotation signal. In this way, simultaneous measurement of the time-resolved total reflectivity and Kerr rotation signals is possible. An optical chopper operating at 373 Hz placed in th e path of the pump beam provides a reference signal for the digital signal processing lock -in amplifiers (Stanford Research Systems, SR830) which record the modulated signal in a phase sensitive manner. All experiments were carried out under ambient condit ions of temperature and pressure. 4 3. Results and Discussion 3.1 Theoretical models for ultrafast demagnetization The phenomenon of optically induced ultrafast demagnetization starts with the irradiation of the magnetic sample with a brief and intense optical laser pulse, exciting electrons momentarily a few electron volts above the Fermi level. Though the exact sequence of events following the initial excitation is difficult to trace due to the highly nonequilibrium conditions created by it, a qualitative overview of the complete demagnetization process is fairly well established. The laser excitation generates a nonthermal pool of excited electrons which thermalize rapidly within several femtoseconds via electron -electron interactions. Spin -dependent scatter ing events taking place during this transient regime lead to a sharp drop in magnetization observable around a few hundred femtoseconds in the experimental Kerr rotation signal. Subsequently, the thermalized electrons may release their excess energy via a variety of relaxation channels, such as by excitation of phonons or magnons. This results in a partial recovery of the magnetization beyond which heat dissipation into the environment promotes further recovery on a longer timescale. The 3TM posits that the thermodynamics of the demagnetization phenomenon can be described simply by considering energy exchange between three thermal reservoirs [33], each of which is assigned a temperature: the electrons at temperature Te, the lattice at temperature Tl and the electronic spin reservoir at temperature Ts. Since the reservoirs are in thermal contact and the overall process is adiabatic, equilibration of the excited electrons with the spin and lattice reservoirs via energy transfer may be described by coupled rate equations in the following manner: (1) where, Ce, Cl and Cs denote the specific heats of the electron, lattice and spin reservoirs respectively, while Gel, Ges, and Gsl denote the inter -reservoir coupling parameters. The term P (t) describes the action of the laser pulse as a source term driving the excitation of the electron reservoir to high temperatures. The thermal diffusion term describes heat dissipation occurring via thermal conduction along the sample thicknes s. Under this description, the observed demagnetization is attributed to a rise in the spin temperature Ts occurring shortly after the electron temperature rise. The coupling strengths between the electron -spin and electron -lattice subsystems qualitatively determine the efficiency of energy transfer between them and hence influence the timescales associated with the demagnetization and fast relaxation. However, the 3TM is purely phenomenological and does not explicitly consider any microscopic mechanisms un derlying the phenomenon it describes. On the other hand, the M3TM proposed by Koopmans et al. [23] provides a “spin -projected” perspective [27] to explain the ultrafast transfer of angular momentum highlighting the role of Elliot -Yafet type ultrafast spin-flip scattering in the demagnetization process. The initial excitation by the laser pulse disturbs the electronic subsystem from equilibrium which leads to an imbalance in spin-up and spin-down scattering rates, resulting in the observed loss of magnetic order. The process is mediated by spin-orbit interactions leading to the formation of hot spots in the band structure where spin-up and spin-down channels are intermixed. An electron scattered into these hot spots via a phonon - or impurity -mediated scatteri ng will flip its spin with a finite probability. The individual scattering events are characterized by a parameter asf across the sample, identified with the probability of spin-flip due to electron -phonon scattering. The magnitude of this parameter will directly depend on the extent of spin-orbit coupling and hence is expected to be comparable in materials with similar spin-orbit coupling strengths. The M3TM retains the coupled rate equations for the electron and lattice temperatures, similar to the thermal description provided by the 3TM. However the fundamental difference from the 3TM is that in the framework of the M3TM, the spin bath is formed by a collection of two-level systems obeying Boltzmann statistics. Instead of assigning a temperature to the spin bath, the normalized magnetization is directly calculated from the associated exchange splitting. The rate of change of the magnetization, derived analytically considering an Elliot -Yafet scattering -driven demagnetization, is parametrized by asf and c oupled to the electron and the lattice subsystem temperatures. The assignment of a characteristic temperature to the spin subsystem is replaced in the M3TM by an evolution equation for the magnetization: 5 (2) The quantity R is a mate rial-specific factor which influences the demagnetization rate and is proportional to asfTc2/μat where TC is the material Curie temperature and µat is the atomic magnetic moment. Though the two models differ in their approaches, one can immediately discern certain similarities in their domains of validity. Both models are appropriate only when th e nonlocal mechanisms driving ultrafast demagnetization such as superdiffusive spin transport can be neglected. We note here that it has been reported that spin transport is not a major contributor to the ultrafast demagnetization in transition metals [34] . We nevertheless use an insulating SiO 2-coated Si substrate for our samples to minimize spin transport effects such that analysis with the local models described above should suffice in our case. Any additional contributions arising from the gold capping layer would be uniform across all samples investigated and therefore unlikely to impact the main results of our comparative study. Moreover, since the thickness of the capping layer is much smaller than the penetration depth of both 400 nm and 800 nm light in gold [35], the pump excitation fully penetrates down to the magnetic layer ensuring that the effect of direct laser excitation of the ferromagnet is probed in our case. Thus, we set up numerical calculations based on the models described above in order to extract microscopic information from the experimentally obtained demagnetization traces. 3.2 Ultrafast demagnetization in cobalt, nickel, and permalloy thin films We proceed by performing time -resolved measurements of the polar magneto -optical Kerr effect in the cobalt, nickel and permalloy thin film samples as a function of the laser fluence. Measurements are carried out under a strong enough external magnetic field kept constant at around 2 kOe tilted at a small angle from the sample plane to saturate the magnetization of the samples. The pump fluence is varied between 0.8 -8.7 mJ/cm2 by varying the power of the pump pulse. The results are presented in Figure 2. To ascertain that the measured Kerr signal reflects the true magnetization dynamics without any spurious contribution from optical effects triggered in the initial stages of laser excitation, we also examine the transient reflectivity signal for each sample. The fluence dependent variation in the reflectivity can be found in Figure S1 of the Supplementary Materials, demonstrating that at any given fluence the amplitude of the reflectivity signal is negligibly small compared to that of the Kerr rotation. We nevertheless restrict ourselves to the low fluence regime to avoid nonlinear effects a nd sample damage. For all our experiments, the probe fluence is kept constant at a value about half that of the lowest pump fluence used to prevent additional contribution to the spin dynamics by probe excitation. As seen in Figure 2, the ultrafast demagne tization completes within 1 ps for all three samples considered which is followed by a fast recovery of the magnetization, all observed within the experimental time window of 4 ps. These experimental traces clearly exhibit the “Type -I” or “one -step” demagn etization expected for transition metal thin films at room temperature and under low -to- moderate pump fluence [23]. The amplitude of the maximum quenching of the Kerr rotation signal increases with the laser fluence, allowing us to rule out nonlinear effec ts [36]. Closer inspection of the traces also reveals an increase of the time taken to demagnetize the samples with increasing fluence for all three samples. To quantify this increase, we fit our demagnetization traces to a phenomenological expression based on the 3TM and valid in the low laser fluence regime [37]: (3) where Θ(t) is the Heaviside step function, δ(t) is the Dirac delta function and Γ(t) is the Gaussian laser pulse. The constant A1 represents the value of the normalized magnetization after remagnetization has completed and equilibrium between the electron, spin and lattice reservoirs has been re-established. A2 is proportional to the initial rise in electron temperature and hence to the maximum magnetization quenching. A3 represents the magnitude of state -filling effects present during the onset of the demagnetiz ation response, which is negligible in our case. τM and τE are the demagnetization time and fast relaxation time, respectively. Prior to the fitting, all the experimental traces were normalized by hysteresis measurements of the Kerr rotation signal under the saturating magnetic field in the absence of la ser excitation. We find that within the range of fluence values considered, permalloy exhibits the largest magnetization quenching of 54.6%, followed by a 23.7% quenching achieved in nickel, while the magnetization of cobalt the least, only about 8% for the largest applied 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 attributed to enhanced spin -fluctuation s at elevated spin temperatures for higher fluences [38]. At a fluence of 4.8 mJ/cm2, the extracted demagnetization times are 276.6 ± 3.41 fs for cobalt, 187.3 ± 2.89 fs for nickel and 236.8 ± 2.45 fs for permalloy. The timescale for the magnetization recovery τE also increases with increasing pump fluence. The variation of these characteristic timescales with laser fluence is shown in Figure 3. These fluence -dependent trends in τM and τE hint at a spin-flip process -dominated ultrafast demagnetization in our stu died systems [23, 39, 40]. The values of τM extracted from our experiments lie within the typical range of 100-300 fs consistent with previous reports of the ultrafast demagnetization times in these metals [17, 23], and are too large to represent a superdi ffusive transport -driven demagnetization [41]. For the 3TM and M3TM simulations, we choose a laser pump term given by proportional to the pump fluence F and following a Gaussian temporal profile. The maximum rise of the electron temperature a nd thus also the extent of demagnetization depends sensitively on this term which is hence adjusted to reproduce the maximum quenching observed experimentally. We use a pulse width τp = 100 fs determined by the pump -probe cross -correlation in all calculations. Intrinsic to both the models we consider is the assumption that electron thermalization occurs extremely fast. The thermal diffusion term can be neglected in our case since th e thicknesses of the films we study are kept slightly greater than the optical penetration depth of 400 nm pump beam in those films. This ensures uniform heating of the films in the vertical direction while also avoiding laser penetration into the substrat e in which case heat dissipation into the substrate would have to be taken into account. Besides, the timescales associated with heat dissipation are generally tens to hundreds of picoseconds, much longer than the demagnetization and fast relaxation times, and hence unlikely to significantly influence our observations at these timescales. Since both models we consider are thermal in their approach, choosing correct values for the reservoir specific heats is vital for a proper simulation of the demagnetizati on. For the electronic specific heat Ce, we assume a linear dependence on the electronic temperature Ce(Te) = γTe derived from the Sommerfeld free -electron approximation where γ is determined by the electronic density of states at the Fermi level [42]. The value of γ for permalloy is approximated as a weighted average of the individual γ values of nickel and iron in permalloy. The lattice specific heat Cl is calculated at each value of the lattice temperature according to the following relation derived from Debye theory: (4) where NA is the Avogadro’s number, kB is the Boltzmann’s constant and θD is the Debye temperature. Finally, we fix the spin specific heat Cs to its value at room temperature for the 3TM calculations, obtained by subtracting the electronic and lattice contributions from the experimental values of the total specific heat found in the literature [43]. Considering a spin-temperature -dependent form of Cs was not found to significantly affect our conclusions as described in Section IV of the Supplementary Materials. The fixed parameter set used in our calculations have been listed in Table 1. To relate the experimental demagnetization to the temperature of the spin subsystem under the 3TM framework, the spin temperature Ts is mapped to the magnetization of the system via the Weiss’ mean field theory [44], which is then fitted with the experimental magnetization traces to obtain the empirical inter -reservoir coupling parameters Gel and Ges consistent with the observed dynamics. We neglect the spin-lattice coupling parameter Gsl for the 3TM simulations since in ferromagnetic transitio n metals the energy transfer between electrons and lattice is far greater than that between lattice and spins [45]. Table 1 . Fixed parameter set used in the calculations. Literature values have been used for all parameters listed [30, 31, 34, 35]. For the 3TM simulations, we proceeded to extract Gel and Ges by first fitting the demagnetization data at the lowe st fluence to the model. However, fitting the higher fluence data using identical values of the coupling parameters as extracted at the lowest fluence did not result in a good match to our experimental results. The coupling parameters extracted from the lo w fluence data led to an overestimation 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) Ni 627 450 1065 3.07 × 105 0.62 Co 1388 445 714 1.59 × 105 1.72 Py 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 time. On the other h and, the remagnetization dynamics is most sensitive to the Gel parameter so that the overall dynamics is best reproduced only by adjusting both Gel and Ges. As shown in Figure 4, the resulting fit shows excellent agreement with the experimental data. This exercise reveals the crucial role played by the electron -spin relaxation channels in determining the timescale associated with the initial demagnetization while the magnetization recovery is primarily mediated by the electron -lattice interaction. We also f ind that the mismatch between model and experiment can be resolved by considering an increasing trend of Gel and Ges with pump fluence arising from a faster demagnetization process for the same percentage quenching as compared to the model predictions with in the studied fluence range. The values of the microscopic parameters extracted from the least -squares fits with their corresponding error bounds can be found in Supplementary Tables S1 -S3. Since the exact values of the coupling parameters extracted from the fits naturally depend on the values chosen for the fixed parameters, the interpretation of the results from these fits is best limited to a comparative one. For the M3TM simulations, the demagnetization traces are fitted directly to Equation 2 yielding Gel and asf as fit parameters. In this case, asf plays a role in determining the maximum extent and the associated timescale of the demagnetization via the scaling factor R while Gel continues to influence mainly the magnetization recovery process. How ever, the demagnetization time is less sensitive to changes in asf than it is to Ges in the 3TM case. This results in somewhat higher values of Gel and a sharper rise with pump fluence than those extracted from the 3TM simulations, in order to compensate for the overestimation of demagnetization time that results from the model if Gel at the lowest fluence is used 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 for permalloy. The value of asf we have ext racted for nickel is an order of magnitude lower than the value asf = 0.185 first reported by Koopmans et al. [23] but quite close to the value of 0.08 reported by Roth et al. [39]. This discrepancy is expected, as the artificially high value of 0.185 aros e due to an overestimation of the electronic specific heat in the original work, avoided here by considering experimentally determined γ values reported in the literature. The observation that asf [Co] <asf [Ni] is consistent with previous reports [23, 40]. With regard to thermal treatments of ultrafast demagnetization like the 3TM and M3TM, there is some contention centered on the role o f the nonthermal electrons generated within the first 50-100 femtoseconds after laser excitation [9, 48, 49]. Within this time frame, the excited hot electrons have not yet thermalized and their energies cannot be described by a Fermi -Dirac distribution. However, both the 3TM and the M3TM completely neglect the finite electron thermalization time and consider only the energy exchange between the thermalized electron population with the lattice and/or the spin subsystem as the initial nonequilibrium distribu tion is assumed to last for much shorter durations (≳ 10 fs) than the timescale over which demagnetization typically occurs (≳ 100 fs). However, strictly speaking, this assumption is valid only when the thermalization proceeds very rapidly such as can be ensured with high fluence excitations [50]. In the low fluence regime, electron thermalization is slower, hence the influence of nonthermal electrons on the measured dynamics may become more important with decreasing pump fluence. The increasing of Gel with fluence deduced from our calculations may be an indication of a decrease in the efficiency of electron -lattice interactions at low laser fluences when nonthermal electrons prevail. However, the specific values of Gel and Ges values we have obtained for th e different systems remain comparable to values previously reported for these metals [11, 23, 37]. After completion of the fitting routine, we set up simulations using the optimized values of the free parameters to obtain the temporal evolution of Te, Tl and Ts as a function of the pump -probe delay. The calculated profiles of Te, Tl and Ts for four different values of the pump fluence are presented in Figure 5 for nickel, showing good agreement between the results predicted by the 3TM and the M3TM. Simil ar profiles for cobalt and permalloy are given in Figs. S2 and S3 of the Supplementary Materials. As the pump fluence is increased, laser excitation deposits a greater amount of energy in the electronic reservoir which leads to a proportionate increase in the initial electron temperature rise. However, the maximum electron temperature reached for the highest applied pump fluence remains well below the Fermi temperature for the respective material, ensuring the assumption of the linear temperature dependence of the electronic specific heat remains valid [42]. Given the fundamentally different origins of the 3TM and M3TM, the close agreement between the temperature profiles calculated using the two models and among the extracted values of the electron -lattice coupling parameter Gel is remarkable. This observation reflects that, given the strength of the laser excitation determining the initial temperature excursion of the electron system, the Gel parameter common to both the 3TM and the M3TM 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 both models. While the electron temperature is coupled to the spin temperat ure via the Ges parameter in the 3TM, it enters into the evolution equation for the magnetization via Equation 2 in the M3TM case. From Figure 5, a qualitative difference in the temporal profiles of the electron, spin and lattice temperatures is also appa rent. Referring to the figures corresponding to the 3TM, while Te exhibits a sharp peak within the first few hundred femtoseconds, Ts exhibits a shorter and broader peak at slightly longer time delays. The delayed peak in Ts occurs when excited electrons t ransfer energy to the spin degrees of freedom [51] through scattering events mediated by the coupling parameter Ges, which qualitatively determines the maximum spin temperature value [11]. These scattering events can result in the generation of magnons, St oner excitations and spin fluctuations which may play a part in the observed demagnetization [52]. Lastly, the lattice temperature gradually increases over the first few picoseconds after excitation, reaching an equilibrium value at which it stays nearly constant over several picoseconds. In Figure 6 we present a comparison between the experimental demagnetization curves and calculated electron temperature profiles for each of our three samples. Under the same applied pump fluence, different amplitudes of magnetization quenching and electron temperature rise are observed. It is clearly seen that nickel quenches more strongly than cobalt, an observation which can be explained by band structure effects and a more efficient laser -induced electronic excitation i n nickel owing to its much lower Curie temperature [34, 40, 53], reflected in the M3TM as a higher extracted value of the spin -flip probability asf in nickel. On the other hand, a lower electronic specific heat capacity in cobalt as compared to nickel [42] predictably leads to a much greater rise in its electron temperature for the same pump fluence. In the case of permalloy, alloying with iron can reduce the electronic density of states at the Fermi level as compared to nickel, decreasing γ. Because o f the larger electron temperatures reached as a result, a value of asf comparable to that of nickel is sufficient to drive a much larger reduction in the magnetization. Permalloy also shows a somewhat larger demagnetization time at a given pump fluence as compared to nickel. Since a standard TR -MOKE optical pump -probe measurement cannot be used to probe the demagnetization response in alloys in an element -specific manner, interpreting this finding is difficult. It has been reported previously using element -specific measurements in a nickel -iron alloy that the demagnetization of nickel proceeds faster than that of iron [54]. Our observation of a slower demagnetization in permalloy may be reflective of this result. Finally, the demagnetization times for the three systems at any given pump fluence is seen to sensitively depend on the ratio of TC/µat, confirming its role as a ‘figure of merit’ (FOM) for the demagnetization time [23]. As shown in Figure 8 ( a), the demagnetization proceeds slower for a lower value of the FOM. Thus at a given incident laser pump fluence, the demagnetization times are found to correlate with the values of this quantifying parameter, rather than the material Curie temperature directly. 3.3 Precessional dynamics and correlation of Gilbert damping parameter with ultrafast demagnetization time To gain further insight into the microscopic mechanism underlying the ultrafast demagnetization, we study also the precessional magnetization dynamics in our thin film systems. A typical time -resol ved Kerr rotation data showing the different temporal regimes of laser -induced magnetization dynamics is given in Figure 1( d). Regime I and II represent the ultrafast demagnetization and magnetization recovery processes respectively, while the oscillatory signal of regime III represents the magnetization precession with the Gilbert damping characteristic of the ferromagnetic material. To extract the Gilbert damping factor α, the background -subtracted oscillatory Kerr signal is first fitted with a damped sinusoidal function given by: (5) where f is the precessional frequency, τ is the relaxation time, and φ is the initial phase of oscillation. The depende nce of f on applied bias magnetic field H is fitted to the Kittel formula relevant for ferromagnetic systems: (6) where γ0 = gµB/ℏ is the gyromagnetic ratio and Meff is the effective saturation magnetization of the probed volume. From the Kittel fit, the values of g and Meff can be extracted as fit parameters. 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 emu/cm3 for permalloy (Section V of the Supplementary Materials). Subsequently, the effective damping factor α is calculated depending on the extracted value of τ and Meff as: (7) Although the timescale of precessional dynamics and magnetic damping in ferromagnetic thin films differ by many orders of magnitude, similar underlying phenomena may oft en determine the characteristics of either process. Through several independent investigations relating these two distinct processes, the correlation between the demagnetization time τM and the Gilbert damping factor α emerged as a critical indicator of the dominant microscopic contribution to the ultrafast demagnetization. Koopmans et al. in 2005 analytically derived an inversely proportional relationship between the Gilbert damping factor and ultrafast demagnetization time based on quantum mechanical considerations [16]. However, the validity of the model could not be confirmed experimentally when τM and α were tuned by transition metal and rare earth doping [18]. Later it was proposed that the Gilbert damping parameter and ultrafast demagnetization time can be either directly or inversely proportional to each other depending on the predominant micr oscopic mechanism contributing to the magnetic damping [55]. A proportional dependence would suggest a dominating conductivity -like or “breathing Fermi surface” contribution to the damping due to the scattering of intraband electrons and holes, while an inverse dependence may arise in materials with a dominant resistivity -like damping arising from inter-band scattering processes. Zhang et al. later suggested that an inverse correlation between Gilbert damping factor and ultrafast demagnetization time may arise in the presence of spin transport, which provides an additional relaxation channel resulting in an enhancement of the magnetic damping along with an acceleration of the demagnetization process at the femtosecond timescale [56]. In cases where local spin-flip scattering processes are expected to dominate the demagnetization process, the breathing Fermi surface model is valid predicting a proportional relationship between τM and α. To study the correlation between ultrafast demagnetization and damping in a system, both τM and α have to be systematically varied. For a given material, the exciting pump fluence can be used to modulate the demagnetization time. As discussed in the previous section, we observe an increment of the demagnetization time with the pump fluence for the systems under investigation. An enhancement in α is also observed as the pump fluence is increased from 0.8 mJ/cm2 to 8.7 mJ/cm2 for each sample, associ ated with a corresponding decrease in the precessional relaxation time. Within the experimental fluence range, α is found to be enhanced from 0.0089 to 0.0099 for cobalt, from 0.0346 to 0.0379 for nickel and from 0.0106 to 0.0119 for permalloy. This enhanc ement of the damping can be attributed to the transient rise in the system temperature as the laser pump fluence is increased [57]. As a laser -induced increase in electron temperature also drives the demagnetization process at the femtosecond timescale, one can qualitatively correlate the demagnetization time with the magnetic damping at various excitation energies even for a single sample. Clearly, a direct correlation between demagnetization time and the Gilbert damping factor is obtained in our samples b y measuring the magnetization dynamics at various pump fluences, as shown in Figure 7 for the nickel thin film. Thus, we infer that similar relaxation processes likely drive the laser induced transient magnetization dynamics at the two different timescales . At a low fluence value of 2.1 mJ/cm2, the extracted value of α is the highest for the nickel film, and relatively low for cobalt and permalloy. The observed trend in Figure 8 (b) can be directly correlated with the values of the Sommerfeld coefficient γ given in Table 1. This observation is in accordance with Kamber sky’s spin flip scattering theory [58], wherein α is directly proportional to the density of states at the Fermi level, which in turn governs the value of γ. 4. Summary To summarize, we have investigated fluence -dependent ultrafast demagnetization in identi cally -grown cobalt, nickel and permalloy thin films using an all-optical time-resolved magneto -optical Kerr effect technique. We study both the femtosecond laser -induced ultrafast demagnetization and the magnetization precession and damping in each of our samples for various pump excitation fluences. Using a phenomenological 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 with a spin-flip scattering -dominated demagnetization in our samples. Two distinct ultrafast demagnetization models can each reproduce our experimental demagnetization traces remarkably well. By directly fitting our experimental data to numerical solutions of these models, we have extracted values for the electron - lattice and electron -spin coupling parameters and an empirical estimate of the spin flip scattering probability for each of our samples and demonstrated the fluence -dependent variation in calculated electron, spin, and lattice temperatures as a function of the pump -probe delay. We conjecture that the increasing trend of the extracted electron -lattice coupling parameter with fluence may indicate a decrease in the efficiency of electron -lattice interactions in the low laser fluences when nonthermal electrons may play a role in influencing the magnetization dynamics. We have compared the experimental and theoretical results obtained from each of the systems under investigation and confirmed that the ratio of the Curie temperature to the atomic magnetic moment acts as a figure of merit sensitive to the demagnetization time for these systems. Our study demonstrates the mutual agreement between the phenomenological three temperature model and the microscopic three temperature model for the systems we study and highlights the possibility of a fluence -dependent enhancement in the inter-reservoir interactions in the low laser fluence regime. 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( c) A schematic of the samples used in our measurements and the measurement geometry. The 400 nm pump and 800 nm probe are incident non- collinearly on the sample surface. (d) Typical TR-MOKE data showing different temporal regimes of the magnetization dynamics. 15 Figure 2: Ultrafast demagnetization traces obtained from TR-MOKE measurements on (a) cobalt, (b) nickel and ( c) permalloy thin films at va rious pump fluences. The downward arrows represent increased pump fluence. The symbols represent experimental data points and the solid lines represent fit to Equation 3. 16 Figure 3: Variation of ( a) demagnetization time τM and ( b) fast relaxation time τE with the pump fluence for 20-nm-thick cobalt, nickel and permalloy films. Both timescales show an increasing trend with increasing pump fluence. Figure 4: Sensitivity of the demagnetization traces to the various adju stable fit parameters. Symbols represent data points and solid lines represent fits to the 3TM. Data shown is for the cobalt 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; for F =2.1 mJ/cm2, Gel=1.1 × 1018 Js−1m−3K−1 and Ges=8 × 1017 Js−1m−3K−1. 17 Figure 5: Temporal evolution of the reservoir temperatures of nickel Te (blue), Tl (red), and Ts (green) calculated using the (a) phenomenological three temperature model and (b) the microscopic three temperature model. Figure 6: Comparison between (a) the experimental demagnetization and (b) the simulated electron temperature profiles for cobalt, nickel and permalloy at a pump fluence of 4.8 mJ/cm2. 18 Figure 7: (a) Fluence -dependent variation of precessional dynamics for the nickel thin film. Symbols are experimental data points and the solid lines are fits to Equation 5. (b) Variation of f with pump fluence. (c) Variation of α with pump fluence. (d) Direct correlation between τM and α. Figure 8: (a) Inverse variation of the demagnetization time with the ratio of Curie temperature to atomic magnetic moment and ( b) inverse correlation of the demagnetization time with ma gnetic damping across three different materials. 19 Supplementary Materials S1 I. Transient reflectivity measurements Time -resolved transient reflectivity signal is recorded by TR-MOKE measurements simultaneously with the Kerr rotation signal quantifying the magnetic response.At any given fluence, the maximum amplitude of the transient reflectivity change is about a factor of 100 smaller as compared to that of the Kerr rotation. This validates that the Kerr rotation signal reflects the genuine magnetic response with negligible nonmagnetic contribution. Figure S1: Time -resolved transient reflectivity measured by TR -MOKE for cobalt, nickel, and permalloy. The upward arrows represent increased pump fluence. II. Calculation of electron, spin, an d lattice temperature profiles The calculated temporal evolution of the reservoir temperatures for the 20-nm-thick cobalt and permalloy thin films are represented in Fig. S2 and Fig. S3, respectively. The electron temperature shows a rapid and dramatic increase within one picosecond of the laser excitation, beyond which it decays to an equilibrium value over a few picoseconds. Over this period, the lattice subsystem is gradually heated above the equilibrium temperature while the heating of the spin subsyste m directly leads to the demagnetization. The maximum electron, spin and lattice temperatures increase with increasing laser fluence. S2 Figure S2: Electron, lattice, and spin temperature profiles for the cobalt thin film calculated using (a) the three tem perature model and (b) the microscopic three temperature model for various laser fluences (blue: Te, dashed, red: Tl and dotted, green: Ts). Figure S3: Electron, lattice, and spin temperature profiles for the permalloy thin film calculated using (a) the three temperature model and (b) the microscopic three temperature model for various laser fluences). III. Extraction of inter -reservoir coupling constants The least -squares fitting routines set up based on the three temperature model and the microscopic three temperature model return the values of the inter-reservoir coupling S3 constants that best reproduce the experimental demagnetization traces. The microscopic three temperature model additionally returns the value of spin-flip probability asf associated with the spin-flip scattering events which directly lead to the magnetization quenching. The extracted values of these microscopic parameters are given in Tables S1 -S3. Fig. S4 shows a comparison of the values of the electron -lattice coupling parameter Gel as extracted from the fits to the three temperature model and the microscopic three temperature model. For reasons discussed in the main text, the spin-lattice coupling parameter Gsl is taken to be zero for the three temperature model simulations. Fig. S5 shows that, when Gsl is set equal to 3 × 1016 Jm−1s−1m−3K−1 as considered in Phys. Rev. Lett. 76, 4250 (1996), the simulated trace and calculated electron temperature profile are nearly indistinguishable from the case in which Gsl is neglected. Figure S4: Values of the electron -lattice coupling parameter Gel extracted from fitting to the three temperature model and microscopic three temperature model. Figure S5: (a) Results of three temperature modelling and (b) calculated reservoir temperature profiles for two values of Gsl. Pump fluence (mJ/cm2) Gel[3TM] (×1018) (Js-1m-3K-1) Gel[M3TM] (×1018) (Js-1m-3K-1) Ges(×1017) (Js-1m-3K-1) asf (×10-1) 0.8 0.82±0.081 0.85±0.097 0.69±0.037 0.19±0.023 2.1 1.08±0.063 1.18±0.105 0.8±0.029 0.16±0.014 3.5 1.26±0.057 1.48±0.106 0.91±0.028 0.16±0.009 4.8 1.2±0.037 1.57±0.096 0.97±0.015 0.17±0.008 6.0 1.21±0.042 1.49±0.072 1.02±0.022 0.17±0.008 7.3 1.56±0.076 1.98±0.087 1.15±0.041 0.16±0.004 8.7 1.65±0.045 2.18±0.066 1.28±0.036 0.17±0.003 Tabl e S1: Microscopic parameters extracted from the 3TM and M3TM s imulations for cobalt. S4 Pump fluence (mJ/cm2) Gel[3TM] (×1018) (Js-1m-3K-1) Gel[M3TM] (×1018) (Js-1m-3K-1) Ges(×1017) (Js-1m-3K-1) asf (×10-1) 0.8 1.29±0.154 1.41±0.172 2.1±0.182 0.53±0.03 2.1 1.52±0.061 1.74±0.077 2.84±0.087 0.55±0.012 3.5 1.53±0.0 51 1.77±0.069 2.93±0.076 0.56±0.011 4.8 1.65±0.047 1.9±0.06 2.86±0.064 0.51±0.008 6.0 1.49±0.036 1.73±0.039 2.74±0.048 0.57±0.007 7.3 1.58±0.028 1.88±0.033 2.87±0.038 0.59±0.006 8.7 1.58±0.048 1.77±0.039 2.77±0.062 0.62±0.008 Tabl e S2: Microscopic pa rameters extracted from the 3TM and M3TM simulations for nickel . Pump fluence (mJ/cm2) Gel[3TM] (×1018) (Js-1m-3K-1) Gel[M3TM] (×1018) (Js-1m-3K-1) Ges(×1017) (Js-1m-3K-1) asf (×10-1) 0.8 1.62±0.144 1.86±0.184 2.16±0.145 0.31±0.016 2.1 1.79±0.074 2.14± 0.084 2.34±0.073 0.34±0.007 3.5 2.05±0.081 2.46±0.065 2.65±0.084 0.39±0.005 4.8 2.08±0.068 2.55±0.038 2.81±0.071 0.45±0.006 6.0 2.02±0.053 2.5±0.065 2.72±0.058 0.49±0.007 7.3 2.07±0.083 2.67±0.052 2.87±0.089 0.56±0.009 8.7 2.11±0.084 2.71±0.069 2.87±0 .086 0.64±0.01 Tabl e S1: Microscopic parameters extracted from the 3TM and M3TM simulations for permalloy . IV. Temperature dependence of spin specific heat Considering the theory of Manchon et al. ( Phys. Rev. B 85, 064408 (2012)) as adopted in Sci. Rep. 10:6355 (2020), the specific heat Cs of the spin subsystem in the 3TM framework can be calculated as a function of the spin temperature Ts as: (S1) Fig. S6 below shows the trends in the inter-reservoir coupling parameters for two scenar ios: considering the room temperature value of Cs(T0) or an explicit temperature (and hence time) dependence of Cs. Figure S6: Fluence -dependent variation of (a) Gel and (b) Ges considering two different forms for the spin specific heat Cs. S5 ∼ V. Bias fiel d dependence of precessional frequency and Gilbert damping The bias magnetic field-dependent variation of precessional dynamics in cobalt, nickel and permalloy is shown in Fig. S7 - Fig. S9 for a pump fluence of 4.8 mJ/cm2. The oscillatory Kerr signal is fitted with Eq. 5 of the main text to extract the precessional frequency f and relaxation time τ. The bias field dependence of the extracted frequency, shown in Fig. S7(b) for cobalt, is fitted to the Kittel formula (Eq. 6) to extract the effective saturation magnetization Meff . Once Meff have been derived, the Gilbert damping factor α is calcula ted using Eq. 7 of the main text. The extracted α does not show any systematic variation with bias field. Figure S7: (a) Bias field -dependent variation of precessional dynamics for the cobalt thin film. Symbols are experimental data points and the solid lines are fits to Eq. 5 of the main text. (b) Variation of f with H. The solid red line is a fit to the Kittel formula. (c) Variation of τ with H. (d) Variation of α with H. The solid line is a guide to the eye. VI. Pump fluence dependence of precessional frequency and Gilbert damping The variation of precessional dynamics in cobalt, nickel and permalloy with the laser pump fluence is shown in Fig. S10 for the cobalt and in Fig. S11 for the permalloy thin film at a saturating field of 2 kOe. The oscilla tory Kerr signal is fitted with Eq. 5 of the main text to extract the precessional frequency f and relaxation time τ. With increasing pump fluence, the precessional frequency decreases and the magnetic damping is enhanced. A direct correlation between the demagnetization time τM and α is observed. S6 Figure S8: (a) Bias field -dependent variation of precessional dynami cs for the nickel thin film. Symbols are experimental data points and the solid lines are fits to Eq. 5 of the main text. (b) Variation of f with H. The solid red line is a fit to the Kittel formula. (c) Variation of τ with H. (d) Variation of α with H. The solid line is a guide to the eye. Figure S9: (a) Bias field -dependent variation of precessional dynamics for the permalloy thin film. Symbols are experimental data points and the solid lines are fits to Eq. 5 of the main text. (b) Variation of f with H. The solid red line is a fit to the Kittel formula. (c) Variation of τ with H. (d) Variation of α with H. The solid line is a guide to the eye. S7 Figure S10: (a) Fluence -dependent variation of precessional dynamics for the cobalt thin film. Symbo ls are experimental data points and the solid lines are fits to Eq. 5 of the main text. (b) Variation of f with pump fluence. (c) Variation of α with pump fluence. (d) Direct correlation between τM and α. Figure S11: (a) Fluence -dependent variation of precessional dynamics for the permalloy thin film. Symbols are experimental data points and the solid lines are fits to Eq. 5 of the main text. (b) Variation of f with pump fluence. (c) Variation of α with pump fluence. (d) Direct correlation between τM and α.

2023-01-30

Following the demonstration of laser-induced ultrafast demagnetization in ferromagnetic nickel, several theoretical and phenomenological propositions have sought to uncover its underlying physics. In this work we revisit the three temperature model (3TM) and the microscopic three temperature model (M3TM) to perform a comparative analysis of ultrafast demagnetization in 20-nm-thick cobalt, nickel and permalloy thin films measured using an all-optical pump-probe technique. In addition to the ultrafast dynamics at the femtosecond timescales, the nanosecond magnetization precession and damping are recorded at various pump excitation fluences revealing a fluence-dependent enhancement in both the demagnetization times and the damping factors. We confirm that the Curie temperature to magnetic moment ratio of a given system acts as a figure of merit for the demagnetization time, while the demagnetization times and damping factors show an apparent sensitivity to the density of states at the Fermi level for a given system. Further, from numerical simulations of the ultrafast demagnetization based on both the 3TM and the M3TM, we extract the reservoir coupling parameters that best reproduce the experimental data and estimate the value of the spin flip scattering probability for each system. We discuss how the fluence-dependence of inter-reservoir coupling parameters so extracted may reflect a role played by nonthermal electrons in the magnetization dynamics at low laser fluences.

Investigation of Ultrafast Demagnetization and Gilbert Damping and their Correlation in Different Ferromagnetic Thin Films Grown Under Identical Conditions

2301.12797v1

Theory of magnon motive force in chiral ferromagnets Utkan G ung ord uand Alexey A. Kovalev Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA We predict that magnon motive force can lead to temperature dependent, nonlinear chiral damping in both conducting and insulating ferromagnets. We estimate that this damping can signicantly in uence the motion of skyrmions and domain walls at nite temperatures. We also nd that in systems with low Gilbert damping moving chiral magnetic textures and resulting magnon motive forces can induce large spin and energy currents in the transverse direction. PACS numbers: 85.75.-d, 72.20.Pa, 75.30.Ds, 75.78.-n Emergent electromagnetism in the context of spintron- ics [1] brings about interpretations of the spin-transfer torque [2, 3] and spin-motive force (SMF) [4{10] in terms of ctitious electromagnetic elds. In addition to pro- viding beautiful interpretations, these concepts are also very useful in developing the fundamental understanding of magnetization dynamics. A time-dependent magnetic texture is known to induce an emergent gauge eld on electrons [5]. As it turns out, the spin current gener- ated by the resulting ctitious Lorentz force (which can also be interpreted as dynamics of Berry-phase leading to SMF) in uences the magnetization dynamics in a dis- sipative way [5, 6, 11{16], aecting the phenomenologi- cal Gilbert damping term in the Landau-Lifshitz-Gilbert (LLG) [17] equation. Inadequacy of the simple Gilbert damping term has recently been seen experimentally in domain wall creep motion [18]. Potential applications of such studies include control of magnetic solitons such as domain walls and skyrmions [19{31], which may lead to faster magnetic memory and data storage devices with lower power requirements [32{34]. Recently, phe- nomena related to spin currents and magnetization dy- namics have also been studied in the context of energy harvesting and cooling applications within the eld of spincaloritronics [35{39]. Magnons, the quantized spin-waves in a magnet, are present in both conducting magnets and insulating mag- nets. Treatment of spin-waves with short wavelengths as quasiparticles allows us to draw analogies from systems with charge carriers. For instance, the ow of thermal magnons generates a spin transfer torque (STT) [40{42] and a time-dependent magnetic texture exerts a magnon motive force. According to the Schr odinger-like equa- tion which governs the dynamics of magnons in the adia- batic limit [41], the emergent \electric" eld induced by the time-dependent background magnetic texture exerts a \Lorentz force" on magnons, which in turn generates a current by \Ohm's law" (see Fig. 1). Despite the sim- ilarities, however, the strength of this feedback current has important dierences from its electronic analog: it is inversely proportional to the Gilbert damping and grows ugungordu@unl.eduwith temperature. In this paper, we formulate a theory of magnon feed- back damping induced by the magnon motive force. We nd that this additional damping strongly aects the dy- namics of magnetic solitons, such as domain walls and skyrmions, in systems with strong Dzyaloshinskii-Moriya interactions (DMI). We also nd that the magnon mo- tive force can lead to magnon accumulation (see Fig. 1), non-vanishing magnon chemical potential, and large spin and energy currents in systems with low Gilbert damp- ing. To demonstrate this, we assume diusive transport of magnons in which the magnon non-conserving relax- ation time, , is larger compared to the magnon conserv- ing one,m(m). For the four-magnon thermal- ization,m=~=(kBT)(Tc=T)3, and for LLG damping, =~=kBT, this leads to the constraint (Tc=T)31 [43, 44]. Emergent electromagnetism for magnons. We initially assume that the magnon chemical potential is zero. The validity of this assumption is conrmed in the last sec- tion. Eects related to emergent electromagnetism for magnons can be captured by considering a ferromagnet well below the Curie temperature. We use the stochastic LLG equation: s(1 +n)_n=n(He+h); (1) wheresis the spin density along n,He=nF[n] is the eective magnetic eld, F[n] =R d3rF(n) is the free energy and his the random Langevin eld. It is convenient to consider the free energy density F(n) = J(@in)2=2 +^Dei
(n@in) +H
n+Kun2 zwhereJis the exchange coupling, ^Dis a tensor which describes the DMI [47],H=Ms0Haezdescribes the magnetic eld, Kudenotes the strength of uniaxial anisotropy, Msis the saturation magnetization, Hais the applied magnetic eld, and summation over repeated indices is implied. At suciently high temperatures, the form of anisotropies is unimportant for the discussion of thermal magnons and can include additional magnetostatic and magnetocrys- talline contributions. Linearized dynamics of magnons can be captured by the following equation [48]: s(i@t+ns
At) = J(@i=ins
[AiDi=J])2+' ; (2)arXiv:1605.01694v2 [cond-mat.mes-hall] 13 Jul 20162 μ/μ0 -0.3-0.2-0.100.10.20.3 μ/μ0 -0.75-0.50-0.2500.250.500.75 FIG. 1. (Color online) A moving magnetic texture, such as a domain wall (top) or an isolated skyrmion (bottom), gen- erates an emergent electric eld and accumulates a cloud of magnons around it. In-plane component of nsand electric eldEare represented by small colored arrows and large black arrows, respectively. Magnon chemical potential is mea- sured in0=~vD=J
for domain wall and 0=~vD=JR for skyrmion, where is the magnon diusion length. Soli- tons are moving along + xaxis with velocity v,nz=1 at x=1 for domain wall and nz= 1 at the center for the isolated hedgehog skyrmion. Material parameters for Co/Pt (J= 16pJ/m, D= 4mJ/m2,Ms= 1:1MA/m,= 0:03, at room temperature [45]) were used for domain wall lead- ing to
7nm, and Cu 2OSeO 3parameters ( J= 1:4pJ/m, D= 0:17mJ/m2,s= 0:5~=a3,a= 0:5nm with= 0:01, at T50K [46]) for skyrmion leading to R50nm. System size is taken to be 6
2
for domain wall and 6 R6Rfor skyrmion. where'absorbs eect of anisotropies, DMI and the magnetic eld, =nf
(e0 x+ie0 y) describes uctu- ationsnf=nnsq 1n2 faround slow component ns(jnj=jnsj= 1,ns?nf) in a rotated frame in whiche0 z=ns,Di=^Dei, andA ^R@^RTcorre- sponds to the gauge potential with =x;y;z;t . Note that in the rotated frame, we have n!n0=^Rnand @!(@A0 ) withA0 = (@^R)^RT. In deriv- ing Eq. 2, we assumed that the exchange interaction is the dominant contribution and neglected the coupling between the circular components of and ydue to anisotropies [41, 49]. The gauge potential in Eq. (2) leads to a reactive torque in the LLG equation for the slow dynamics [50]. Alternatively, one can simply average Eq. (1) over the fast oscillations arriving at the LLG equation with the magnon torque term [40]: s(1 +ns)_nsnsHs e=~(j
D)ns;(3) whereHs e=nsF[ns] is the eective eld for theslow magnetization calculated at zero temperature [51], ji= (J=~)hns
(nf@inf)iis the magnon current and Di=@i+ (^Dei=J)is the chiral derivative [48, 52, 53]. Magnon feedback damping. The magnon current jis induced in response to the emergent electromagnetic po- tential, and can be related to the driving electric eld Ei=~ns
(@tnsDins) by local Ohm's law j=E whereis magnon conductivity. The induced elec- tric eld Ecan be interpreted as a magnon generaliza- tion of the spin motive force [6]. The magnon feed- back torque=~(E
D)nshas dissipative eect on magnetization dynamics and leads to a damping tensor ^emf=(nsDins) (nsDins) in the LLG equation with=~2=s[54]. A general form of the feedback damping should also include the contribution from the dissipative torque [40]. Here we introduce such -terms phenomenologically which leads to the LLG equation: s(1 +ns[^+^])_nsnsHs e=; (4) whereis the magnon torque term and we separated the dissipative ^ and reactive ^ contributions: ^=+^emf2Dins Dins; (5) ^ =[(nsDins) DinsDins (nsDins)]; where in general the form of chiral derivatives in the - terms can be dierent. Given that andare typically small for magnon systems, the term ^ emfwill dominate the feedback damping tensor. An unusual feature of the chiral part of the damping is that it will be present even for a uniform texture. While the DMI prefers twisted magnetic structures, this can be relevant in the presence of an external magnetic eld strong enough to drive the system into the ferromagnetic phase. In conducting ferromagnets, charge currents also lead to a damping tensor of the same form where the strength of the damping is characterized by e=~2e=4e2swith eas the electronic conductivity [11, 13, 15], which should be compared to in conducting ferromagnets where both eects are present. Since the magnon feed- back damping grows as/1=, the overall strength of magnon contribution can quickly become dominant con- tribution in ferromagnets with small Gilbert damping. Under the assumption that magnon scattering is domi- nated by the Gilbert damping such that the relaxation time is given by = 1=2!, magnon conductivity is given by3D1=62~in three-dimensions and 2D 1=4~in two-dimensions [40] where =p ~J=skBTis the wavelength of the thermal magnons. For Cu 2OSeO 3 in ferromagnetic phase, we nd 2nm2. Similarly, for a Pt/Co/AlO xthin lm of thickness t= 0:6nm yield 1nm2at room temperature. This shows that the magnon feedback damping can become signicant in fer- romagnets with sharp textures and strong DMI. Domain wall dynamics. We describe the domain wall prole in a ferromagnet with DMI by Walker ansatz3 tan((x;t)=2) = exp([xX(t)]=
) whereX(t) and (t) denote the center position and tilting angle of the domain wall [55],
=p J=K 0is the domain wall width, K0=Ku0M2 s=2 includes the contributions from uniaxial anisotropy as well as the demagnetizing eld and ^D=D(sin 11 + cos ez) contains DMI due to bulk and structure inversion asymmetries whose relative strength is determined by . After integrating the LLG equation, we obtain the equations of motion for a domain wall driven by external perpendicular eld [56]: XX_X=
+ _=FX;__X=
=F;(6) where XX=+(D=J)2sin2( +)=3 and = +[2=3
2+ (D=2J
) cos( +) + (D=J)2cos2( + )] are dimensionless angle-dependent drag coecients, FX=H=s andF= [Ksin 2+ sin( +)D=2
]=sare generalized \forces" associated with the collective coordi- natesXand,Kis the strength of an added anisotropy corresponding, e.g., to magnetostatic anisotropy K= Nx0M2 s=2 whereNxis the demagnetization coecient. In deriving these equations, we have neglected higher or- der terms in and[57]. Time-averaged domain wall velocity obtained from numerical integration of the equations of motion for a Co/Pt interface with Rashba-like DMI is shown in Fig. 2. Thermal magnon wavelength at room tempera- ture (0:3nm) is much shorter than the domain wall size
=p J=K 07nm, so the quasiparticle treatment of magnons is justied. We observe that damping reduces the speed at xed magnetic eld, and this eect is en- hanced with increasing DMI strength Dand diminishing the Gilbert damping (see Fig. 2). Another important observation is that in the presence of the feedback damping, the relation between applied eld and average domain wall velocity becomes nonlin- ear. This is readily seen from steady state solution of the equations of motion before the Walker breakdown with =0which solves sin( +0)D=2s
=[H=s][+ (D=J)2sin2( +0)=3]1(noting that D=
K, implying a N eel domain wall [56, 58]) and X=vt, leading to the cubic velocity-eld relation ( sv=
)(+ [2sv=J ]2=3) =Hfor eld-driven domain wall motion. The angle0also determines the tilting of Eas seen in Fig. 1. Skyrmion dynamics. Under the assumption that the skyrmion retains its internal structure as it moves, we treat it as a magnetic texture ns=ns(rq(t)) with q(t) being the time-dependent position (collective coor- dinate [61]) of the skyrmion. We consider the motion of a skyrmion under the temperature gradient r= rT=T, which exerts a magnon torque: = (1 +Tns)(Lr
D)ns; (7) whereLis the spin Seebeck coecient and Tis the \- type" correction. These are given by L3DkBT=62 andT3=2 in three-dimensions and L2DkBT=4 andTin two-dimensions within the relaxation time D=4mJ/m2 D=2mJ/m2 D=1mJ/m2 0 10 20 30 4002004006008001000 μ0Ha(mT)〈X〉(m/s)Co/Pt D=1.5mJ/m2D=1mJ/m2D=0.5mJ/m2 0 1 2 3 402004006008001000 μ0Ha(mT)〈X〉(m/s)Pt/CoFeB/MgOFIG. 2. (Color online) Domain wall velocity as a func- tion of the magnetic eld and varying strength of DMI for Co/Pt and Pt/CoFeB/MgO lms. Solid (dashed) lines cor- respond to dynamics at zero (room) temperature. We used material parameters Ms= 1:1MA/m,J= 16pJ/m, K0= 0:34MJ/m3,= 0:03 [45] for Co/Pt, and Ms= 0:43MA/m, J= 31pJ/m, K0= 0:38MJ/m3,= 4103[31, 59, 60] for Pt/CoFeB/MgO. approximation [41, 42]. Multiplying the LLG equation Eq. (4) withR d2r@qjns
nsand substituting _ns= _qi@qins, we obtain the equation of motion for v=_q: s(WQz)v+ (TDQz)Lr=F:(8) Above,W=0+0can be interpreted as the con- tribution of the renormalized Gilbert damping, Q=R d2rns
(@xns@yns)=4is the topological charge of the skyrmion,0is the dyadic tensor, Dis the chiral dyadic tensor which is0for isolated skyrmions and vanishes for skyrmions in SkX lattice [48] (detailed denitions of these coecients are given in the Supplemental Material [62]). The \force" term F=rU(q) due to the eec- tive skyrmion potential U(q) is relevant for systems with spatially-dependent anisotropies [63], DMI [64], or mag- netic elds. In deriving this equation, we only considered the dominant feedback damping contribution ^ emfwhich is justied for small and. For temperature gradients and forces along the x-axis we obtain velocities: vx=L@x(Q2+WTD) +FxW s(Q2+W2); vy=L@xQ(TDW) +FxQ s(Q2+W2): (9) The Hall angle dened as tan H=vy=vxis strongly af- fected by the renormalization of Wsince tanH=Q=W for a \force" driven skyrmion and tan H(T0 W)=Qfor a temperature gradient driven skyrmion. Sim- ilar to the domain wall velocity in Fig. 2, the Hall eect will depend on the overall temperature of the system. We nd that for a skyrmion driven by @x, the Hall an- gleHmay ip the sign in magnets with strong DMI as the temperature increases. We estimate this should happen in Cu 2OSeO 3atT50K using a typical radial prole for a rotationally symmetric skyrmion given by Usov ansatz cos( =2) = (R2r2)=(R2+r2) forrR andR2J=D52nm. Magnon pumping and accumulation. The motion of skyrmions induces a transverse magnon current across4 the sample. This eect can be quantied by the av- erage magnon current due to magnon motive force per skyrmion: j=Z d2rE=R2= (vez)4~2Q=R2: (10) The current can only propagate over the magnon diu- sion length; thus, it can be observed in materials with large magnon diusion length or small Gilbert damping. μ/μ0 -1.5-1.0-0.500.51.01.5 FIG. 3. (Color online) An array of moving skyrmions (only 3 shown in the gure) induces a transverse current and accumu- lation of magnons along the edges . is obtained by numer- ically solving the diusion equation using material parame- ters for Pt/CoFeB/MgO given in the caption of Fig. 2 with R= 35nm. System height and distance between skyrmion centers are taken to be 3 R. So far, we have assumed a highly compressible limit in which we disregard any build up of the magnon chem- ical potential . In a more realistic situation the build up of the chemical potential will lead to magnon diu- sion. To illustrate the essential physics, we consider a situation in which the temperature is uniform. For slow magnetization dynamics in which magnons quickly estab- lish a stationary state (i.e. R=vfor skyrmions and
=_Xfor domain walls, which is satised at high enough temperatures) we write a stationary magnon dif- fusion equation: r2= 2+r
E; (11) where==2is the magnon diusion length and we used the local Ohm's law r=j=E. Renor- malization of magnon current in Eq. (3) then follows from solution of the screened Poisson equation j=E+(=4)rR d3r0(r0
E)ejrr0j==jrr0jin three dimen- sions andj=E+ (=2)rR d2r0(r0
E)K0(jrr0j=) in two dimensions where K0is the modied Bessel func- tion for an innitely large system [65]. By analyzing the magnon current due to magnon accumulation ana- lytically and numerically, we nd that renormalization becomes important when the length associated with the magnetic texture is much smaller than the magnon dif- fusion length. Finally, we numerically solve Eq. (11) for isolated soli- tons (see Fig. 1) and for an array of moving skyrmions (see Fig. 3). Given that the width of the strip in Fig. 3 is comparable to the magnon diusion length one can have substantial accumulation of magnons close to the boundary. Spin currents comparable to the estimate in Eq. (10) can be generated in this setup and further de- tected by the inverse spin Hall eect [66]. From Eq. (10), for a skyrmion with R= 35nm moving at 10m/s in Pt/CoFeB/MgO with D= 1:5mJ/m2[31], we obtain an estimate for spin current js=j~107J/m2which roughly agrees with the numerical results. This spin cur- rent will also carry energy and as a result will lead to a temperature drop between the edges. Conclusion. We have developed a theory of magnon motive force in chiral conducting and insulating ferro- magnets. The magnon motive force leads to tempera- ture dependent, chiral feedback damping. The eect of this damping can be seen in the non-linear, temperature dependent behavior of the domain wall velocity. In ad- dition, observation of the temperature dependent Hall angle of skyrmion motion can also reveal this additional damping contribution. We have numerically conrmed the presence of the magnon feedback damping in nite- temperature micromagnetic simulations of Eq. (1) using MuMaX3 [67]. Magnon pumping and accumulation will also result from the magnon motive force. Substantial spin and en- ergy currents can be pumped by a moving chiral tex- ture in systems in which the size of magnetic textures is smaller or comparable to the magnon diusion length. 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B 92, 064405 (2015). [51] Note2, we disregard O(hn2 fi) corrections assuming tem- peratures well below the Curie temperature. These cor- rections can be readily reintroduced. [52] K.-W. Kim, H.-W. Lee, K.-J. Lee, and M. D. Stiles, Phys. Rev. Lett. 111, 216601 (2013). [53] Y. Tserkovnyak and S. A. Bender, Phys. Rev. B 90, 014428 (2014). [54] Note3, in a quasi two-dimensional or two-dimensional system, surface spin density should be used which can be obtained from the bulk spin density as s2D=s3Dt wheretis the layer thickness. [55] Note4, while a rigorous analysis of domain wall dynamics in the presence of a strong DMI should take domain wall tilting into account in general, in the particular case of a domain wall driven by a perpendicular eld, the ( X;) model remains moderately accurate for studying the ef- fects of feedback damping [16, 68]. [56] A. Thiaville, S. Rohart, E. Ju e, V. Cros, and A. Fert, Europhys. Lett. 100, 57002 (2012). [57] Note5, this equation is similar to the equation obtained for electronic feedback damping in [16] but for magnons XXand are determined by D=J rather than ~ R (a parameter which is taken to be independent from D), which leads to dierent conclusions. In [16], the chiral derivative associated with SMF is parameterized by ~ R asDi=@i+ (~Rez)ei. For magnons, D=J corre- sponds to ~R. [58] F. J. Buijnsters, Y. Ferreiros, A. Fasolino, and M. I. Kat- snelson, Phys. Rev. Lett. 116, 147204 (2016). [59] M. Yamanouchi, A. Jander, P. Dhagat, S. Ikeda, F. Mat- sukura, and H. Ohno, IEEE Magn. Lett. 2, 3000304 (2011). [60] X. Liu, W. Zhang, M. J. Carter, and G. Xiao, J. Appl. Phys. 110, 033910 (2011). [61] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). [62] Note6, see Supplemental Material at the end.[63] G. Yu, P. Upadhyaya, X. Li, W. Li, S. K. Kim, Y. Fan, K. L. Wong, Y. Tserkovnyak, P. K. Amiri, and K. L. Wang, Nano Lett. 16, 1981 (2016). [64] S. A. D az and R. E. Troncoso, arXiv:1511.04584 (2015). [65] Note7, as such, dynamics of magnetic solitons depend on temperature through and. [66] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao, et al., Phys. Rev. Lett. 111, 176601 (2013). [67] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv. 4, 107133 (2014). [68] O. Boulle, S. Rohart, L. D. Buda-Prejbeanu, E. Ju e, I. M. Miron, S. Pizzini, J. Vogel, G. Gaudin, and A. Thiaville, Phys. Rev. Lett. 111, 217203 (2013).Supplemental Material for Theory of Magnon Motive Force in Chiral Ferromagnets Utkan G¨ ung¨ ord¨ u and Alexey A. Kovalev Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA THIELE’S EQUATION OF MOTION FOR SKYRMION We consider the motion of a rotationally symmetric skyrmion under the influence of applied temperature gradient. Assuming that skyrmion drifts without any changes to its internal structure, ns(r,t) =ns(r−q(t)) where qis the position of the skyrmion, we multiply the LLG equation with the operator/integraltext d2r∂qjns·ns×and integrate over the region containing the skyrmion and obtain the following equation motion: s(W−Q/primez×)v+ (βTηD−Qz×)L∂χ=F. (1) where v=˙qis the skyrmion velocity, W=η0α+η(α0−β2α2),Lis the spin Seebeck coefficient, χ= 1/T,F=−∇U, Qis the topological charge defined in the main text and Q/prime=Q−ηβα 1. In terms of the polar coordinates ( θ,φ) of ns, the dyadic tensor η0and the chiral dyadic tensor ηDare given by η0=π/integraldisplayR 0dr/parenleftbigg(r∂rθ)2+ sin2θ r/parenrightbigg , ηD=η0+D Jπ/integraldisplayR 0dr(sinθcosθ+r∂rθ), (2) The damping terms αican be expanded in powers of D/J asαi=/summationtext jαβi,Dj(D/J)j, whereαβi,Djis given by αβ0,D2=π/integraldisplayR 0dr(r∂rθ)2cos2θ+ sin2θ r αβ0,D=π/integraldisplayR 0dr(r∂rθ)r∂rθtanθcos2θ+ sin2θ r2 αβ0,D0=π/integraldisplayR 0dr(r∂rθ)22 sin2θ r3(3) αβ,D2=2π/integraldisplayR 0dr(∂rθ) sinθ(cos2θ+ 1) αβ,D=4π/integraldisplayR 0dr(∂rθ) sinθcos2θtanθ+r∂rθ r αβ,D0=2π/integraldisplayR 0dr(∂rθ) sinθcos2θtan2θ+ (r∂rθ)2 r2(4) αβ2,D2=π/integraldisplayR 0dr/parenleftbiggcos2θsin2θ+ (r∂rθ)2 r/parenrightbigg αβ2,D=2π/integraldisplayR 0dr/parenleftbiggcos2θsin2θtanθ+ (r∂rθ)3 r2/parenrightbigg αβ2,D0=π/integraldisplayR 0dr/parenleftbiggcos2θsin2θtan2θ+ (r∂rθ)4 r3/parenrightbigg (5) The integrals can be evaluated by using an approximate radial profile for θ(r). Using Usov ansatz yields the values enumerated in Table I for αβi,Dj. Only the dominant terms are kept in the main text.2 1 D/J (D/J)2 1496π 15R252π 5R16π 5 βC R2472π 15R16π 3 β2 5056 105R22804π 105R464π 105 TABLE I. List of feedback damping coefficients αβi,Djfor a rotationally symmetric skyrmion using Usov ansatz cos( θ/2) = (R2−r2)/(R2+r2) forr≤Rand 0 forr > R . Rows correspond to α0,α1andα2, expanded in powers of D/J. Above, C≈449. Remaining parameters are given as η0= 16π/3,ηD=η0+ (4πR/3)(D/J). TRANSPORT COEFFICIENTS Texture-independent part of the transport coefficients can be obtained using the Boltzmann equation within the relaxation-time approximation in terms of the integral [1, 2] Jij n=1 (2π)3/planckover2pi1/integraldisplay d/epsilon1τ(/epsilon1)(/epsilon1−µ)n(−∂/epsilon1f0)/integraldisplay dS/epsilon1vivj |v|(6) asσ=J0and Π =−J1/J0. Above,τ(/epsilon1) is the relaxation time, /epsilon1(k) =/planckover2pi1ωk,vi=∂ωk/∂ki,dS/epsilon1is the area d2k corresponding to a constant energy surface with /epsilon1(k) =/epsilon1,f0is the Bose-Einstein equilibrium distribution. Under the assumption that the scattering processes are dominated by Gilbert damping, we set τ(/epsilon1)≈1/2αω. By evaluating the integral after these substitutions, we obtain σ2D≈F−1/6π2λ/planckover2pi1αin three dimensions ( d= 3), where λ=/radicalbig /planckover2pi1J/skBT is the wavelength of the thermal magnons, F−1=/integraltext∞ 0d/epsilon1/epsilon1d/2e/epsilon1+x/(/epsilon1+x)(e/epsilon1+x−1)2∼1 evaluated at the magnon gap x=/planckover2pi1ω0/kBT. Similarly for d= 2, we obtain σ2D≈F−1/4π/planckover2pi1α. The spin Seebeck coefficient Lis given by−/planckover2pi1σΠ = /planckover2pi1J1, for which we obtain L3D≈F0kBT/6π2λαin 3D and L2D≈F0kBT/4παin 2D, where F0=/integraltext∞ 0d/epsilon1/epsilon1d/2/(e/epsilon1+x−1)2∼1. Ford >2 and small x, the numerical factor F0 can be expressed in terms of Riemann zeta function and Euler gamma function as ζ(d/2)Γ(d/2 + 1) [3]. In the main text, the numerical factors F−1andF0are omitted. [1] N. Ashcroft and N. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976). [2] A. A. Kovalev and Y. Tserkovnyak, EPL (Europhysics Lett. 97, 67002 (2012). [3] R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, 1996), 2nd ed.

2016-05-05

We predict that magnon motive force can lead to temperature dependent, nonlinear chiral damping in both conducting and insulating ferromagnets. We estimate that this damping can significantly influence the motion of skyrmions and domain walls at finite temperatures. We also find that in systems with low Gilbert damping moving chiral magnetic textures and resulting magnon motive forces can induce large spin and energy currents in the transverse direction.

Theory of magnon motive force in chiral ferromagnets

1605.01694v2

arXiv:1111.1219v1 [cond-mat.mes-hall] 4 Nov 2011Tunable magnetization relaxation in spin valves Xuhui Wang∗and Aurelien Manchon Physical Science & Engineering Division, KAUST, Thuwal 239 55-6900, Kingdom of Saudi Arabia (Dated: June 24, 2018) In spin values the damping parameters of the free layer are de termined non-locally by the entire magnetic configuration. In a dual spin valve structure that c omprises a free layer embedded be- tween two pinned layers, the spin pumping mechanism, in comb ination with the angular momentum conservation, renders the tensor-like damping parameters tunable by varying the interfacial and dif- fusive properties. Simulations based on the Landau-Lifshi tz-Gilbert phenomenology for a macrospin model are performed with the tensor-like damping and the rel axation time of the free layer mag- netization is found to be largely dependent on while tunable through the magnetic configuration of the source-drain magnetization. PACS numbers: 75.70.Ak, 72.25.Ba, 75.60.Jk, 72.25.Rb A thorough knowledge of magnetization relaxation holds the key to understand magnetization dynamics in response to applied fields1and spin-transfer torques.2,3 In the framework of Landau-Lifshitz-Gilbert (LLG) phe- nomenology, relaxation is well captured by the Gilbert damping parameterthat is usuallycited asa scalarquan- tity. As pointed out by Brown half a century ago,4the Gilbert damping for a single domain magnetic particle is in general a tensor. When a ferromagnetic thin film is deposited on a nor- mal metal substrate, an enhanced damping has been ob- served ferromagnetic resonance experiments.5This ob- servation is successfully explained by spin pumping:6,7 The slow precession of the magnetization pumps spin current into the adjacent normal metal where the dis- sipation of spin current provides a non-local mechanism to the damping. The damping enhancement is found to be proportional to spin mixing conductance, a quan- tity playing key roles in the magneto-electronic circuit theory.7,8 Thepumped spincurrent Ip∝M×˙Misalwaysin the planeformedbythefreelayermagnetizationdirection M and the instantaneous axis about which the magnetiza- tion precesses. Therefore, in a single spin valve, when M is precessingaround the source(drain) magnetization m, the pumpingcurrentisalwaysinthe planeof mandM.9 Let us assume an azimuth angle θbetween mandM. In such anin-plane configuration, the pumping current Ip has a component Ipsinθthat is parallel to m. The spin transfer torque acting on the source (drain) ferromagnet mis the component of spin current that is in the plane and perpendicular to m. To simplify the discussion, we consider it to be completely absorbed by m. The lon- gitudinal (to m) component experiences multiple reflec- tion at the source (drain) contact, and cancels the damp- ing torque by an amount proportional to Ipsin2θbut is still aligned along the direction of M×˙M. Therefore the total damping parameter has an angle θdependence but still picks up a scalar (isotropic) form. This is the well-known dynamic stiffness explained by Tserkovnyak et al.9In the most general case, when the precessing axis of the free layer is mis-aligned with m, there is always anout-of-plane pumping torque perpendicular to the plane. In the paradigmof Slonczewski, this out-of-plane compo- nent is not absorbed at the interface of the source (drain) ferromagnetic nodes, while the conservation of angular momentum manifests it as a damping enhancement that shows the tensor form when installed in the LLG equa- tion. Studies in lateral spin-flip transistors have suggested a tensor form for the enhanced damping parameters.10 In spin valves, works based on general scattering the- ory have discussed the damping in the framework of fluctuation-dissipation theorem11and shown that the Gilbert dampingtensorcanbe expressedusingscattering matrices,12thus enabling first-principle investigation.13 But explicit analytical expressions of the damping ten- sor, its dependence on the magnetic configuration as well as the material properties and particularly its impact on the magnetization relaxation are largely missing. In this paper, we investigate the Gilbert damping pa- rameters of the free layer in the so-called dual spin valve (DSV).14–16We analyze the origin of the damping tensor and derive explicit analytical expressions of its non-local dependence on the magnetic configuration and materials properties. A generalization of our damping tensor to a continuous magnetic texture agrees well with the results in earlier works. Particularly, we show, in numeric sim- ulations, that by tuning the magnetic configurations of the entire DSV, the relaxation time of the free layer can be increased or decreased. mLM mR Reservoir Reservoir FIG. 1: A dual spin valve consists of a free layer (with magne- tization direction M) sandwiched by two fixed ferromagnetic layers (with magnetization directions mLandmR) through two normal metal spacers. The fixed layer are attached to reservoirs.2 To analyze the spin and charge currents in a DSV, we employ the magneto-electronic circuit theory and spin pumping,7,8in combination with diffusion equations.17 Pillar-shaped metallic spin valves usually consist of normal-metal ( N) spacers much shorter than its spin-flip relaxation length, see for example Ref.[3,15]. To a good approximation, in the Nnodes, a spatially homogeneous spin accumulation is justified and the spin current ( Ii) conservation dictates/summationtext iIi= 0 (where subscript iindi- cates the source of spin current). A charge chemical potential ( µ) and a spin accumula- tion (s) are assigned to every ForNnode. In a transi- tion metalferromagnet,astrongexchangefieldalignsthe spin accumulation to the magnetization direction. At ev- eryF|Ninterface, the charge and spin currents on the N side are determined by the contact conductance and the charge and spin distributions on both sides of the con- tact. For example, at the contact between the left lead ferromagnet to the left normal metal N1, called L|N1 thereafter, the currents are8 IL=e 2hGL[(µ1−µL)+PL(s1−sL)·mL], IL=−GL 8π[2PL(µ1−µL)mL+(s1−sL)·mLmL +ηL(s1−s1·mLmL)]. (1) We have used the notation G=g↑+g↓is the sum of the spin- σinterface conductance gσ. The contact polar- isationP= (g↑−g↓)/(g↑+g↓). The ratio η= 2g↑↓/G is between the real part of the spin-mixing conductance g↑↓and the total conductance G. The imaginary part ofg↑↓is usually much smaller than its real part, thus discarded.18The spin-coherence length in a transition metalferromagnetisusuallymuchshorterthanthethick- ness of the thin film,19which renders the mixing trans- mission negligible.7The precession of the free layer mag- netization Mpumps a spin current Ip= (/planckover2pi1/4π)g↑↓ FM× ˙Minto the adjacent normal nodes N1andN2, which is given by the mixing conductance g↑↓ Fat theF|N1(2) interface (normal metals spacers are considered identical on both sides of the free layer). A back flow spin current at the F|N1interface reads I1=−GF 8π[2PF(µ1−µF)M+(s1−sF)·MM +ηF(s1−s1·MM)] (2) on theN1side. Therefore, a weak spin-flip scattering inN1demands IL+I1+Ip= 0, which is dictated by angularmomentum conservation. The sameconservation law rules in N2, whereIR+I2+Ip= 0. For the ferromagnetic ( F) nodes made of transition metals, the spin diffusion is taken into account properly.9 In a strong ferromagnet, any transverse components de- cay quickly due to the large exchange field, thus the longitudinal spin accumulation sν=sνmν(withν= L,R,F) diffuses and decays exponentially at a length scale given by spin diffusion length ( λsd) as∇2 xsν=sν/λsd. The difference in spin-dependent conductivty of majority and minority carriers is taken into account by enforcing the continuity of longitudinal spin current mν·Iν=−(D↑ ν∇xs↑ ν−D↓ ν∇xs↓ ν) at the every F|Nin- terface. We assume vanishing spin currents at the outer interfaces to reservoirs. The diffusion equations and current conservation de- termine, self-consistently, the spin accumulations and spin currents in both NandFnodes . We are mainly concerned with the exchange torque9T=−M×(IL+ IR)×Macting on M. A general analytical formula is attainable but lengthy. In the following, we focus on two scenarios that are mostly relevant to the state-of-the-art experiments in spin valves and spin pumping: (1) The free layer has a strong spin flip (short λsd) and the thick- nessdF≥λsd, for which the permalloy (Py) film is an ideal candidate;15(2) The free layer is a half metal, such as Co2MnSi studied in a recent experiment.20 Strong spin flip in free layer. We assume a strong spin flip scattering in the free layer i.e., dF≥λsd. We leave the diffusivity properties in the lead Fnodes arbitrary. The total exchange torque is partitioned into two parts: Anisotropic part that is parallel to the direction of the Gilbert damping M×˙Mand ananisotropic part that is perpendicular to the plane spanned by mL(R)andM(or the projection of M×˙Mto the direction mL(R)×M), i.e., T=/planckover2pi1g↑↓ F 4π(DL is+DR is)/parenleftBig M×˙M/parenrightBig +/planckover2pi1g↑↓ F 4πM×/bracketleftBig (DL anˆAL,an+DR anˆAR,an)˙M/bracketrightBig ,(3) where the material-dependent parameters DL(R) isand DL(R) anare detailed in the Appendix A. Most interest is in the anisotropic damping described by a symmetric tensor with elements ˆAij an=−mimj (4) wherei,j=x,y,z(we have omitted the lead index Lor R). The elements of ˆAanare given in Cartesian coordi- nates of the source-drain magnetization direction. The anisotropic damping appears as M׈Aan˙Mthat is al- ways perpendicular to the free layer magnetization direc- tion, thus keeping the length of Mconstant.11It is not difficult to show that when Mis precessing around m, the anisotropic part vanishes due to ˆAan˙M= 0. We generalizeEq.(4) toa continuousmagnetictexture. Consider here only one-dimensional spatial dependence and the extension to higher dimensions is straightfor- ward. The Cartesian component of vector U≡M× ˆAan˙MisUi=−εijkMjmkml˙Ml(whereεijkis the Levi- Civita tensor and repeated indices are summed). We as- sume the fixed layer and the free layer differ in space by a lattice constant a0, which allows mk≈Mk(x+a0). A Taylor expansion in space leads to U=−a2 0M×(ˆD˙M), wherethematrixelements ˆDkl= (∂xM)k(∂xM)landwe3 have assumed that the magnetization direction is always perpendicular to ∂xM. In this case, three vectors ∂xM, M×∂xMandMare perpendicular to each other. A rotation around Mbyπ/2 leavesMand˙Munchanged while interchanging ∂xMwithM×∂xM, we have ˆDkl= (M×∂xM)k(M×∂xM)l, (5) which agrees with the so-called differential damping ten- sor Eq.(11) in Ref.[21]. Eq.(3) suggests that the total exchange torque on the free layer is a linear combination of two independent ex- change torques arsing from coupling to the left and the rightFnodes. This form arises due to a strong spin- flip scattering in the free layer that suppresses the ex- change between two spin accumulations s1ands2in the Nnodes. In the pursuit of a concise notation for the Gilbert form, the exchange torque can be expressed as T=M×← →α˙Mwith a total damping tensor given by ← →α=/planckover2pi1g↑↓ F 4π/parenleftBig DL is+DR is+DL anˆAL,an+DR anˆAR,an/parenrightBig .(6) The damping tensor← →αis determined by the entire mag- netic configuration of the DSV and particularly by the conductance of F|Ncontacts and the diffusive proper- ties theFnodes. Half metallic free layer . This special while experimen- tally relevant20case means PF= 1. Half-metallicity in combination with the charge conservation enforces a longitudinal back flow that is determined solely by the bias current: The spin accumulations in Nnodes do not contribute to the spin accumulation inside the free layer, thus an independent contribution due to left and right leads is foreseen. We summarize the material spe- cific parameters in the Appendix A. When spin flip is weak in the source-drain ferromagnets, ξL≈0 leads to Dis≈0. In this configuration, by taking a (parallel or anti-parallel)source-drainmagnetization direction as the precessingaxis,thetotaldampingenhancementvanishes, which reduces to the scenario of ν= 1 in Ref.[9]. Magnetization relaxation . To appreciate the impact of an anisotropic damping tensor on the magnetization re- laxation, we perform a simulation, for the free layermag- netization, using Landau-Lifshitz-Gilbert (LLG) equa- tion augmented by the tensor damping, i.e., dM dt=−γM×Heff+α0M×dM dt +γ µ0MsVM×← →αdM dt.(7) α0is the (dimensionless) intrinsic Gilbert damping pa- rameter. Symbol γis the gyromagnetic ratio, Msis the saturation magnetization, and Vis the volume of the free layer. µ0stands for the vacuum permeability. The dynamics under the bias-driven spin transfer torque is not the topic in this paper, but can be included in a straightforward way.22We give in the Appendix B the expressions of the bias-driven spin torques.We are mostly interested in the relaxation of the mag- netization, instead of particular magnetization trajecto- ries, in the presence of a tensor damping. The follow- ing simulation is performed for the scenario where the free layer has a strong spin flip, i.e., Case (1). We em- ploy the pillar structure from Ref.[15] while consider- ing the free layer (Py) to be 8nm thick (a thicker free layer favors a better thermal stability.15) The source- drain ferromagnets are cobalt (Co) and we expect the results are valid for a larger range of materials selec- tions. The Py film is elliptic with three axes given by 2a= 90 nm, 2 b= 35 nm,15andc= 8 nm. The de- magnetizing factors Dx,y,zin the shape anisotropy en- ergyEdem= (1/2)µ0M2 sV/summationtext i=x,y,zDiM2 iareDx= 0.50, Dy= 0.37 and Dz= 0.13. An external field Haleads to a Zeeman splitting EZee=−Vµ0MsHa·M. For Py films, we neglect the uniaxial anisotropy. The total free energyET=EZee+Edemgives rise to an effective field Heff=−(1/VMsµ0)∂ET/∂M. The spin-dependent conductivities in the bulk of Co andthe spin diffusion length λCo≈60nm aretaken from the experimental data.24For Py, we take λPy≈4 nm.25 To have direct connection with experiments, the above mentioned bare conductance has to be renormalized by the Sharvin conductance.26For Py/Cu the mixing con- ductance, we take the value g↑↓ FS−1≈15 nm−2,26which givesRL(R)F≈1.0. 56789100.950.960.970.980.991(a) Bz = 50 G; I.P. y−axis. (y,y)(y,x)(x,x)(x,z)(y,z)(z,z)369(b) Relaxation time (x,x) (y,y) (z,z) FIG. 2: (Color online) Mzas a function of time (in ns) in presence of differentsource-drain magnetic configurations and applied fields. (a) The external magnetic field Bz= 50 Gauss is applied along z-axis. The blue (dashed), red (solid) and black (dotted dash) curves correspond to source-drain magn e- tization in configurations ( y,y), (x,x), and (z,z) respectively. (b) Magnetization relaxation times (in the unit of ns)versu s source-drain magnetic configurations at different applied fi eld alongz-axis.:Bz= 10 G (red /square),Bz= 50 G (blue /circlecopyrt), Bz= 200 G (green ▽),Bz= 800 G (black ♦). Lines are a guide for the eyes. The initial position (I.P.) of the free la yer is taken along y-axis. The relaxation time τris extracted from the sim- ulations by demanding at a specific moment τrthe |Mz−1.0|<10−3, i.e., reaches the easy axis. In the absence of bias, panel (a) of Fig.2 shows the late stage of magnetization relaxation from an initial position ( y-4 axis) in the presence of an tensor damping, under various source-drain (SD) magnetic configurations. The results are striking: Under the same field, switching the SD con- figurations increases or decreases τr. In panel (b), the extracted relaxation times τrversus SD configurations under various fields are shown. At low field Bz= 10 G (red/square), when switching from ( z,z) to (y,y),τris im- proved from 8.0 ns to 6.3 ns, about 21%. At a higher fieldBz= 800 G (black ♦), the improvement is larger from 5.2 at ( z,z) to 3.6 at ( y,y), nearly 31%. To a large trend, the relaxation time improvement is more signifi- cant at higher applied fields. In conclusion, combining conservation laws and magneto-electronic circuit theory, we have analyzed the Gilbert damping tensor of the free layer in a dual spin valve. Analytical results of the damping tensor as func- tions of the entire magnetic configuration and material properties are obtained. Numerical simulations on LLG equation augmented by the tensor damping reveal a tun- able magnetization relaxation time by a strategic selec- tion of source-drain magnetization configurations. Re- sults presented in this paper open a new venue to the design and control of magnetization dynamics in spin- tronic applications. X.Wang is indebted to G. E. W. Bauer, who has brought the problem to his attention and offered invalu- able comments. Appendix A: Material dependent parameters In this paper, RL(R)F≡g↑↓ L(R)/g↑↓ Fis the mixing con- ductance ratio and χL(R)≡mL(R)·M. The diffusivity parameter ξL(R)=φL(R)(1−P2 L(R))/ηL(R), where for the leftFnode φL=1 1+(σ↑ L+σ↓ L)λLe2 4hSσ↑ Lσ↓ Ltanh(dL/λL)GL(1−P2 L)(A1) wherehthe Planck constant, Sthe area of the thin film, ethe elementary charge, λLthe spin diffusion length, dL the thickness of the film, and σ↑(↓)the spin-dependent conductivity. φRisobtainedbysubstituting all LbyRin Eq.(A1). Parameter ξFis given by ξF= (1−P2 F)φF/ηF with φF=1 1+(σ↑ F+σ↓ F)λFe2 4hSσ↑ Fσ↓ FGF(1−P2 F).(A2) The material dependent parameters as appearing in the damping tensor Eq.(6) are: (1) In the case of a strongspin flip in free layer, DL(R) is=RL(R)F LL(R)F/bracketleftBig ξL(R)RL(R)F+ξL(R)ξF(1−χ2 L(R)) +ξF(1−χ2 L(R))χ2 L(R)/bracketrightBig , DL(R) an=RL(R)F LL(R)F(ξL(R)−1)[ξF(1−χ2 L(R))+RL(R)F] 1+RL(R)F, LL(R)F=(1+RL(R)Fχ2 L)ξF(1−χ2 L(R)) +RL(R)F/bracketleftBig (1−χ2 L(R))(1+ξL(R)ξF) +ξL(R)RL(R)F+ξFχ2 L(R)/bracketrightBig ; (A3) (2)In the case of a half metallic free layer DL(R) is=RL(R)FξL(R) (1−χ2 L(R))+ξL(R)(χ2 L+RL(R)F), DL(R) an=RL(R)F 1+RL(R)F ×ξL(R)−1 (1−χ2 L(R))+ξL(R)(χ2 L+RL(R)F).(A4) Appendix B: Bias dependent spin torques The full analytical expression of bias dependent spin torques are rather lengthy. We give here the expres- sions, under a bias current I, for symmetric SD fer- romagnets (i.e., φL=φR=φthusξL=ξR=ξ) with parallelor anti-parallelmagnetization direction. (1) With a strong spin flip in the free layer, the parallel SD magnetization leads to vanishing bias-driven torque T(b) ⇑⇑= 0; WhentheSDmagnetizationsareanti-parallelly (i.e.,mL=−mR≡m), T(b) ⇑⇓=I/planckover2pi1Pφ e(1+R)L/bracketleftbig (ξF+RξFχ2+R)(1−χ2) +R(R+ξF(1−χ2)+χ2)/bracketrightbig mF×(m×mF). (B1) (2) When the free layer is half metallic, for symmetric SD ferromagnets , T(b) ⇑⇑= 0 and T(b) ⇑⇓=I/planckover2pi1 eφP (1−ξ)(1−χ2)+ξ(χ2+R)mF×(m×mF). (B2) ∗Electronic address: xuhui.wang@kaust.edu.sa 1L. D. Landau and E. M. Lifshitz, Statistical Physics ,Part 2(Pergamon, Oxford, 1980); T. L. Gilbert, IEEE. Trans. Mag.40, 2443 (2004).2J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 54, 9353 (1996). 3E. B. Myers, et al., Science 285, 867 (1999); J. A. Katine, et al., Phys. Rev. Lett. 84, 3149 (2000); S. I. Kiselev, et5 al., Nature (London) 425, 380 (2003). 4W. F. Brown, Phys. Rev. 130, 1677 (1963). 5Mizukami et al., Jpn. J. Appl. Phys. 40, 580 (2001); J. Magn. Mater. Magn. 226, 1640 (2001). 6Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002); Phys. Rev. B 66, 224403 (2002). 7Y. Tserkovnyak, et al., Rev. Mod. Phys. 77, 1375 (2005). 8A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep. 427, 157 (2005). 9Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B67, 140404(R) (2003). 10X. Wang, G. E. W. Bauer, and A. Hoffmann, Phys. Rev. B73, 054436 (2006). 11J. Foros, et al., Phys. Rev. B 78, 140402(R) (2008); Phys. Rev. B79, 214407 (2009). 12A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett. 101, 037207 (2008); arXiv:1104.1625. 13A.A.Starikov, et al., Phys.Rev.Lett. 105, 236601 (2010). 14L. Berger, J. Appl. Phys. 93, 7693 (2003). 15G. D. Fuchs, et al., Appl. Phys. Lett. 86, 152509 (2005).16P. Bal´ aˇ z, M. Gmitra, and J. Barna´ s, Phys. Rev. B 80, 174404 (2009); P. Yan, Z. Z. Sun, and X. R. Wang, Phys. Rev. B83, 174430 (2011). 17T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993); A. A. Kovalev, A. Brataas, and G. E. W. Bauer Phys. Rev. B 66, 224424 (2002). 18K. Xia,et al., Phys. Rev. B 65, 220401(R) (2002). 19M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 (2002). 20H. Chudo, et al., J. Appl. Phys. 109, 073915 (2011). 21S. Zhang and S. -L. Zhang, Phys. Rev. Lett. 102, 086601 (2010). 22J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 72, 014446 (2005). 23J. Osborn, Phys. Rev. 67, 351 (1945). 24J. Bass and W. P. Pratt, J. Magn. Magn. Mater. 200, 274 (1999). 25A. Fert and L. Piraux, J. Magn. Magn. Mater. 200, 338 (1999). 26G. E. W. Bauer, et al., Phys. Rev. B 67, 094421 (2003).

2011-11-04

In spin values the damping parameters of the free layer are determined non-locally by the entire magnetic configuration. In a dual spin valve structure that comprises a free layer embedded between two pinned layers, the spin pumping mechanism, in combination with the angular momentum conservation, renders the tensor-like damping parameters tunable by varying the interfacial and diffusive properties. Simulations based on the Landau-Lifshitz-Gilbert phenomenology for a macrospin model are performed with the tensor-like damping and the relaxation time of the free layer magnetization is found to be largely dependent on while tunable through the magnetic configuration of the source-drain magnetization.

Tunable magnetization relaxation in spin valves

1111.1219v1

Unidirectional magnetic coupling H. Y. Yuan,1R. Lavrijsen,2and R. A. Duine1, 2 1Institute for Theoretical Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands 2Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands (Dated: September 29, 2022) We show that interlayer Dzyaloshinskii-Moriya interaction in combination with non-local Gilbert damping gives rise to unidirectional magnetic coupling. That is, the coupling between two magnetic layers | say the left and right layer | is such that dynamics of the left layer leads to dynamics of the right layer, but not vice versa. We discuss the implications of this result for the magnetic sus- ceptibility of a magnetic bilayer, electrically-actuated spin-current transmission, and unidirectional spin-wave packet generation and propagation. Our results may enable a route towards spin-current and spin-wave diodes and further pave the way to design spintronic devices via reservoir engineering. Introduction. | Non-reciprocal transmission of elec- trical signals lies at the heart of modern communication technologies. While semi-conductor diodes, as an exam- ple of an electronic component that underpins such non- reciprocity, have been a mature technology for several decades, new solutions are being actively pursued [1, 2]. Such research is spurred on by the emergence of quan- tum technologies that need to be read out electrically but should not receive unwanted back-action from their electronic environment. Complementary to these developments, spintronics has sought to control electronic spin currents and, more recently, spin currents carried by spin waves | i.e., magnons | in magnetic insulators [3]. Devices that im- plement non-reciprocal spin-wave spin currents have been proposed [4{7]. Most of these proposals rely on dipolar interactions [8{11] or Dzyaloshinskii-Moriya interactions (DMI) [12{16]. Other proposals involve the coupling of the spin waves to additional excitations such that the spin waves are endowed with non-reciprocity. Examples are the coupling of the spin waves to magnetoelastic, optical, and microwave excitations [17{22]. Most of these proposals have in common that they con- sider spin-wave dispersions that are asymmetric in wave vector. For example, due to the DMI spin waves at one particular frequency have dierent wave numbers and ve- locities for the two dierent directions. There are there- fore spin waves travelling in both directions. This may be detrimental for some applications. For example, one would like to shield quantum-magnonic technologies from spin-current noise [23], and completely quench the spin- current transmission in one of the two directions along a wire. Here, we propose a set-up that realizes unidirectional magnetic coupling between two magnetic layers or be- tween two magnetic moments. The ingredients are DMI and dissipative coupling between the two layers or mo- ments. The dissipative coupling takes the form of a non- local Gilbert damping and may arise, for example, from the combined action of spin pumping and spin transfer.Then, one magnet emits spin current when it precesses, which is absorbed by the other. The resulting dissipa- tive coupling turns out to, for certain parameters, pre- cisely cancel the DMI in one direction. As a result, an excitation of one of the magnets leads to magnetization dynamics of the other, but not vice versa. This yields spin-wave propagation that is truly uni-directional: for specic direction and magnitude of the external eld, all spin waves travel in one direction only. Minimal model. | Let us start with the minimal set- up that demonstrates the unidirectional coupling. We rst consider two identical homogeneous magnetic lay- ers that are coupled only by an interlayer DMI with Dzyaloshinskii vector Dand by interlayer spin pumping (see Fig. 1). The magnetization direction in the layers is denoted by mi, wherei2f1;2glabels the two lay- ers. We also include an external eld H. The magnetic energy is given by E[m1;m2] =D
(m1m2)0MsH
(m1+m2);(1) whereMsis the saturation magnetization of both layers and0is the vacuum susceptibility. The magnetization dynamics of layer 1 is determined by the Landau-Lifshitz- Gilbert (LLG) equation @m1 @t= Msm1E m1+nlm1@m2 @t; (2) where is the gyromagnetic ratio and nlcharacterizes the strength of the non-local damping that in this set-up results from the combination of spin pumping and spin transfer torques, as described in the introduction. The equation of motion for the magnetization dynamics of the second layer is found by interchanging the labels 1 and 2 in the above equation. Working out the eective eldsarXiv:2209.14179v1 [cond-mat.mes-hall] 28 Sep 20222 FM FM m2 m1xy z, H, D FIG. 1. Schematic of two magnetic moments coupled by an interlayer DMI and by interlayer spin pumping. The dynam- ics of m1induces the motion of m2, but not vice versa for appropriate parameters. E=miyields @m1 @t= Msm1(m2D0MsH) +nlm1@m2 @t; (3a) @m2 @t= Msm2(Dm10MsH) +nlm2@m1 @t; (3b) where the sign dierence in eective-eld contribution from the DMI stems from the asymmetric nature of the DMI. We show now that depending on the magnitude and direction of the eective eld, this sign dierence leads for one of the layers to cancellation of the torques due to interlayer DMI and non-local damping. As the can- cellation does not occur for the other layer, and because the DMI and non-local damping are the mechanisms that couple the layers in the model under consideration, this leads to uni-directional magnetic coupling. Taking the external eld to be much larger than the interlayer DMI, i.e., 0jHjjDj=Ms, and taking nl 1, we may replace @mi=@tby 0miHon the right- hand side of Eqs. (3) because the external eld then is the dominant contribution to the precession frequency. For the eld H=D=nl0Ms, one then nds that @m1 @t= nlMsm1D; (4a) @m2 @t=2 Msm2(Dm1) nlMsm2D:(4b) Hence, the coupling between the two magnetic layers is unidirectional at the eld H=D=nl0Ms: the magne- tization dynamics of layer 1 leads to dynamics of layer 2 as evidenced by Eq. (4b), but not vice versa as im- plied by Eq. (4a). This one-way coupling is reversed by changing the direction of the eld to Hor the sign of the non-local coupling nl. Magnetic susceptibility. | Let us now take into ac- count the Gilbert damping within the layers, exchange, and anisotropies and discuss the in uence of the unidi- rectional coupling on the magnetic susceptibility. Theenergy now reads E[m1;m2] =Jm1
m2+D
(m1m2) 0MsH
(m1+m2)K 2 m2 1;z+m2 2;z
;(5) with the constant Kcharacterizing the strength of the anisotropy and Jthe exchange. We shall focus on the ferromagnetic coupling ( J >0) without loss of generality. The LLG equation now becomes @m1 @t= Msm1@E @m1+m1@m1 @t +nlm1@m2 @t; (6) withthe Gilbert damping constant of each layer, and where the equation for the second layer is obtained from the above by interchanging the labels 1 and 2. We take the external eld in the same direction as the Dzyaloshinskii vector and D=D^z,H=H^z, while 0MsH;KD, so that the magnetic layers are aligned in the ^z-direction. Linearizing the LLG equation around this direction we write mi= (mi;x;mi;y;1)Tand keep terms linear in mi;xandmi;y. Writingi=mi;ximi;y, we nd, after Fourier transforming to frequency space, that 1(!)1(!) 2(!) = 0: (7) To avoid lengthy formulas, we give explicit results below for the case that J= 0, while plotting the results for J6= 0 in Fig. 2. The susceptibility tensor ij, or magnon Green's function, is given by (!) =1 ((1 +i)!!0)2( D=Ms)22 nl!2) (1 +i)!!0i( D=Msnl!) i( D=Ms+nl!) (1 +i)!!0 ;(8) with!0= (0H+K=Ms) the ferromagnetic-resonance (FMR) frequency of an individual layer. The poles of the susceptibility determine the FMR frequencies of the coupled layers and are, for the typical case that ; nl 1, given by !=!r;i!r;; (9) with resonance frequency !r;= (0H+K=MsD=Ms): (10) When 0H= (1nl)D=(nlMs)K=Ms D=(nlMs)K=Mswe have for J= 0 that12(!r;) = 0 while21(!r;)6= 0, signalling the non-reciprocal cou- pling. That is, the excitation of layer 1 by FMR leads to response of magnetic layer 2, while layer 1 does not respond to the excitation of layer 2. For opposite direc- tion of eld the coupling reverses: the excitation of layer3 |χ21(J=0)| |χ21(J=0.5D)| |χ21(J=15D)| |χ12(J=0)| |χ12(J=0.5D)| |χ12(J=15D)| 0.96 0.98 1.00 1.02 1.040100200300400 ω/ωH FIG. 2. Magnetic susceptibilities of two magnetic layers as a function of frequency at dierent exchange couplings. !H (0H+K=M s). The resonance frequencies are located at the peak positions. The parameters are D=! H= 0:001;nl= 0:001;= 0:002. 2 by FMR leads in that case to response of magnetic layer 1, while layer 2 does not respond to the excitation of layer 1. As is observed from Fig. 2, for nite but smallJD, the coupling is not purely unidirectional anymore but there is still a large non-reciprocity. For JD, this non-reciprocity is washed out. Electrically-actuated spin-current transmission. | In practice, it may be challenging to excite the individual layers independently with magnetic elds, which would be required to probe the susceptibility that is determined above. The two layers may be more easily probed inde- pendently by spin-current injection/extraction from ad- jacent contacts. Therefore, we consider the situation that the two coupled magnetic layers are sandwiched between heavy-metal contacts (see Fig. 3(a)). In this set-up, spin current may be transmitted between the two contacts through the magnetic layers. Following the Green's function formalism developed by Zheng et al. [24], the spin-current from the left (right) lead to its adjacent magnetic layer is determined by the transmission function of the hybrid system T12(T21) given by Tij(!) = Trh i(!)G(+)(!)j(!)G()(!)i : (11) Here,G(+)(!) is the retarded Green's function for magnons in contact with the metallic leads that is de- termined by Dyson's equation G(+)1(!) =1(!) (+) 1(!)(+) 2(!), where the retarded self energy ~(+) i(!) accounts for the contact with the metallic lead i. These self energies are given by ~(+) 1(!) =i~0 1 !0 0 0 ; (12)and ~(+) 2(!) =i~0 20 0 0! : (13) The rates for spin-current transmission from the heavy metal adjacent to the magnet iinto it, are given by i(!) =2Imh (+) i(!)i =~. The couplings 0 i= Re[g"# i]=4Msdiare proportional to the real part of the spin-mixing conductance per area g"# ibetween the heavy metal and the magnetic layer i, and further depend on the thickness diof the magnetic layers. Finally, the ad- vanced Green's function is G()(!) = G(+)y. In the analytical results below, we again restrict our- selves to the case that J= 0 for brevity, leaving the caseJ6= 0 to plots. Using the above ingredients, Eq. (11) is evaluated. Taking identical contacts so that 0 1=0 20, we nd that T12=4(0)2!2( D=Ms+nl!)2 jC(!)j2; (14) while T21=4(0)2!2( D=Msnl!)2 jC(!)j2; (15) with C(!) = [!H(1 +i(nl+0))!]
[!H(1 +i(+nl+0))!]( D=Ms)2:(16) From the expression for C(!) it is clear that, since ; nl;01, the transmission predominantly occurs for frequencies equal to the resonance frequencies !r; from Eq. (9). Similar to the discussion of the suscepti- bilities, we have for elds 0H=D= nlK=Msthat the transmission T12(!=D= nl)6= 0, while T21(!= D= nl) = 0. As a result, the spin-current transmis- sion is unidirectional at these elds. For the linear spin- conductances Gij, given byGij=R ~!(N0(~!))Tij(!), we also have that G126= 0, while G21= 0. Here, N(~!) = [e~!=kBT1]1is the Bose-Einstein distri- bution function at thermal energy kBT. For the oppo- site direction of external eld we have G12= 0, while G216= 0. Like in the case of the susceptibility discussed in the previous section, a nite but small exchange cou- pling makes the spin current transport no longer purely unidirectional, while maintaining a large non-reciprocity (see Fig. 3(b)). Spin-wave propagation. | Besides the unidirectional coupling of two magnetic layers, the above results may be generalized to a magnetic multilayer, or, equivalently, an array of coupled magnetic moments that are labeled by the index isuch that the magnetization direction of thei-th layer is mi. This extension allows us to engi- neer unidirectional spin-wave propagation as we shall see4 m1 m2 Lead Lead FM FM(a) (b) T21(J=0) T21(J=0.5D) T21(J=15D) T12(J=0) T12(J=0.5D) T12(J=15D) 0.96 0.98 1.00 1.02 1.040.0000.0020.0040.0060.008 ω/ωH FIG. 3. (a) Schematic of the system that the two coupled magnetic layers are sandwiched between two heavy-metal con- tacts. (b) Transmission of the hybrid system as a function of frequency. below. We consider the magnetic energy E[m] =X k[D
(mkmk+1)0MsH
mk];(17) and nd | within the same approximations as for our toy model above | for the magnetization dynamics that @mk @t=2 Msmk(Dmk1) nlMsmkD;(18) for the eld H=D=nl0Ms. This shows that for these elds the magnetic excitations travel to the right | cor- responding to increasing index k| only. The direction of this one-way propagation is reversed by changing the magnetic eld to Hor by changing the sign of the non- local damping. To study how spin waves propagate in an array of cou- pled magnetic moments described by the Hamiltonian in Eq. (17). We start from the ground state mk= (0;0;1)T and perturb the left-most spin ( k= 0) to excite spin waves. Since the dynamics of this spin is not in uenced by the other spins for the eld H=D= nl0Ms, its small-amplitude oscillation can be immediately solved as0(t) =0(t= 0) exp(i!0t!0t) withk= mk;ximk;yas used previously. The dynamics of the spins to the right of this left-most spin is derived by solv- ing the LLG equation (18) iteratively, which yields k(t) =0(t= 0)ei!0te!0t k!(2nl!0t)k;(19)wherek= 0;1;2;:::N1. To guarantee the stability of the magnetization dynam- ics, the dissipation matrix of the N-spin system should be negative-denite, which imposes a constraint on the relative strength of Gilbert damping and non-local damp- ing, i.e., > 2nlcos N+1. For an innitely-long chain N!1 , we have>2nl. Physically, this means that the local dissipation of a spin has to be strong enough to dissipate the spin current pumped by its two neighbors. For a spin chain with nite number of spins, = 2jnljis always sucient to guarantee the stability of the system. Taking this strength of dissipation simplies Eq. (19) to k(t) =0(t= 0)eit=()et= k!(t=)k; (20) where1=!0is the inverse lifetime of the FMR mode. This spatial-temporal prole of spins is the same as a Poisson distribution with both mean and variance equal to=t=except for a phase modulation, and it can be further approximated as a Gaussian wavepacket on the time scale t, i.e. (x) =0(t= 0)eit=() p 2e(x)2 2: (21) Such similarity suggests that any local excitation of the left-most spin will generate a Gaussian wavepacket prop- agating along the spin chain. The group velocity of the moving wavepacket is v=a=, whereais the distance between the two neighboring magnetic moments. The width of the wavepacket spreads with time as ap t=, which resembles the behavior of a diusive particle. Af- ter suciently long time, the wavepacket will collapse. On the other hand, the excitation is localized and can- not propagate when the right-most spin ( k=N1) is excited, because its left neighbor, being in the ground state, has zero in uence on its evolution. These results demonstrate the unidirectional properties of spin-wave transport in our magnetic array. Discussion, conclusion, and outlook. | We have shown that the ingredients for unidirectional coupling be- tween magnetic layers or moments are that they are cou- pled only by DMI and non-local Gilbert damping. While in practice it may be hard to eliminate other couplings, the DMI and non-local coupling need to be suciently larger than the other couplings to observe unidirectional coupling. There are several systems that may realize the unidi- rectional coupling we propose. A rst example is that of two magnetic layers that are coupled by a metallic spacer. Such a spacer would accommodate non-local coupling via spin pumping and spin transfer. For a spacer that is much thinner than the spin relaxation length, we nd, following Refs. [25{27], that nl= ~Re[~g"#]=4dMs, with ~g"#the spin-mixing conductance of the interface between the magnetic layers and the spacer, dthe thick- ness of the magnetic layers. For simplicity, we took the5 magnetic layers to have equal properties. The two mag- netic layers may be coupled by the recently-discovered interlayer DMI [28, 29], tuning to a point (as a function of thickness of the spacer) where the ordinary RKKY ex- change coupling is small. We estimate nl= 4:5103 ford= 20 nm, Re[~g"#] = 4:561014 1m2and Ms= 1:92105A=m (YIGjPt). The required mag- netic eld for unidirectional magnetic coupling is then around 4.5 T for D= 1 mT. Another possible platform for realizing the unidirectional coupling is the system of Fe atoms on top of a Pt substrate that was demonstrated recently [30]. Here, the relative strength of the DMI and exchange is tuned by the interatomic distance between the Fe atoms. Though not demonstrated in this experi- ment, the Pt will mediate non-local coupling between the atoms as well. Hence, this system may demonstrate the unidirectional coupling that we proposed. The non-local damping is expected to be generically present in any magnetic material and does not require special tuning, though it may be hard to determine its strength experimentally. Hence, an attractive implemen- tation of the unidirectional coupling would be a magnetic material with spins that are coupled only via DMI, with- out exchange interactions. While such a material has to the best of our knowledge not been discovered yet, it is realized transiently in experiments with ultrafast laser pulses [31]. Moreover, it has been predicted that high-frequency laser elds may be used to manipulate DMI and exchange, even to the point that the former is nonzero while the latter is zero [32, 33]. Possible applications of our results are spin-wave and spin-current diodes and magnetic sensors, where a weak eld signal can be amplied and transported through the unidirectional coupling to the remote site to be read out without unwanted back-action. Finally, we remark that the unidirectional magnetic coupling that we pro- pose here may be thought of as reservoir engineering, cf. Ref. [34]. In our proposal, the reservoir is made up by the degrees of freedom that give rise to the non-local damp- ing, usually the electrons. We hope that this perspective may pave the way for further reservoir-engineered mag- netic systems Acknowledgements. | It is a great pleasure to thank Mathias Kl aui and Thomas Kools for discus- sions. H.Y.Y acknowledges the European Union's Hori- zon 2020 research and innovation programme under Marie Sk lodowska-Curie Grant Agreement SPINCAT No. 101018193. R.A.D. is member of the D-ITP consor- tium that is funded by the Dutch Ministry of Education, Culture and Science (OCW). R.A.D. has received fund- ing from the European Research Council (ERC) under the European Union's Horizon 2020 research and inno- vation programme (Grant No. 725509). This work is in part funded by the project \Black holes on a chip" with project number OCENW.KLEIN.502 which is nanced by the Dutch Research Council (NWO).[1] Directions for non-reciprocal electronics, Nat. Electron. 3, 233 (2020). [2] S.-W. Cheong, D. Talbayev, V. Kiryukhin, and A. Sax- ena, Broken symmetries, non-receprocity, and multifer- roicity, npj Quant. Mater. 3, 19 (2018). [3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Magnon spintronics, Nat. Phys. 11, 453 (2015). [4] M. Jamali, J. H. Kwon, S.-M. Seo, K.-J. Lee and H. Yang, Spin wave nonreciprocity for logic device applications, Sci. Rep. 3, 3160 (2013). [5] J. Lan, W. Yu, R. Wu, and J. Xiao, Spin-Wave Diode, Phys. Rev. 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Yu, Chi- ral spin-wave velocities induced by all-garnet interfacial Dzyaloshinskii-Moriya interaction in ultrathin yttrium iron garnet lms, Phys. Rev. Lett. 124, 027203 (2020). [17] Y.-P. Wang, J. W. Rao, Y. Yang, P.-C. Xu, Y. S. Gui, B. M. Yao, J. Q. You, and C.-M. Hu, Nonreciprocity and Unidirectional Invisibility in Cavity Magnonics, Phys. Rev. Lett. 123, 127202 (2019). [18] T. Yu, Y. M. Blanter, and G. E. W. Bauer, Chiral Pump- ing of Spin Waves, Phys. Rev. Lett. 123, 247202 (2019).6 [19] S. Tateno and Y. Nozaki, Highly Nonreciprocal Spin Waves Excited by Magnetoelastic Coupling in a Ni/Si Bilayer, Phys. Rev. Applied 13, 034074 (2020). [20] P. J. Shah, D. A. Bas, I. Lisenkov, A. Matyushov, N. X. Sun, and M. R. Page, Giant nonreciprocity of surface acoustic waves enabled by the magnetoelastic interaction, Sci. Adv. 6, eabc5648 (2020). [21] X. Zhang, A. Galda, X. Han, D. Jin, and V. M. 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Bauer, Dynamic exchange cou- pling in magnetic bilayers, Phys. Rev. Lett. 90, 187601 (2003).[27] G. Woltersdorf, O. Mosendz, B. Heinrich, and C. H. Back, Magnetization dynamics due to pure spin currents in magnetic double layers, Phys. Rev. Lett. 99, 246603 (2015). [28] A. Fernandez-Pacheco, E. Vedmedenko, F. Ummelen, R. Mansell, D. Petit, and R. P. Cowburn, Symmetry- breaking interlayer Dzyaloshinskii-Moriya interactions in synthetic antiferromagnets, Nat. Mater. 18, 679 (2019). [29] D. S. Han, K. Lee, J.-P. Hanke, Y. Mokrousov, K.-W. Kim, W. Yoo, Y. L. W. van Hees, T.-W. Kim, R. Lavri- jsen, C.-Y. You, H. J. M. Swagten, M.-H. Jung, and M. Kl aui, Long-range chiral exchange interaction in syn- thetic antiferromagnets, Nat. Mater. 18, 703 (2019). [30] M. Steinbrecher, R. Rausch, K. T. That, J. Hermenau, A. A. Khajetoorians, M. Pottho, R. Wiesendanger, and J. Wiebe, Non-collinear spin states in bottom-up fabricated atomic chains, Nat. Commun. 9, 2853 (2018). [31] N. Kerber, D. 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2022-09-28

We show that interlayer Dzyaloshinskii-Moriya interaction in combination with non-local Gilbert damping gives rise to unidirectional magnetic coupling. That is, the coupling between two magnetic layers -- say the left and right layer -- is such that dynamics of the left layer leads to dynamics of the right layer, but not vice versa. We discuss the implications of this result for the magnetic susceptibility of a magnetic bilayer, electrically-actuated spin-current transmission, and unidirectional spin-wave packet generation and propagation. Our results may enable a route towards spin-current and spin-wave diodes and further pave the way to design spintronic devices via reservoir engineering.

Unidirectional magnetic coupling

2209.14179v1

arXiv:1709.04401v1 [math.AP] 13 Sep 2017Life-span of blowup solutions to semilinear wave equation with space-dependent critical damping Masahiro Ikeda∗and Motohiro Sobajima† Abstract. This paper is concerned with the blowup phenomena for initia l value problem of semilinear wave equation with critical space-dependent damping term ∂2 tu(x,t)−∆u(x,t)+V0|x|−1∂tu(x,t)=|u(x,t)|p,(x,t)∈RN×(0,T), u(x,0)=εf(x), x∈RN, ∂tu(x,0)=εg(x), x∈RN,(DW: V0) where N≥3,V0∈[0,(N−1)2 N+1),fandgare compactly supported smooth functions and ε > 0 is a small parameter. The main result of the present paper is to give a so lution of (DW: V0) and to provide a sharp estimate for lifespan for such a solution whenN N−1<p≤pS(N+V0), where pS(N) is the Strauss exponent for (DW:0). The main idea of the proof is due to the technique o f test functions for (DW:0) originated by Zhou–Han (2014, MR3169791). Moreover, we find a new threshol d value V0=(N−1)2 N+1for the coefficient of critical and singular damping |x|−1. Mathematics Subject Classification (2010): Primary: 35L70. Key words and phrases : wave equation with singular damping, Small data blow up, St rauss exponent, Critical and subcrit- ical case, the Gauss hypergeometric functions . 1 Introduction In this paper we consider the blowup phenomena for initial va lue problem of semilinear wave equation with scale-invariant damping term of space-dependent type as follows: ∂2 tu(x,t)−∆u(x,t)+a(x)∂tu(x,t)=|u(x,t)|p,(x,t)∈RN×(0,T), u(x,0)=εf(x), x∈RN, ∂tu(x,0)=εg(x), x∈RN,(1.1) where N≥3,a(x)=V0|x|−1(V0≥0),ε > 0 is a small parameter and f,gare smooth nonnegative functions satisfying g/nequivalence0 with supp( f,g)⊂B(0,R0)={x∈RN;|x|≤R0} for some R0>0. Note that by taking uλ(x,t)=λ2 p−1u(λx,λt) withλ=R0, we can always assume R0=1 without loss of generality. The study of blowup phenomena for (1.1) with N=3 and V0=0 was initially started by F. John in [5] for 1<p<1+√ 2. Strauss conjectured in [ 9] that the number p0(N) given by the positive root of the quadratic equation (N−1)p2−(N+1)p−2=0 ∗Department of Mathematics, Faculty of Science and Technolo gy, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan /Center for Advanced Intelligence Project, RIKEN, Japan, E- mail: masahiro.ikeda@keio.jp/masahiro.ikeda@riken.jp †Department of Mathematics, Faculty of Science and Technolo gy, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan, E-mail: msobajima1984@gmail.com 1is the threshold for dividing the following two situations: blowup phenomena at a finite time for arbitrary small initial data and global existence of small solutions. The conjecture of Strauss was completely solved until Yordanov–Zhang [ 11] and Zhou [ 12]. After that the lifespan of solutions to nonlinear wave equat ions ((1.1) with V0=0) with small initial data has been considered by many authors. If 1 <p<p0(N), then by Sideris [ 8] and Di Pomponio– Georgiev [ 2] we have the two-sided estimates for lifespan of solution wi th small initial data as cε2p(p−1) 2+(N+1)p−(N−1)p2+δ≤LifeSpan( u)≤Cε2p(p−1) 2+(N+1)p−(N−1)p2 with arbitrary small δ>0. For the critical case p=p0(N), Takamura–Wakasa [ 10] succeeded in proving sharp upper bound of lifespan exp[cε−p(p−1)]≤LifeSpan( u)≤exp[Cε−p(p−1)] for remaining case N=4, and by [ 10] the study of the lifespan for blowup solutions to nonlinear wave equations with small data has been completed (for the other c ontributions see e.g, [ 10] and its references therein). In the connection to the previous paper, we have to remark that Zhou–Han [ 13] gave a short proof for verifying the sharp upper bound of lifespan by usin g an estimate established in [ 11] and a kind of test functions including the Gauss hypergeometric funct ions. In this paper, we mainly deal with the problem (1.1) with N≥3 and V0>0. Because of the strong singularity of damping term at the origin, the study of (1.1) has not been considered so far. Since the problem has a scaling-invariant structure, one can expect t hat some threshold for V0appears. The first purpose of this paper is to clarify the local wellpos edness of (1.1) for 1 <p<N−2 N−4in solutions in H2(RN). The second is to show an upper bound of the lifespan of solut ions to (1.1) with respect to small parameter ε > 0 and to pose a threshold number for V0dividing completely di fferent situations. The first assertion of this paper is for local wellposedness o f (1.1). Proposition 1.1. Let N≥3, V0≥0and 1<p<∞ if N=3,4 1<p<N−2 N−4if N≥5. For 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 strong solution of (1.1) in the following class: u∈ST=C2([0,T];L2(RN))∩C1([0,T];H1(RN))∩C([0,T];H2(RN)) Moreover, one has for every t ≥0, supp u(t)⊂B(0,R0+t). Definition 1.1. We denote LifeSpan( u) as the maximal existence time for solution of (1.1), that is , LifeSpan( u)=sup{T>0 ;u∈ST&uis a solution of (1.1) in (0 ,T)} Definition 1.2. We introduce the following quadratic polynomial γ(n;p)=2+(n+1)p−(n−1)p2 2and denote p0(n) as the positive root of the quadratic equation γ(n;p)=0 as in Introduction. We also put V∗=(N−1)2 N+1 and for areas for ( p,V0) as follows: Ω0={(p,V0) ;p=p0(N+V0),0≤V0<V∗} (1.2) Ω1=/braceleftigg (p,V0) ; max/braceleftigg p0(N+2+V0),2 N−1−V0/bracerightigg ≤p<p0(N+V0),0≤V0<V∗/bracerightigg (1.3) Ω2=/braceleftigg (p,V0) ;2(N+1) N+1+V0<p<2 N−1−V0,(N+1)(N−2) N+2<V0<V∗/bracerightigg (1.4) Ω3=/braceleftigg (p,V0) ; max/braceleftiggN N−1,N+3+V0 N+1+V0/bracerightigg <p<max/braceleftigg p0(N+2+V0),2(N+1) N+1+V0/bracerightigg/bracerightigg (1.5) V0 pV∗ p0(N) p0(N+2)N+1 N−1N N−1Ω1Ω2 Ω3Ω0 Figure 1: the regions Ω0,Ω1,Ω2andΩ3 Now we are in a position to state our main result in this paper a bout upper bound of lifespan of solutions to ( 1.1). Theorem 1.2. LetN N−1<p<∞if N=3,4andN N−1<p<N−2 N−4if N≥5. Fix (f,g)satisfying f≥0, g≥0, g/nequivalence0andsupp( f,g)⊂B(0,1). Let uεbe the solution of (1.1) in Proposition 1.1 with the parameterε>0. If(p,V0)∈Ω0∪Ω1∪Ω2∪Ω3, then LifeSpan( uε)<∞. Moreover, one has LifeSpan( uε)≤exp[Cε−p(p−1)] if(p,V0)∈Ω0, C′ δε−2p(p−1)/γ(N+V0;p)−δif(p,V0)∈Ω1, C′′ δε−2(p−1) 2N−(N−1+V)p−δif(p,V0)∈Ω2, C′′′ δε−1−δif(p,V0)∈Ω3,(1.6) 3whereδcan be chosen arbitrary small and C δ, C′ δ, C′′ δand C′′′ δare positive constants which depend on all parameters without ε. Remark 1.1.We emphasize the following two facts. The proof of [ 13] depends on an estimate established by [11] (for detail, see [ 11, (2,5’)]), however, our proof does not depend on that. The pr oof of Theorem 1.2 can be applicable to weaker solutions of (1.1) belonging toC([0,T));H1(RN)∩Lp((0,T)×RN). Remark 1.2.Taking the threshold value V0=V∗formally, we have γ(N+V∗;p)=2(1+N p)/parenleftigg 1−N−1 N+1p/parenrightigg and therefore p0(N+V∗)=N+1 N−1=1+2 N−1. On the one hand, critical exponent for the blowup phenomena for the problem ∂2 tu(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), u(x,0)=εf(x), x∈Ω, ∂tu(x,0)=εg(x), x∈Ω,(1.7) is given by pF(α)=1+2 N−α,α∈[0,1) which is so-called Fujita exponent (see e.g., Ikehata–To dorova– Yordanov [ 4] and also Ikeda–Ogawa [ 3]). We formally put again a threshold value α=1. Then one can find p0(N+V0)=pF(1). The left-hand side comes from the blowup phenomena for nonli near wave equation and the right-hand side comes from the one for nonlinear heat equation. In this c onnection, we would conjecture that if V0>V∗, then the threshold of blowup phenomena is given by the Fujit a exponent pF(1). Remark 1.3.If (p,V0)∈Ω0∪Ω1, then Theorem 1.2 seems to give a sharp lifespan of solutions to (1.1) with small initial data. In the case ( p,V0)∈Ω3, we cannot derive the estimates for lifespan with ε−τ withτless than one. So the estimate in Ω3seems not to be sharp. For the case ( p,V0)∈Ω2the effect of diffusion structure seems to appear in the estimate. The present paper is organized as follows. In Section 2, we fir st give existence and uniqueness of local-in-time solutions to (1.1) if p≤N−2 N−4by using the standard semigroup properties. In Section 3, we construct special solutions of linear wave equation with an ti-damping term−V0 |x|∂tu. In this point we use the idea due to [ 13] (they only considered the case V0=0), which will be a test function for proving blowup phenomena. In Section 4, we prove blowup phenomena by dividing two cases p<p0(N+V0) andp=p0(N+V0). 2 Local solvability of nonlinear wave equation with singula r damping In this section we construct a solution of (1.1) with initial data belonging to H2(RN)×H1(RN). To do so, we first treat the linear problem ∂2 tu(x,t)−∆u(x,t)+a(x)∂tu(x,t)=0,(x,t)∈RN×(0,∞), u(x,0)=u0(x), ∂ tu(x,0)=u1(x), x∈RN.(2.1) 42.1 C0-Semigroup for linear wave equation with singular damping Now we start with the usual N-dimensional Laplacian Au:=−∆u,D(A)=H2(RN). We note that Aism-accretive in L2(RN), that is, ( I+A)D(A)=L2(RN) and (−∆u,u)≥0 for every u∈H2(RN). SetH=H1(RN)×L2(RN) and A=/parenleftigg0−1 A0/parenrightigg ,D(A)=H2(RN)×H1(RN) and B=/parenleftigg0 0 0|x|−1/parenrightigg ,D(B)=H1(RN)×{v∈L2(RN) ;|x|−1v∈L2(RN)} and put Aκ=A+κB,D(Aκ)=D(A)∩D(B) Then in view of Hille–Yosida theorem, we have the following C0-semigroup onH(see e.g., Pazy [ 7]). Lemma 2.1. Let N≥3. For every V≥0,1 2I+Aκis m-accretive inH. Therefore−Aκgenerates a C 0- semigroup{Tκ(t)}t≥0onH. Moreover, if supp( u0,u1)⊂B(0,R), then supp[ Tκ(t)(u0,u1)]⊂B(0,R+t). Proof. By Hardy’s inequality we have D(A)⊂D(B). This means that D(Aκ)=D(A)∩D(B)=D(A). (Accretivity) By interation by parts, we have /parenleftigg A/parenleftiggu v/parenrightigg ,/parenleftiggu v/parenrightigg/parenrightigg H=/parenleftigg/parenleftigg−v −∆u/parenrightigg ,/parenleftiggu v/parenrightigg/parenrightigg H=/integraldisplay RN/parenleftig −vu−∇v·∇u−(∆u)v/parenrightig dx=−/integraldisplay RNuv dx. SinceκBis clearly accretive, we have the accretivity of Aκ. (Maximality) LetF=(f,g)∈H. ThenλU+AU+κBU=Fis equivalent to the system λu−v=f, λ v−∆u+κ|x|−1v=g. Substituting v=λu−f, we see that λ2u−∆u+λκ|x|−1u=g+λf+k|x|−1f=fλ,κ. Taking /tildewideu(y)=u(λ−1y) and/tildewidestfλ,κ(y)=fλ,κ(λ−1y) yields that /tildewideu−∆/tildewideu+k|y|−1/tildewideu=λ−2/tildewidestfλ,κ. This is nothing but the resolvent problem of the Schr¨ odinge r operator with positive Coulomb potentials. Therefore there exists /tildewideuλ,k∈H2(RN) such that /tildewideuλ,k−∆/tildewideuλ,k+k|y|−1/tildewideuλ,k=λ−2/tildewidestfλ,k. Putting uλ,k(x)=/tildewideuλ,k(λ2x)∈H2(RN), we obtain λ2uλ,k−∆uλ,k+λk|x|−1uλ,k=fλ,k=g+λf+k|x|−1f. Finally setting vλ,k=λuλ,k−f∈H1(RN), we obtain ( λI+A+kB)(uλ,k,vλ,k)=(f,g). The finite propagation property follows from the standard ar gument for wave equation with regular damping term. The proof is complete. /square 52.2 Local solvability of nonlinear problem We consider (1.1) with initial data ( u(0),∂tu(0))=U0=(u0,u1)∈H2(RN)×H1(RN) which is equivalent to the following problem ∂t/parenleftiggu(t) v(t)/parenrightigg +Aκ/parenleftiggu(t) v(t)/parenrightigg =/parenleftigg0 N(u(t),v(t))/parenrightigg withN(u,v)=(0,|u|p). Here we construct the corresponding mild solution in H2(RN)×H1(RN) given by/parenleftiggu(t) v(t)/parenrightigg =Tκ(t)/parenleftiggu0 u1/parenrightigg +/integraldisplayt 0Tκ(t−s)/parenleftigg0 N(u(s),v(s))/parenrightigg ds, where{Tκ(t)}t≥0is determined in Lemma 2.1. Lemma 2.2. (i) The following metric space XT=/braceleftig U∈C([0,T];H)∩L∞([0,T];D(Aκ)) ; sup 0<t<T/ba∇dblU(t)/ba∇dblD(Aκ)≤M/bracerightig ,M:=2(/ba∇dblU0/ba∇dblD(Aκ)+1) with the distance d(U1,U2) :=max 0≤t≤T/vextenddouble/vextenddouble/vextenddoubleU(t)−U2(t)/vextenddouble/vextenddouble/vextenddoubleH,U1,U2∈XT. is complete. (ii)Ifsupp( u0,u1)⊂B(0,R), then the following metric space YT=/braceleftig U∈C([0,T];H)∩L∞([0,T];D(Aκ)) ; sup 0<t<T/ba∇dblU(t)/ba∇dblD(Aκ)≤M,supp( u(t),v(t))⊂B(0,R+t)/bracerightig with the same distance d is also complete. Proof. Take a Cauchy sequence {Un}n∈NinXT. The completeness of C([0,T];H) yields that there exists U∞∈C([0,T];H) such that Un→U∞strongly in C([0,T];H) as n→∞. Moreover, we can subtract a subsequence Unjand/tildewideU∈L∞([0,T];D(Aκ)) such that sup 0<t<T/ba∇dbl/tildewideU(t)/ba∇dblD(Aκ)≤M and Unj→/tildewideU∗-weakly in L∞([0,T];D(Aκ)) as j→∞. Therefore we have U∞=/tildewideU. Therefore XTis a complete metric space. If supp Un(t)⊂{x;|x|≤R+t}, then by strong convergence we have supp U∞(t)⊂{x;|x|≤R+t}. This means that YTis also complete. /square Lemma 2.3. There exists T 0such thatΨ:XT0→XT0andΨ:YT0→YT0are both well-defined, and Ψ is contractive in X T0and in Y T0. Proof. First observe that by finite propagation property in Lemma 2. 1, we can deduce supp ΨU(t)⊂ B(0,R+t) when supp U0⊂B(0,R). Since XTandYTare endowed with the same distance, It su ffices to prove the assertion for XT. We recall that the norms in D(Aκ) and in H2(RN)∩H1(RN) are equivalent. 6(well-defined) IfU=(u,v)∈XT, then /ba∇dblN(U(t))/ba∇dbl2 D(Aκ)≤C2 κ/ba∇dblN(U(t))/ba∇dbl2 H2×H1 =C2 κ/vextenddouble/vextenddouble/vextenddouble|u(t)|p/vextenddouble/vextenddouble/vextenddouble2 H1 =C2 κ/integraldisplay RN/parenleftig |u(t)|p+|∇(|u(t)|p)|2/parenrightig dx =C2 κ/parenleftigg/integraldisplay RN|u(t)|2pdx+p2/integraldisplay RN|u(t)|2(p−1)|∇u(t)|2dx/parenrightigg ≤C2 κ/parenleftigg/integraldisplay RN|u(t)|N(p−1)dx/parenrightigg2 N/parenleftigg/integraldisplay RN|u(t)|2N N−2dx/parenrightigg1−2 N +p2C2 κ/parenleftigg/integraldisplay RN|u(t)|N(p−1)dx/parenrightigg2 N/parenleftigg/integraldisplay RN|∇u(t)|2N N−2dx/parenrightigg1−2 N . Since p≤N−2 N−4, we have N(p−1)≤(1 2−2 N) and therefore /ba∇dblN(U(t))/ba∇dblD(Aκ)≤C′ κ/ba∇dblu(t)/ba∇dblH2 ≤C′′ κ/ba∇dblU(t)/ba∇dblH2×H1 ≤C′′ κ/ba∇dblU(t)/ba∇dblD(Aκ) ≤C′′ κM. Therefore we have for 0 ≤t≤T≤log 2, /ba∇dblΨU(t)/ba∇dblD(Aκ)≤/ba∇dblTκ(t)U0/ba∇dblD(Aκ)+/integraldisplayt 0/ba∇dblTκ(t−s)[N(U(s))]/ba∇dblD(Aκ)ds ≤et/2/ba∇dblU0/ba∇dblD(Aκ)+/integraldisplayt 0e(t−s)/2/ba∇dblN(U(s))/ba∇dblD(Aκ)ds ≤√ 2/ba∇dblU0/ba∇dblD(Aκ)+√ 2C′′ κMt. Therefore there exists T1∈(0,log 2] such that sup0<t<T1/ba∇dblΨU(t)/ba∇dblD(Aκ)≤M. Next we prove continuity of ΨUon [0,T]. For 0≤t1<t2≤T, /ba∇dblΨU(t1)−ΨU(t2)/ba∇dblH≤/ba∇dbl[Tκ(t1)−Tκ(t2)]U0/ba∇dblH +/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt1 0T(t1−s)N(U(s))ds−/integraldisplayt2 0T(t2−s)N(U(s′))ds′/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH ≤|t1−t2|/ba∇dblA kU0/ba∇dblH +/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt1 0[I−T(t2−t1)]T(t1−s)N(U(s))ds/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH+/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/integraldisplayt2 t1T(t2−s)N(U(s′))ds′/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH ≤|t1−t2|/ba∇dblA kU0/ba∇dblH +eT/2|t2−t1|/integraldisplayt1 0/ba∇dblAκN(U(s))/ba∇dblHds+eT/2/integraldisplayt2 t1/ba∇dblN(U(s′))/ba∇dblHds′ ≤M′|t1−t2|. 7(contractivity) IfU1=(u1,v1),U2=(u2,v2)∈XT, then /ba∇dblN(U1(t))−N(U2(t))/ba∇dbl2 H=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/parenleftigg0 |u1(t)|p−|u2(t)|p/parenrightigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 H ≤p2/integraldisplay RN(|u1(t)|+|u2(t)|)2(p−1)|u1(t)−u2(t)|2dx ≤p2/vextenddouble/vextenddouble/vextenddouble|u1(t)|+|u2(t)|/vextenddouble/vextenddouble/vextenddouble2(p−1) LN(p−1)/ba∇dblu1(t)−u2(t)/ba∇dbl2 L2N N−2 ≤C(/ba∇dblu1(t)/ba∇dblH2+/ba∇dblu2(t)/ba∇dblH2)2(p−1)/ba∇dblu1(t)−u2(t)/ba∇dbl2 H1 ≤(C′)2M2(p−1)/ba∇dblU1(t)−U2(t)/ba∇dbl2 H. This implies that /ba∇dblΨU1(t)−ΨU2(t)/ba∇dblH≤/integraldisplayt 0/ba∇dblTκ(t−s)[N(U1(s))−N(U2(s))]/ba∇dblHds ≤C′Mp−1eT/2/integraldisplayt 0/ba∇dblU1(s)−U2(s)/ba∇dblHds. Consequently, taking T0∈(0,T1] satisfying 2C′Mp−1TeT/2≤1, we obtain d(ΨU1,ΨU2)≤1 2d(U1,U2), that is,Ψis contractive in XT0and also in YT0./square Proof of Proposition 1.1. By Lemma 2.3, we can find a unique fixed point U∞ofΨinYT0. Moreover, combining the previous arguments implies /ba∇dblN(U∞(t1))−N(U∞(t2))/ba∇dblH≤C′Mp−1/ba∇dblU∞(t1)−U∞(t2)/ba∇dblH ≤C′′Mp|t1−t2|. ThusN(U∞(·)) is Lipschitz continuous on [0 ,T]. By [ 7, Corollary 4.2.11(p.109)], we verify that Ut+ AκU=F(U∞) has a unique strong solution U∗ ∞given by U∗ ∞(t)=Tκ(t)U0+/integraldisplayt 0Tκ(t−s)N(U∞(s))ds,t∈[0,T0]. Since U∞is a fixed point ofΨ, we obtain U∗ ∞(t)=U∞(t) for t∈[0,T0]. Since U∞is a strong solution, we have∂tU∞=−AκU∞+N(U∞)∈L∞(0,T;H) a.e. on [0,T]. This gives us that ∂tu=v∈L∞(0,T;H2(RN))∩W1,∞(0,T;H1(RN))∩C([0,T];H1(RN)), and ∂tv=∆u−κ |x|v+|u|p∈L∞(0,T;H1(RN))∩C([0,T];L2). Hence u∈C2([0,T];L2(RN)) is nothing but a strong solution of ∂2 tu−∆u+κ |x|∂tu=|u|p on [0,T]. Uniqueness of local solutions is due to a proof similar to t he contractivity of Ψand the finite propagation property follows from the use of YT0. /square 83 Special solutions of linear damped wave equation In this section we construct special solutions of linear dam ped wave equation which will be test functions for proving blowup properties. The following function plays an essential role in the proof o f upper bound of lifespan of solutions to (1.1). Similar test functions appear in Zhou–Han [ 13]. Definition 3.1. Forβ>0, set Φβ(x,t)=(|x|+t)−βF/parenleftigg β,N−1+V0 2,N−1;2|x| 2+t+|x|/parenrightigg , where F(a,b,c;z) is the Gauss hypergeometric function given by F(a,b,c;z)=∞/summationdisplay n=0(a)n(b)n (c)nzn n! with ( d)0=1 and ( d)n=/producttextn k=1(d+k−1) for n∈N. (For further properties of F(·,·,·;z), see e.g., Chaper 8 in Beals–Wong [ 1]). For the reader’s convenience we would give a derivation the G auss hypergeometric function from the wave equation. Lemma 3.1. Forβ>0,Φβsatisfies the wave equation with the anti-damping term ∂2 tΦβ−∆Φβ−V0 |x|∂tΦβ=0,inQ={(x,t)∈RN×(0,∞) ;|x|<2+t}. Proof. We can putΦ(x,t)=Φβ(x,t−2) for t>0. We start with the desired equation ∂2 tΦ(x,t)−∆Φ(x,t)−V0 |x|∂tΦ(x,t)=0,in{(x,t)∈RN×(0,∞) ;|x|<t}. (3.1) Put u(x,t)=(|x|+t)−βϕ/parenleftigg2|x| |x|+t/parenrightigg , Then setting z=2|x| |x|+t=2−2t |x|+t, we have|x|+t=2t 2−zand therefore u(x,t)=(2t)−β(2−z)βϕ(z). Observing that ∂z ∂t=−2|x| (|x|+t)2=−z(2−z) 2t, we have ∂tu=−2β(2t)−β−1(2−z)βϕ(z)+(2t)−β[−β(2−z)β−1ϕ(z)+(2−z)βϕ′(z)]∂z ∂t =−2β(2t)−β−1(2−z)βϕ(z)−(2t)−β−1[−β(2−z)β−1ϕ(z)+(2−z)βϕ′(z)]z(2−z). =−(2t)−β−1(2−z)β+1/bracketleftig βϕ(z)+zϕ′(z)/bracketrightig 9and also ∂2 tu=(2t)−β−2(2−z)β+2/bracketleftig (β+1)(βϕ(z)+zϕ′(z))+z(βϕ(z)+zϕ′(z))′/bracketrightig =(2t)−β−2(2−z)β+2/bracketleftig β(β+1)ϕ(z)+2(β+1)zϕ′(z)+z2ϕ′′(z)/bracketrightig . On the other hand, for radial derivative, we see from∂z ∂r=2t (|x|+t)2=(2−z)2 2tthat ∂ru=(2t)−β/bracketleftig −β(2−z)β−1ϕ(z)+(2−z)βϕ′(z)/bracketrightig∂z ∂r =(2t)−β−1(2−z)β+1/bracketleftig −βϕ(z)+(2−z)ϕ′(z)/bracketrightig and ∂2 ru=(2t)−β−2(2−z)β+2/bracketleftig (β+1)βϕ(z)−2(β+1)(2−z)ϕ′(z)+(2−z)2ϕ′′(z)/bracketrightig . Combining these equalities andt r(2−z)−1=1 z, we obtain 0=/parenleftigg ∂2 tu−∂2 ru−N−1 r∂ru+V0 r∂tu/parenrightigg (2t)β+2(2−z)−β−2 =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−1 r(2t)(2−z)−1/bracketleftig −βϕ(z)+(2−z)ϕ′(z)/bracketrightig −V0 r(2t)(2−z)−1/bracketleftig −βϕ(z)−zϕ′(z)/bracketrightig =−4(1−z)ϕ′′(z)+4(β+1)ϕ′(z)+2β(N−1+V0) zϕ(z)+2(N−1) z/bracketleftig −(2−z)ϕ′(z)/bracketrightig −2V0ϕ′(z) =−4 z/bracketleftigg (1−z)zϕ′′(z)+/bracketleftigg N−1−/parenleftigg 1+β+N−1+V0 2/parenrightigg z/bracketrightigg ϕ′(z)−β(N−1+V0) 2ϕ(z)/bracketrightigg . This is nothing but the Gauss hypergeometric di fferential equation z(1−z)ϕ′′(z)+(c−(1+a+b)z)ϕ′(z)−abϕ(z)=0 with (a,b,c)=/parenleftigg β,N−1+V0 2,N−1/parenrightigg . This implies that ϕ(z)=F(β,N−1+V0 2,N−1;z). /square Lemma 3.2. (i) For everyβ>0and(x,t)∈Q, ∂tΦβ(x,t)=−βΦβ+1(x,t). (ii)If0<β<N−1−V0 2, then there exists a constant c β>0such that for every (x,t)∈Q, cβ(2+t)−β≤Φβ(x,t)≤c−1 β(2+t)−β. (iii)Ifβ>N−1−V0 2, then there exists a constant c′ β>0such that for every (x,t)∈Q, cβ(2+t)−β/parenleftigg 1−|x| t+2/parenrightiggN−1−V0 2−β ≤Φβ(x,t)≤c−1 β(2+t)−β/parenleftigg 1−|x| t+2/parenrightiggN−1−V0 2−β . 10Proof. (i)In view of the proof of Lemma 3.1, we have ∂tΦβ(x,t)=−(2t)−β−1(2−z)β+1[βϕ(z)+zϕ′(z)] with s=2|x| 2+t+|x|. It suffices to show that βϕ(z)+zϕ′(z)=βF/parenleftigg β+1,N−1+V0 2,N−1;z/parenrightigg ,z∈(0,1). (3.2) Putψ(z)=βϕ(z)+zϕ′(z) for z∈[0,1). Then by the definition of F(·,·,·;z), we haveψ(0)=β. On the other hand, we see from the gauss hypergeometric equation wi tha=β,b=N−1+V0 2andc=N−1 that (1−z)ψ′(z)=(1−z)/parenleftig (β+1)ϕ′(z)+zϕ′′(z)/parenrightig =(β+1)(1−z)ϕ′(z)+z(1−z)ϕ′′(z) =(a+1)(1−z)ϕ′(z)−(c−(1+a+b)z)ϕ′(z)+abϕ(z) =(a+1−c)ϕ′(z)+bzϕ′(z)+abϕ(z) =(a+1−c)ϕ′(z)+bψ(z) and therefore (1−z)ψ′(z)−bψ(z)=(a+1−c)ϕ′(z). The definition of ψyields z(1−z)ψ′(z)−bzψ(z)=(a+1−c)zϕ′(z) =(a+1−c)ψ(z)−(a+1−c)aϕ(z) Differentiating the above equality, we have z(1−z)ψ′′(z)+(1−(2+b)z)ψ′(z)−bψ(z)=(a+1−c)ψ′(z)−(a+1−c)aϕ′(z) =(a+1−c)ψ′(z)−a/parenleftig (1−z)ψ′(z)−bψ(z)/parenrightig . Hence 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 of this equation near 0 can be written by ψ(z)=hF(a+1,b,c;z) with h∈R. Combining the initial value ψ(0)=β, we obtain (3.2). The remaining assertions (ii)and(iii)are a direct consequence of the integral representation for mula F(a,b,c,z)=1 B(c,c−a)/integraldisplay1 0sa−1(1−s)c−a−1(1−zs)−bds,0≤z<1 when c>0 and c−a>0. The proof is complete. /square 4 Proof of blowup phenomena In this section we prove upper bound of the lifespan of soluti ons to (1.1) and its dependence of εunder the condition 0≤V0<V∗=(N−1)2 N+1. 114.1 Preliminaries for showing blowup phenomena We first state a criterion for derivation of upper bound for li fespan. Lemma 4.1. Let H∈C2([σ0,∞)be nonnegative function. (i)Assume that there exists positive constants c ,C,C′such that C[H(σ)]p≤H′′(σ)+cH′(σ) σ with H (σ)≥εpCσ2and H′(σ)≥εpCσ. Then H blows up before σ=C′′ε−p−1 2for some C′′>0. (ii)Assume that there exists positive constants c ,C,C′such that Cσ1−p[H(σ)]p≤H′′(σ)+2H′(σ) with H (σ)≥εpCσand H′(σ)≥εpC. Then H blows up before σ=C′′ε−p(p−1)for some C′′>0. Proof. The assertion follows from [ 13, Lemma 2.1] with the argument in [ 13, Section 3]. /square We focus our eyes to the following functionals. Definition 4.1. Forβ∈(0,N−1−V0 2), define the following three functions Gβ(t) :=/integraldisplay RN|u(x,t)|pΦβ(x,t)dx,t≥0, Hβ(t) :=/integraldisplayt 0(t−s)(2+s)Gβ(t)ds,t≥0, Jβ(t) :=/integraldisplayt 0(2+s)−3Hβ(t)ds,t≥0. Note that we can see from Lemma 3.2 (ii)thatGβ(t)≈(2+t)−β/ba∇dblu(t)/ba∇dblp Lp(RN). Lemma 4.2. If uεis a solution of (1.1) in Proposition 1.1 with parameter ε >0, then Jβdoes not blow up until LifeSpan( uε). Proof. It follows from the embedding H2(RN)→Lp(RN) (given by Gagliardo-Nirenberg-Sobolev in- equalities) that/ba∇dbluε(t)/ba∇dblLpis continuous on [0 ,LifeSpan( uε)) and also Gβ(t). This means that Jβ(t) is finite for all t∈[0,LifeSpan( uε)). /square Lemma 4.3. For everyβ>0and t≥0, (2+t)2Jβ(t)=1 2/integraldisplayt 0(t−s)2Gβ(s)ds. Proof. This can be verified by integration by parts twice, by noting t hat d ds/parenleftig (t−s)2(1+s)−1/parenrightig =2(1+t)2 (1+s)3. /square 12Lemma 4.4. Let u be a solution of (1.1) . Then for every β>0and t≥0, εEβ,0+εEβ,1t+/integraldisplayt 0(t−s)Gβ(s)ds=/integraldisplay RNu(x,t)Φβ(x,t)dx+2β/integraldisplayt 0/integraldisplay RNu(x,s)Φβ+1(x,s)dx ds +V0/integraldisplayt 0/integraldisplay RN1 |x|u(x,t)Φβ(x,t)dx ds, (4.1) where Eβ,0=/integraldisplay RNf(x)Φβ(x,0)dx>0. Eβ,1=/integraldisplay RNg(x)Φβ(x,0)dx+β/integraldisplay RNf(x)Φβ+1(x,0)dx+V0/integraldisplay RN1 |x|f(x)Φβ(x)dx>0. Proof. By the equation in (1.1) we see from integration by parts that Gβ(t)=/integraldisplay RN/parenleftigg ∂2 tu(t)−∆u(t)+V0 |x|∂tu(t)/parenrightigg Φβ(t)dx =/integraldisplay RN/parenleftigg ∂2 tu(t)+V0 |x|∂tu(t)/parenrightigg Φβ(t)dx−/integraldisplay RNu(t)(∆Φβ(t))dx. Using Lemma 3.1, we have Gβ(t)=/integraldisplay RN/parenleftigg ∂2 tu(t)+V0 |x|∂tu(t)/parenrightigg Φβ(t)dx−/integraldisplay RNu(t)/parenleftigg ∂2 tΦβ(t)−V0 |x|∂tΦβ(t)/parenrightigg dx =d dt/bracketleftigg/integraldisplay RN/parenleftig ∂tu(t)Φβ(t)−u(t)∂tΦβ(t)/parenrightig dx+V0/integraldisplay RN1 |x|u(t)Φβ(t)dx/bracketrightigg . Noting that Lemma 3.2 (i)(the formula ∂tΦβ=−βΦβ+1), we have εEβ,1+/integraldisplayt 0Gβ(s)ds=/integraldisplay RN/parenleftig ∂tu(t)Φβ(t)−u(t)∂tΦβ(t)/parenrightig dx+V0/integraldisplay RN1 |x|u(t)Φβ(t)dx =d dt/bracketleftigg/integraldisplay RNu(t)Φβ(t)dx/bracketrightigg +2β/integraldisplay RNu(t)Φβ+1(t)dx+V0/integraldisplay RN1 |x|u(t)Φβ(t)dx. Integrating it again, we obtain (4.1). /square The following lemma makes sense when2 N−1−V0<p0(N+V0) which is equivalent to 0 ≤V0<(N−1)2 N+1. Lemma 4.5. AssumeN N−1<p<∞and0≤V0<(N−1)2 N+1.(i)Let q>1satisfy max{p,2 N−1−V0}<q<∞ and put β=N−1−V0 2−1 q∈/parenleftigg 0,N−1−V0 2/parenrightigg . Then there exists a positive constant C 1>0such that εEβ,0+εEβ,1t+/integraldisplayt 0(t−s)Gβ(s)ds≤C1/bracketleftigg /ba∇dblu(t)/ba∇dblLp(2+t)N p′−β+/integraldisplayt 0/ba∇dblu(s)/ba∇dblLp(2+s)N p′−N−1−V0 2−1 p′ds/bracketrightigg . (ii)If p>2 N−1−V0, then setting β0=N−1−V0 2−1 p∈/parenleftig 0,N−1−V0 2/parenrightig ,one has /integraldisplayt 0(t−s)Gβ(s)ds≤C1/bracketleftigg /ba∇dblu(t)/ba∇dblLp(2+t)N p′−β+/integraldisplayt 0/ba∇dblu(s)/ba∇dblLp(2+s)N p′−β0−1(log(2+s))1 p′ds/bracketrightigg . 13Proof. By Lemma 4.4 with finite propagation property, we have εEβ,0+εEβ,1t+/integraldisplayt 0(t−s)Gβ(s)ds=Iβ,1(t)+2βIβ,2(t)+V0Iβ,3(t). where Iβ,1(t)=/integraldisplay B(0,1+t)u(x,t)Φβ(x,t)dx Iβ,2(t)=/integraldisplayt 0/parenleftigg/integraldisplay B(0,1+t)u(x,s)Φβ+1(x,s)dx/parenrightigg ds Iβ,3(t)=/integraldisplayt 0/parenleftigg/integraldisplay B(0,1+t)1 |x|u(x,s)Φβ(x,s)dx/parenrightigg ds. Using Lemma 3.2 (ii), we have Iβ,1(t)≤/parenleftigg/integraldisplay B(0,1+t)|u(x,t)|pdx/parenrightigg1 p/parenleftigg/integraldisplay B(0,1+t)Φβ(x,t)p′dx/parenrightigg1 p′ ≤c−1 βN−1 p′|SN−1|1 p′/ba∇dblu(t)/ba∇dblLp(2+t)N p′−β. and I′ β,3(t)≤/parenleftigg/integraldisplay B(0,1+t)|u(x,t)|pdx/parenrightigg1 p/parenleftigg/integraldisplay B(0,1+t)1 |x|p′Φβ(x,t)p′dx/parenrightigg1 p′ ≤c−1 β(N−p′)−1 p′|SN−1|1 p′/integraldisplayt 0/ba∇dblu(t)/ba∇dblLp(2+t)N p′−β−1ds. Noting that β+1=N−1−V0 2−1 q′, we see from Lemma 3.2 (iii)that I′ β,2(t)≤/parenleftigg/integraldisplay B(0,1+t)|u(x,t)|pdx/parenrightigg1 p/parenleftigg/integraldisplay B(0,1+t)Φβ+1(x,t)p′dx/parenrightigg1 p′ ≤(c′ β)−1/ba∇dblu(t)/ba∇dblLp(2+t)−β−1/integraldisplay B(0,1+t)/parenleftigg 1−|x| t+2/parenrightigg(N−1−V0 2−β−1)p′ dx1 p′ =(c′ β)−1|SN−1|1 p′/ba∇dblu(t)/ba∇dblLp(2+t)−β−1/integraldisplay1+t 0/parenleftbigg 1−r t+2/parenrightbigg−p′ q′ rN−1dr1 p′ =(c′ β)−1|SN−1|1 p′/ba∇dblu(t)/ba∇dblLp(2+t)N p′−β−1/integraldisplay1 1 2+tρ−p′ q′(1−ρ)N−1dρ1 p′ ≤(c′ β)−1|SN−1|1 p′/parenleftiggp′ q′−1/parenrightigg1 p′ /ba∇dblu(t)/ba∇dblLp(2+t)N p′−β−1+1 q′−1 p′. Thus we have εEβ,0+εEβ,1t+/integraldisplayt 0(t−s)Gβ(s)ds≤C1/bracketleftigg /ba∇dblu(t)/ba∇dblLp(2+t)N p′−β+/integraldisplayt 0/ba∇dblu(s)/ba∇dblLp(2+s)N p′−β−1+1 q′−1 p′ds/bracketrightigg . By the definition of βwe have the first desired inequality. The second is verified by noticing q′/p′=1 in the previous proof. /square 144.2 Proof of Theorem 1.2 for subcritical case max{N N−1,N+3+V0 N+1+V0}<p<p0(N+V0) Proof. Fixq>pas the following way: 1 q∈/parenleftigg 0,N−1−V0 2/parenrightigg ∩/parenleftigg(N−1+V)p−(N+1+V) 2,(N+1+V0)p−(N+3+V0) 2/parenrightigg . The above set is not empty when ( p,V0)∈Ω1∪Ω2∪Ω3; note that for respective cases we can take 1 q=1 p−δ if (p,V0)∈Ω1, N−1−V0 2−δ if (p,V0)∈Ω2, (N+1+V0)p−(N+3+V0) 2−δif (p,V0)∈Ω3 with arbitrary small δ>0. Moreover, this condition is equivalent to q>p, β=N−1−V0 2−1 q>0,&λ=γ(N+V0;p) 2p−1 p+1 q∈(0,p−1). Then we see by Lemma 4.5 (i)that εEβ,0+εEβ,1t+/integraldisplayt 0(t−s)Gβ(s)ds≤C′ 1/bracketleftigg Gβ(t)1 p(2+t)N−β p′+/integraldisplayt 0Gβ(s)1 p(2+s)N−β p′−1 q−1 p′ds/bracketrightigg .(4.2) Observe that N−β p′−1 q−1 p=1 p(p−1−λ)>0, Integrating (4.2) over [0 ,t], we deduce εEβ,0t+εEβ,1 2t2+1 2/integraldisplayt 0(t−s)2Gβ(s)ds ≤C′ 1/bracketleftigg/integraldisplayt 0Gβ(s)1 p(2+s)N−β p′ds+/integraldisplayt 0(t−s)Gβ(s)1 p(2+s)N−β p′−1 q−1 p′ds/bracketrightigg ≤C′ 1/parenleftigg/integraldisplayt 0(2+s)Gβ(s)ds/parenrightigg1 p/parenleftigg/integraldisplayt 0(2+s)(N−β p′−1 p)p′ds/parenrightigg1 p′ +/parenleftigg/integraldisplayt 0(t−s)p′(2+s)(N−β p′−1 q−1 p)p′−1ds/parenrightigg1 p′ ≤C2/parenleftigg/integraldisplayt 0(2+s)Gβ(s)ds/parenrightigg1 p/bracketleftbigg (2+t)N−β p′−1 p+1 p′+(2+t)1+N−β p′−1 q−1 p/bracketrightbigg ≤2C2/parenleftigg/integraldisplayt 0(2+s)Gβ(s)ds/parenrightigg1 p (2+t)1+p−1−λ p. We see from the definition of Hβthat (2C2)−p(2+t)1+λ−2p/parenleftigg εEβ,0t+εEβ,1 2t2/parenrightiggp ≤H′ β(t). Hence H′ β(t)≥C4εp(2+t)1+λ,t≥1. 15Integrating it over [0 ,t], we have for t≥2, Hβ(t)≥/integraldisplayt 0H′ β(s)ds≥/integraldisplayt 1H′ β(s)ds≥C4εp/integraldisplayt 1(2+s)λds≥C4εp 4(2+λ)(2+t)2+λ. We see from the definition of Iβthat for t≥2, J′ β(t)=(2+s)−3Hβ(t)≥C4εp 4(2+λ)(2+t)−1+λ. and for t≥4, Jβ(t)=/integraldisplayt 2J′ β(s)ds≥C4εp 8λ(2+λ)(2+t)λ. On the other hand, we see from Lemma 4.3 that (2C2)−p[Jβ(t)]p≤(2+t)2−λJ′′ β(t)+3(2+t)1−λJ′ β(t)]. Moreover, setting Jβ(t)=/tildewideJβ(σ),σ=2 λ(2+t)λ 2, we see (2+t)1−λ 2J′ β(t)=/tildewideJ′ β(σ),(2+t)2−λJ′′ β(t)+2−λ 2(2+t)1+λJ′ β(t)=/tildewideJ′′ β(σ). Then C−p 5[/tildewideJβ(σ)]p≤/tildewideJ′′ β(σ)+4+λ λσ−1/tildewideJ′ β(σ), σ≥σ0=2 λ2λ 2, /tildewideJ′ β(σ)≥C6εpσ, σ≥σ1=2 λ4λ 2, /tildewideJβ(σ)≥C6εpσ2, σ≥σ2=2 λ6λ 2. Consequently, by Lemma 4.1 (i)we see that /tildewideJβblows up before C7ε−p−1 2and then, Jβblows up before C7ε−p−1 λ. By virtue of Lemma 4.2, we have LifeSpan( uε)≤C7ε−p−1 λ. Finally, we remark that if ( p,V0)∈Ω1, then we can take 1 /q=1/p−δfor arbitrary small δ>0 and thenλ=γ(N+V0;p)/(2p)−1 p+1 q=γ(N+V0;p)/(2p)−δ. This implies that LifeSpan u≤C7ε−2p(p−1) γ(N+V0;p)−δ′ . for arbitrary small δ′>0. The proof is complete. /square 4.3 Proof of Theorem 1.2 for critical case p=p0(N+V0) Proof. In this case we set βδ=N−1−V0 2−1 p+δ∈/parenleftigg 0,N−1−V0 2/parenrightigg . Then by Lemma 4.5 (ii)withβ=βδ, εEβ,0+εEβ,2t≤C1/bracketleftigg /ba∇dblu(t)/ba∇dblLp(2+t)N p′−β+/integraldisplayt 0/ba∇dblu(s)/ba∇dblLp(2+s)N p′−β0−1ds/bracketrightigg ≤K1/bracketleftigg/parenleftig Gβ0(t)/parenrightig1 p(2+t)N−β0 p′+(β0−β)+/integraldisplayt 0/parenleftig Gβ0(t)/parenrightig1 p(2+s)N−β0 p′−1ds/bracketrightigg . 16Noting thatN−β0 p′=1+1 pand integrating it over [0 ,t], we have εEβδ,0t+εEβδ,1 2t2≤K1/bracketleftigg/integraldisplayt 0/parenleftig Gβ0(s)/parenrightig1 p(2+s)1+1 p+(β0−βδ)ds+/integraldisplayt 0(t−s)/parenleftig Gβ0(t)/parenrightig1 p(2+s)1 pds/bracketrightigg ≤K1/parenleftigg/integraldisplayt 0Gβ0(s)(1+s)ds/parenrightigg1 p/parenleftigg/integraldisplayt 0(2+s)p′+(β0−βδ)p′ds/parenrightigg1 p′ +/parenleftigg/integraldisplayt 0(t−s)p′ds/parenrightigg1 p′ ≤K2/parenleftigg/integraldisplayt 0Gβ0(s)(1+s)ds/parenrightigg1 p (2+t)1+1 p′. By the definition of Hβ0, we have for t≥1, H′ β0(t)≥K−p 2εp/parenleftigg Eβδ,0t+Eβδ,1 2t2/parenrightiggp (2+t)1−2p≥K3εp(2+t) and then for t≥2, Hβ0(t)≥/integraldisplayt 1˜G′ β0(s)ds≥K4εp(2+t)2. On the other hand, by Lemma 4.5 (ii)we have /integraldisplayt 0(t−s)Gβ0(s)ds≤C1/bracketleftigg /ba∇dblu(t)/ba∇dblLp(2+t)N p′−β0+/integraldisplayt 0/ba∇dblu(s)/ba∇dblLp(2+s)N p′−β0−1(log(2+s))1 p′ds/bracketrightigg . NotingN−β0 p′=1+1 pagain and integrating it over [0 ,t], we have 1 2/integraldisplayt 0(t−s)Gβ0(s)ds ≤K′ 1/bracketleftigg/integraldisplayt 0Gβ0(s)1 p(2+s)N−β0 p′ds+/integraldisplayt 0(t−s)Gβ0(s)1 p(2+s)N−β0 p′−1(log(2+s))1 p′ds/bracketrightigg ≤K′ 1H′ β0(t)1 p/bracketleftigg/integraldisplayt 0(2+s)p′ds+/integraldisplayt 0(t−s)p′log(2+s)ds/bracketrightigg ≤K′ 2H′ β0(t)1 p(2+t)1+1 p′(log(2+t))1 p′. As in the proof of subcritial case, we deduce (K′ 2)−p(log(2+t))1−pJβ0(t)p≤H′ β0(t)(1+t)−1 ≤(2+t)2J′′ β0(t)+3(2+t)J′ β0(t). Here we take Jβ0(t)=/tildewideJβ0(σ) withσ=log(2+t). Since (2+t)J′ β0(t)=/tildewideJ′ β0(σ),(2+t)2J′ β0(t)+(2+t)J′′ β0(t)=/tildewideJ′′ β0(σ), we obtain for σ≥σ0:=log 2, (K′ 2)−pσ1−p/tildewideJβ0(σ)p≤/tildewideJ′′ β0(σ)+2/tildewideJ′ β0(σ). 17Moreover, we have for σ≥σ1=log 4, /tildewideJ′ β0(σ)=(2+t)J′ β0(t) =(2+t)−2Hβ0(t) ≥K4εp and therefore for σ≥σ2=2 log 4, /tildewideJβ0(σ)≥K4 2εpσ. Applying Lemma 4.1 (ii)we deduce that /tildewideJβ0blows up before σ=K5ε−p(p−1). Then by definition Jβ0 blows up before exp[ K5ε−p(p−1)]. Consequently, using Lemma 4.2, we obtain LifeSpan u≤exp[K5ε−p(p−1)]. The proof is complete. /square Remark 4.1.In particular. in the proof of Theorem 1.2 with p=p(N+V0), we have used two kind of auxiliary parameters 1 /q=N−1−V0 2−1 pand 1/q=N−1−V0 2−1 p+δ. The first choice is for deriving lower bound of the functional Jβ0and the second is for deriving di fferential inequality for Jβ0. The first choice is essentially different from the idea of Yordanov–Zhang [ 11] to prove the lower bound of a functional. Acknowedgements This work is partially supported by Grant-in-Aid for Young S cientists Research (B) No.16K17619 and by Grant-in-Aid for Young Scientists Research (B) No.15K17 571. References [1] R. Beals, R. Wong, “Special functions,” A graduate text. Cambridge Studies in Advanced Mathe- matics 126, Cambridge University Press, Cambridge, 2010. [2] S. Di Pomponio, V . Georgiev, Life-span of subcritical semilinear wave equation , Asymptot. Anal. 28(2001), 91–114. [3] M. Ikeda, T. Ogawa, Lifespan of solutions to the damped wave equation with a crit ical nonlinearity , J. Differential Equations 261(2016), 1880–1903. [4] R. Ikehata, G. Todorova, B. Yordanov, Critical exponent for semilinear wave equations with space - dependent potential , Funkcial. Ekvac. 52(2009), 411–435. [5] F. John, Blow-up of solutions of nonlinear wave equations in three sp ace dimensions , Manuscripta Math. 28(1979), 235–268. [6] N.A. Lai, H. Takamura, K. Wakasa, Blow-up for semilinear wave equations with the scale invari ant damping and super-Fujita exponent , J. Differential Equations 263(2017), 5377–5394. [7] A. Pazy, A. “Semigroups of linear operators and applicat ions to partial differential equations,” Ap- plied Mathematical Sciences 44, Springer-Verlag, New York, 1983. 18[8] T.C. Sideris, Nonexistence of global solutions to semilinear wave equati ons in high dimensions , J. Differential Equations 52(1984), 378–406. [9] W.A. Strauss, Nonlinear scattering theory at low energy , J. Funct. Anal. 41(1981), 110–133. [10] H. Takamura, K. Wakasa, The sharp upper bound of the lifespan of solutions to critica l semilinear wave equations in high dimensions , J. Differential Equations 251(2011), 1157–1171. [11] B. Yordanov, Q.S. Zhang, Finite time blow up for critical wave equations in high dimen sions , J. Funct. Anal. 231(2006), 361–374. [12] Y . Zhou, Blow up of solutions to semilinear wave equations with criti cal exponent in high dimen- sions , Chin. Ann. Math. Ser. B 28(2007), 205–212. [13] Y . Zhou, W. Han, Life-span of solutions to critical semilinear wave equatio ns, Comm. Partial Dif- ferential Equations 39(2014), 439–451. 19

2017-09-13

This paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical space-dependent damping term (DW:$V$). The main result of the present paper is to give a solution of the problem and to provide a sharp estimate for lifespan for such a solution when $\frac{N}{N-1}<p\leq p_S(N+V_0)$, where $p_S(N)$ is the Strauss exponent for (DW:$0$). The main idea of the proof is due to the technique of test functions for (DW:$0$) originated by Zhou--Han (2014, MR3169791). Moreover, we find a new threshold value $V_0=\frac{(N-1)^2}{N+1}$ for the coefficient of critical and singular damping $|x|^{-1}$.

Life-span of blowup solutions to semilinear wave equation with space-dependent critical damping

1709.04401v1

arXiv:1807.06164v1 [math.AP] 17 Jul 2018ON THE BLOW-UP FOR CRITICAL SEMILINEAR WAVE EQUATIONS WITH DAMPING IN THE SCATTERING CASE KYOUHEI WAKASA AND BORISLAV YORDANOV Abstract. We consider the Cauchy problem for semilinear wave equation s with variable coefficients and time-dependent scattering da mping in Rn, where n≥2. It is expected that the critical exponent will be Strauss’ number p0(n), which is also the one for semilinear wave equations without d amping terms. Lai and Takamura [7] have obtained the blow-up part, togethe r with the upper bound of lifespan, in the sub-critical case p < p0(n). In this paper, we extend their results to the critical case p=p0(n). The proof is based on [16], which concerns the blow-up and upper bound of lifespan for cr itical semilinear wave equations with variable coefficients. 1.Introduction We study the blow-up problem for critical semilinar wave equations wit h vari- able coefficients and scattering damping depending on time. The pert urbations of Laplacian are uniformly elliptic operators ∆g=n/summationdisplay i,j=1∂xigij(x)∂xj whose coefficients satisfy, with some α >0,the following: (1.1) gij∈C1(Rn),|∇gij(x)|+|gij(x)−δij|=O(e−α|x|) as|x| → ∞. The admissible damping coefficients are a∈C([0,∞)), such that (1.2) ∀t≥0a(t)≥0 and/integraldisplay∞ 0a(t)dt <∞. Forn≥2 andp >1, we consider the Cauchy problem utt−∆gu+a(t)ut=|u|p, x∈Rn, t >0, (1.3) u|t=0=εu0, ut|t=0=εu1, x∈Rn, (1.4) whereu0, u1∈C∞ 0(Rn) andε >0 is a small parameter. Our results concern only the critical case p=p0(n) with Strauss’ exponent defined in (1.5) below. Let us briefly review previous results concerning (1.3) with gij=δijand various types of damping a. Whena(t) = 1, Todorova and Yordanov [13] showed that the solution of (1.3) blows up in finite time if 1 < p < p F(n), where pF(n) = 1+2 /n is the Fujita exponent known to be the critical exponent for the se milinear heat equation. The same work also obtained global existence for p > pF(n). Finally, Zhang [20] established the blow-up in the critical case p=pF(n). The other typical example of effective damping is a(t) =µ/(1+t)βwithµ >0 andβ∈R. When−1< β <1, Lin, Nishihara and Zhai [9] obtained the expected Date: November 15, 2021. 12 KYOUHEI WAKASA AND BORISLAV YORDANOV blow-up result, if 1 < p≤pF(n), and global existence result, if p > pF(n); see also D’Abbicco, S.Lucente and M.Reissig [2]. In the caseof critical decay β= 1, there areseveralworksabout finite time blow- up and global existence. Wakasugi [17] showed the blow-up, if 1 < p≤pF(n) and µ >1 or 1< p≤1+2/(n+µ−1) and 0 < µ≤1. Moreover, D’Abbicco [1] verified the global existence, if p > pF(n) andµsatisfies one of the following: µ≥5/3 for n= 1,µ≥3 forn= 2 and µ≥n+2 forn≥3. An interesting observation is that the Liouville substitution w(x,t) := (1+ t)µ/2u(x,t) transforms the damped wave equation (1.3) into the Klein-Gordon equation wtt−∆w+µ(2−µ) 4(1+t)2w=|w|p (1+t)µ(p−1)/2. Thus, one expects that the critical exponent for µ= 2 is related to that of the semi- linear wave equation. D’Abbicco, Lucente and Reissig [3] have actua lly obtained the corresponding blow-up result, if 1 < p < p c(n) := max {pF(n), p0(n+2)}and (1.5) p0(n) :=n+1+√ n2+10n−7 2(n−1) is the so-called Strauss exponent, the positive root of the quadra tic equation (1.6) γ(p,n) := 2+( n+1)p−(n−1)p2= 0. Their work also showed the existence of global classical solutions fo r smallε >0, if p > pc(n) and either n= 2 orn= 3 and the data are radially symmetric. Finally, we mention that our original equations (1.3) is related to semilinear wa ve equations in the Einstein-de Sitter spacetime considered by Galstian & Yagdjian [4]. We recall that p0(n) in (1.5) is the critical exponent for the semilinear wave equation conjectured by Strauss [11]. The hypothesis has been ve rified in several cases; see [16] and the references therein. A related problem is to estimate the lifespan, or the maximal existence time Tεof solutions to (1.3), (1.4) in the energy spaceC([0,Tε),H1(Rn))∩C1([0,Tε),L2(Rn)). Lai, Takamura and Wakasa [8] have obtained the blow-up part of St rauss’ con- jecture, together with an upper bound of the lifespan Tε, for (1.3), (1.4) in the casen≥2, 0< µ <(n2+n+ 2)/2(n+ 2) and pF(n)≤p < p0(n+ 2µ). Later, Ikeda and Sobajima [5] were able to replace these conditions by less restrictive 0< µ <(n2+n+2)/(n+2) and pF(n)≤p≤p0(n+µ). In addition, they have derived an upper bound on the lifespan. Tu and Lin [14], [15] have impro ved the estimates of Tεin [5] recently. Forβ≤ −1, the long time behavior of solutions to (1.3), (1.4) is quite different. Whenβ=−1, Wakasugi [18] has obtained the global existence for exponents pF(n)< p < n/ [n−2]+, where [n−2]+:=/braceleftbigg∞ forn= 1,2, m/(n−2) for n≥3. Ikeda and Wakasugi [6] have proved that the global existence ac tually holds for any p >1 whenβ <−1. Forβ >1, we expect the critical exponent to be exactly the Strauss expo nent. In fact, Lai and Takamura [7] have shown that certain solutions of (1.3), (1.4) blow up in finite time when 1 < p < p 0(n). Moreover, Liu and Wang [10] have just obtained the global existence results for n= 3,4 andp > p0(n) on asymptotically Euclidean manifolds.3 IfTεdenotes the lifespan of these solutions, then [7] have also given the upper boundTε≤Cε−2p(p−1)/γ(p,n)forn≥2 and 1< p < p 0(n). This result is probably sharp, since Takamura [12] proved the same type of estimate in the sub-critical case of Strauss’ conjecture for semilinear wave equations without dam ping. However, both the conjecture and lifespan bound remained open problems in t he critical case p=p0(n). The purpose of this paper is to verify the blow-up for p=p0(n) and to give a proof that extends to more general damping, including a(t)∼(1+t)−βwithβ >1. We also succeed to derive an exponential type upper bound on the lif espanTε, which is the same as that of the Strauss conjecture in the conserv ative critical case. Such results are consistent with our knowledge of the linear problem corresponding to (1.3), (1.4); Wirth [19] has shown that energy space solutions sc atter, that is approach solutions to the free wave equations, as t→ ∞. Theorem 1.1. Letn≥2,p=p0(n)anda(t)satisfy (1.2). Assume that both u0∈H1(Rn)andu1∈L2(Rn)are nonnegative, do not vanish identically and have supports in the ball {x∈Rn:|x| ≤R0}, whereR0>1. If (1.3) has a solution (u,ut)∈C([0,Tε),H1(Rn)×L2(Rn)), such that (1.7) supp( u,ut)⊂ {(x,t)∈Rn×[0,Tε) :|x| ≤t+R}, withR≥R0, thenTε<∞. Moreover, there exist constants ε0=ε0(u0,u1,n,p,R,a ) andK=K(u0,u1,n,p,R,a ), such that (1.8) Tε≤exp/parenleftBig Kε−p(p−1)/parenrightBig for0< ε≤ε0. Remark 1.2.The lifespan estimates is the same as that of the Strauss conjectu re in the critical case of semilinear wave equations without damping. For details, see the introduction in [16]. We also note that Liu and Wang [10] have obta ined the sharp lower bound of the lifespan, Tε≥exp(cε−2) ifn= 4 and p=p0(4) = 2. Our proof is based on the approach of Wakasa and Yordanov [16]. Av eraging the solutionwith respecttoasuitabletestfunction, wederiveasecond -orderdissipative ODE which corresponds to equation (1.3). The key point is to establis h lower bounds for the fundamental system of solutions to this ODE; see L emma 2.3. As a consequence, we can follow [16] and obtain the same nonlinear integ ral inequality. The final blow-up argument also repeats the iteration argument of [16]. 2.Test Functions Similarly to the proof of [16], we first consider the following elliptic prob lem: (2.1) ∆ gϕλ=λ2ϕλ, x∈Rn, whereλ∈(0,α/2].Asλ|x| → ∞, theseϕλ(x) are asymptotically given by ϕ(λx), withϕbeing the standard radial solution to the unperturbed equation ∆ ϕ=ϕ: (2.2) ϕ(x) =/integraldisplay Sn−1ex·ωdSω∼cn|x|−(n−1)/2e|x|,|x| → ∞. We recall the following result about the existence and main propertie s ofϕλ. Lemma 2.1. Letn≥2. There exists a solution ϕλ∈C∞(Rn)to (2.1), such that (2.3) |ϕλ(x)−ϕ(λx)| ≤Cαλθ, x∈Rn, λ∈(0,α/2], whereθ∈(0,1]andϕ(x) =/integraltext Sn−1ex·ωdSω∼cn|x|−(n−1)/2e|x|, cn>0,as|x| → ∞.4 KYOUHEI WAKASA AND BORISLAV YORDANOV Moreover, ϕλ(·)−ϕ(λ·)is a continuous L∞(Rn)valued function of λ∈(0,α/2] and there exist positive constants D0, D1andλ0, such that (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, holds whenever 0< λ≤λ0. Proof.See Lemma 2.2 in [16]. /square Givenλ0∈(0,α/2] andq >0, we also introduce the auxiliary functions ξq(x,t) =/integraldisplayλ0 0e−λ(t+R)coshλtϕλ(x)λqdλ, (2.5) ηq(x,t,s) =/integraldisplayλ0 0e−λ(t+R)sinhλ(t−s) λ(t−s)ϕλ(x)λqdλ, (2.6) for (x,t)∈Rn×Rands∈R.Useful estimates are collected in the next lemma. Lemma 2.2. Letn≥2. There exists λ0∈(0,α/2], such that the following hold: (i) if0< q,|x| ≤Rand0≤t, then ξq(x,t)≥A0, ηq(x,t,0)≥B0/an}b∇acketle{tt/an}b∇acket∇i}ht−1; (ii) if0< q,|x| ≤s+Rand0≤s < t, then ηq(x,t,s)≥B1/an}b∇acketle{tt/an}b∇acket∇i}ht−1/an}b∇acketle{ts/an}b∇acket∇i}ht−q; (iii) if(n−3)/2< q,|x| ≤t+Rand0< t, then ηq(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. HereA0andBk,k= 0,1,2,are positive constants depending only on α,qandR, while/an}b∇acketle{ts/an}b∇acket∇i}ht= 3+|s|. Proof.See Lemma 3.1 in [16]. /square The following lemma plays a key role in the proof of Theorem1.1. Lemma 2.3. Letλ >0and introduce the ordinary differential operators La=∂2 t+a(t)∂t−λ2, L∗ a=∂2 s−∂sa(s)−λ2. The fundamental system of solutions {y1(t,s;λ),y2(t,s;λ)}, defined through Lay1(t,s;λ) = 0, y 1(s,s;λ) = 1, ∂ty1(s,s;λ) = 0, Lay2(t,s;λ) = 0, y 2(s,s;λ) = 0, ∂ty2(s,s;λ) = 1, depends continuously on λand satisfies the following estimates, for t≥s≥0: (i)y1(t,s;λ)≥e−/bardbla/bardblL1coshλ(t−s), (ii)y2(t,s;λ)≥e−2/bardbla/bardblL1sinhλ(t−s) λ. Moreover, the conjugate equations and initial conditions h old: (iii)L∗ ay2(t,s;λ) = 0, (iv)y1(t,0;λ) =a(0)y2(t,0;λ)−∂sy2(t,0;λ). Proof.See Section 4. /square5 3.Proof of Theorem 1.1 Letube a weak solution to problem (1.3), defined below, and ηq(x,t,t) be a test function, defined in Section 2, with q >−1. We will show that (3.1) F(t) =/integraldisplay Rnu(x,t)ηq(x,t,t)dx satisfies a nonlinear integral inequality which implies finite time blow-up. Our definition of weak solutions is standard: ( u,ut)∈C([0,Tε),H1(Rn)×L2(Rn)) and ∀φ∈C∞ 0(Rn×[0,Tε)) andt∈(0,Tε) /integraldisplay us(x,t)φ(x,t)dx−/integraldisplay us(x,0)φ(x,0)dx −/integraldisplayt 0/integraldisplay (us(x,s)φs(x,s)−g(x)∇u(x,s)·∇φ(x,s)−a(s)us(x,s)φ(x,s))dxds =/integraldisplay∞ 0/integraldisplay |u(x,s)|pφ(x,s)dxds. In the next result, however, it will be more convenient to work with /integraldisplay (us(x,t)φ(x,t)−u(x,t)φs(x,t)+a(t)u(x,t)φ(x,t))dx −/integraldisplay (us(x,0)φ(x,0)−u(x,0)φs(x,0)+a(0)u(x,0)φ(x,0))dx (3.2) +/integraldisplayt 0/integraldisplay u(x,s)(φss(x,s)−∆gφ(x,s)−(a(s)φ(x,s))s)dxds =/integraldisplay∞ 0/integraldisplay |u(x,s)|pφ(x,s)dxds, which follows from integration by parts. We can also use φ∈C∞(Rn×[0,Tε)), sinceu(·,s) is compactly supported for every s. Proposition 3.1. Let the assumptions in Theorem 1.1 be fulfilled and q >−1. (3.3)/integraldisplay Rnu(x,t)ηq(x,t,t)dx ≥εe−/bardbla/bardblL1/integraldisplay Rnu0(x)ξq(x,t)dx+εe−2/bardbla/bardblL1t/integraldisplay Rnu1(x)ηq(x,t,0)dx +e−2/bardbla/bardblL1/integraldisplayt 0(t−s)/integraldisplay Rn|u(x,s)|pηq(x,t,s)dxds for allt∈(0,Tε). Proof.We will apply (3.2) to φ(x,s) =ϕλ(x)y2(t,s;λ), which satisfies φss(x,s)−∆gφ(x,s)−(a(s)φ(x,s))s= 0, a(0)φ(x,0)−φs(x,0) =ϕλ(x)y1(t,0;λ),6 KYOUHEI WAKASA AND BORISLAV YORDANOV from Lemma 2.3 ( iii) and (iv), respectively. Then we obtain /integraldisplay u(x,t)ϕλ(x)dx=εy1(t,0;λ)/integraldisplay u0(x)ϕλ(x)dx +εy2(t,0;λ)/integraldisplay u1(x)ϕλ(x)dx +/integraldisplayt 0y2(t,s;λ)/parenleftbigg/integraldisplay |u(x,s)|pϕλ(x)dx/parenrightbigg ds, where the initial conditions are determined by (1.4) and the pair {y1,y2}is defined in Lemma 2.3. Making use of estimates ( i) and (ii) in this lemma, we have that /integraldisplay u(x,t)ϕλ(x)dx≥εe−/bardbla/bardblL1cosh(λt)/integraldisplay u0(x)ϕλ(x)dx +εe−2/bardbla/bardblL1sinhλt λ/integraldisplay u1(x)ϕλ(x)dx +e−2/bardbla/bardblL1/integraldisplayt 0sinhλ(t−s) λ/parenleftbigg/integraldisplay |u(x,s)|pϕλ(x)dx/parenrightbigg ds. The lower bound (3.3) follows from multiplying the aboveinequality by λqe−λ(t+R), integrating on [0 ,λ0] and interchanging the order of integration between λandx. Recalling definitions (2.5) and (2.6) for ξqandηq, we complete the proof. /square Similarly to the proof of Proposition 4.2. in [16], we obtain the convenien t iteration frame by using Lemma 2.2. Proposition 3.2. Suppose that the assumptions in Theorem 1.1 are fulfilled and chooseq= (n−1)/2−1/p.IfF(t)is defined in (3.1), there exists a positive constant C=C(n,p,q,R,a ), such that (3.4) F(t)≥C /an}b∇acketle{tt/an}b∇acket∇i}ht/integraldisplayt 0t−s /an}b∇acketle{ts/an}b∇acket∇i}htF(s)p (log/an}b∇acketle{ts/an}b∇acket∇i}ht)p−1ds for allt∈(0,Tε). The finite time blow-up and lifespan estimate (1.8) can now be derived f ollowing Sections 4 and 5 in [16]. 4.Proof of Lemma 2.3 Let us recall that λ >0 andLa= (d/dt)2+a(t)d/dt−λ2.There exists a pair of C2-solutions {y1(t,s;λ),y2(t,s;λ)}which depends continuously on λand satisfies Lay1= 0, y 1(s,s;λ) = 1, y′ 1(s,s;λ) = 0, Lay2= 0, y 2(s,s;λ) = 0, y′ 2(s,s;λ) = 1, fort≥s≥0.We will show that {y1(t,s;λ),y2(t,s;λ)}behaves similarly to the fundamental system of L0, that is {coshλ(t−s),λ−1sinhλ(t−s)}, ast−s→ ∞7 andλ→0.Our proof gives two-sided bounds and relies only on three identities: (y′ 1eA(t))′=λ2y1eA(t),whereA(t) =/integraldisplayt 0a(r)dr, (4.1) /parenleftbigg y1eA(t)−/integraldisplayt sa(r)y1eA(r)dr/parenrightbigg′′ =λ2y1eA(t), (4.2) y′ 2y1−y2y′ 1=eA(s)−A(t)or/parenleftbiggy2 y1/parenrightbigg′ =eA(s)−A(t) y2 1. (4.3) To verify claim ( i), we observe that y1(t0,s;λ) = 0 at some t0> sleads to a contradiction: if t0is the first such number, then y1(t,s;λ)≥0 fort∈[s,t0] and (4.1) imply that y′ 1(t,s;λ)eA(t)=λ2/integraldisplayt sy1(r,s;λ)eA(r)dr≥0,soy′ 1(t,s;λ)≥0 fort∈[s,t0]. Hence,y1(t,s;λ) is increasing on [ s,t0] and 0 = y1(t0,s;λ)≥y1(s,s;λ) = 1 can not hold. The positivity of y1(t,s;λ) also yields, through (4.1), the positivity of its derivative: y′ 1(t,s;λ)≥0 for all t≥s. We can now derive an upper bound on y1using that y′′ 1=λ2y1−ay′ 1≤λ2y1 andy1(s,s;λ) = 1,y′ 1(s,s;λ) = 0: (4.4) y1(t,s;λ)≤coshλ(t−s), t≥s. The lower bound on y1is a consequence of (4.2) and the positivity of y1(t,s;λ): /parenleftbigg y1eA(t)−/integraldisplayt sa(r)y1eA(r)dr/parenrightbigg′′ ≥λ2/parenleftbigg y1eA(t)−/integraldisplayt sa(r)y1eA(r)dr/parenrightbigg . Combining this inequality with the initial values at t=s, /parenleftbigg y1eA(t)−/integraldisplayt sa(r)y1eA(r)dr/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle t=s=eA(s), d dt/parenleftbigg y1eA(t)−/integraldisplayt sa(r)y1eA(r)dr/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle t=s= 0, we obtain that y1(t,s;λ)eA(t)−/integraldisplayt sa(τ)y1eA(τ)dτ≥eA(s)coshλ(t−s). After simplifying, (4.5) y1(t,s;λ)≥eA(s)−A(t)coshλ(t−s), t≥s. SinceA(s)−A(t)≥ −/ba∇dbla/ba∇dblL1and (4.4) holds, claim ( i) follows: (4.6) cosh λ(t−s)≥y1(t,s;λ)≥e−/bardbla/bardblL1coshλ(t−s). To check claim ( ii), we combine (4.4), (4.5) and identity (4.3): eA(t)−A(s) cosh2λ(t−s)≥/parenleftbiggy2 y1/parenrightbigg′ ≥eA(s)−A(t) cosh2λ(t−s). UsingA(s)−A(t)≥ −/ba∇dbla/ba∇dblL1and integration on [ s,t], we have that e/bardbla/bardblL1 λtanhλ(t−s)≥y2(t,s;λ) y1(t,s;λ)≥e−/bardbla/bardblL1 λtanhλ(t−s).8 KYOUHEI WAKASA AND BORISLAV YORDANOV The final result follows from (4.6): e/bardbla/bardblL1sinhλ(t−s) λ≥y2(t,s;λ)≥e−2/bardbla/bardblL1sinhλ(t−s) λ. Finally, we will show the equalities in ( iii) and (iv).Sety1(t) :=y1(t,0;λ) and y2(t) :=y2(t,0;λ). It easy to see that y2(t,s,λ) =y1(t)y2(s)−y1(s)y2(t) y′ 1(s)y2(s)−y1(s)y′ 2(s)= (y1(s)y2(t)−y1(t)y2(s))eA(s). Thus, we can calculate ∂i sy2(t,s,λ) withi= 1,2 as follows: ∂sy2(t,s;λ) = (y′ 1(s)y2(t)−y1(t)y′ 2(s))eA(s)+a(s)y2(t,s;λ), (4.7) ∂2 sy2(t,s;λ) = (y′′ 1(s)y2(t)−y1(t)y′′ 2(s))eA(s) +a(s)(y′ 1(s)y2(t)−y1(t)y′ 2(s))eA(s)+∂s(a(s)y2(t,s;λ)). Noticing that yi(s) withi= 1,2 satisfy the differential equation Layi(s) = 0, we get (iii). To derive ( iv), we just set s= 0 in (4.7) for ∂sy2(t,s;λ) and use the initial conditions for yi(s). The proof is complete. References [1] M.D’Abbicco, The threshold of effective damping for semilinear wave equat ions, Mathemat- ical Methods in Applied Sciences., 38(2015) 1032-1045. [2] M.D’Abbicco, S.Lucente and M.Reissig, Semi-linear wave equations with effective damping , Chin. Ann. Math. Ser. B, 34(2013), 345-380. [3] M.D’Abbicco, S.Lucente and M.Reissig, A shift in the Strauss exponent for semilinear wave equations with a not effective damping , J. Differential Equations., 259(2015), 5040-5073. [4] A.Galstian and K.Yagdjian, Finite lifespan of solutions of the semilinear wave equatio n in the Einstein-de Sitter spacetime , preprint, arXiv:1612.09536v2. [5] M.Ikeda and M.Sobajima, Life-span of solutions to semilinear wave equation with tim e- dependent critical damping for specially localized initia l data, Mathematische Annalen, (2018), 1-24. [6] M.Ikeda and Y.Wakasugi, Global well-posedness for the semilinear wave equation wit h time dependent damping in the overdamping case , accepted for publication in Proceedings of the American Mathematical Society, arXiv:1708.08044v2. [7] N.-A.,Lai and H.Takamura, Blow-up for semilinear damped wave equations with sub-Stra uss exponent in the scattering case , Nonlinear Analysis TMA 168(2018), 222-237. [8] N.-A.Lai, H.Takamura and K. Wakasa, Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent , J. Differential Equations, 263(2017), 5377- 5394. [9] J.Lin, K.Nishihara and J.Zhai, Critical exponent for the semilinear wave equation with tim e- dependent damping , Discrete and Continuous Dynamical Systems - Series A., 32(2012), 4307-4320. [10] M.Liu and C.Wang Global existence of semilinear damped wave equations in rel ation with the Strauss conjecture , preprint, arXiv:1807.05908. [11] W.A.Strauss, Nonlinear scattering theory at low energy , J. Funct. Anal., 41(1981), 110-133. [12] H.Takamura, Improved Kato’s lemma on ordinary differential inequality a nd its application to semilinear wave equations , Nonlinear Analysis TMA 125(2015), 227-240. [13] G.Todorova and B.Yordanov, Critical exponent for a nonlinear wave equation with dampin g, J. Differential Equations, 174(2001) 464-489. [14] Z.Tu and J.Lin, A note on the blowup of scale invariant damping wave equation with sub- Strauss exponent , preprint, arXiv:1709.00866. [15] Z.Tu and J.Lin, Life-Span of Semilinear Wave Equations with Scale-invaria nt Damping: Critical Strauss Exponent Case , preprint, arXiv:1711.00223.9 [16] K.Wakasa and B.Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients , preprint, arXiv:1807.02772. [17] Y.Wakasugi, Critical exponent for the semilinear wave equation with sca le invariant damping , Fourier analysis, 375-390, Trends Math., Birkh¨ auser/Spr inger, Cham, (2014). [18] Y.Wakasugi, Scaling variables and asymptotic profiles for the semilinea r damped wave equa- tion with variable coefficients , J. Math. Anal. Appl., 447(2017), 452-487. [19] J.Wirth, Wave equations with time-dependent dissipation. I. Non-eff ective dissipation , J. Differential Equations, 222(2006), 487-514. [20] Q.S.Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case , C. R. Math. Acad. Sci. Paris, S´ er. I 333 (2001) 109-114. Department of Mathematics, Faculty of Science and Technolo gy, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba, 278-8510, Japan Office of International Affairs, Hokkaido University, Kita 15 , Nishi 8, Kita-ku, Sapporo, Hokkaido 060-0815, Japan and Institute of Mathemat ics, Sofia E-mail address :wakasakyouhei@ma.noda.tus.ac.jp E-mail address :byordanov@oia.hokudai.ac.jp

2018-07-17

We consider the Cauchy problem for semilinear wave equations with variable coefficients and time-dependent scattering damping in $\mathbf{R}^n$, where $n\geq 2$. It is expected that the critical exponent will be Strauss' number $p_0(n)$, which is also the one for semilinear wave equations without damping terms. Lai and Takamura (2018) have obtained the blow-up part, together with the upper bound of lifespan, in the sub-critical case $p<p_0(n)$. In this paper, we extend their results to the critical case $p=p_0(n)$. The proof is based on Wakasa and Yordanov (2018), which concerns the blow-up and upper bound of lifespan for critical semilinear wave equations with variable coefficients.

On the blow-up for critical semilinear wave equations with damping in the scattering case

1807.06164v1

Spin dynamics and relaxation in the classical-spin Kondo-impurity model beyond the Landau-Lifschitz-Gilbert equation Mohammad Sayad and Michael Pottho I. Institut f ur Theoretische Physik, Universit at Hamburg, Jungiusstrae 9, 20355 Hamburg, Germany The real-time dynamics of a classical spin in an external magnetic eld and locally exchange coupled to an extended one-dimensional system of non-interacting conduction electrons is studied numerically. Retardation eects in the coupled electron-spin dynamics are shown to be the source for the relaxation of the spin in the magnetic eld. Total energy and spin is conserved in the non-adiabatic process. Approaching the new local ground state is therefore accompanied by the emission of dispersive wave packets of excitations carrying energy and spin and propagating through the lattice with Fermi velocity. While the spin dynamics in the regime of strong exchange coupling Jis rather complex and governed by an emergent new time scale, the motion of the spin for weakJis regular and qualitatively well described by the Landau-Lifschitz-Gilbert (LLG) equation. Quantitatively, however, the full quantum-classical hybrid dynamics diers from the LLG approach. This is understood as a breakdown of weak-coupling perturbation theory in Jin the course of time. Furthermore, it is shown that the concept of the Gilbert damping parameter is ill-dened for the case of a one-dimensional system. PACS numbers: 75.78.-n, 75.78.Jp, 75.60.Jk, 75.10.Hk, 75.10.Lp I. INTRODUCTION The Landau-Lifshitz-Gilbert (LLG) equation1{3has originally been considered to describe the dynamics of the magnetization of a macroscopic sample. Nowadays it is frequently used to simulate the dynamics of many mag- netic units coupled by exchange or magnetostatic interac- tions, i.e., in numerical micromagnetics.4The same LLG equation can be used on an atomistic level as well.5{9For a suitable choice of units and for several spins Sm(t) at lattice sites m, it has the following structure: dSm(t) dt=Sm(t)B+X nJmnSm(t)Sn(t) +X nmnSm(t)dSn(t) dt: (1) It consists of precession terms coupling the spin at site mto an external magnetic eld Band, via exchange couplingsJmn, to the spins at sites n. Those pre- cession terms typically have a clear atomistic origin, such as the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction10{12which is mediated by the magnetic po- larization of conduction electrons. The non-local RKKY couplingsJmn=J2mnare given in terms of the ele- mentsmnof the static conduction-electron spin suscep- tibility and the local exchange Jbetween the spins and the local magnetic moments of the conduction electrons. Other possibilities comprise direct (Heisenberg) exchange interactions, intra-atomic (Hund's) couplings as well as the spin-orbit and other anisotropic interactions. The re- laxation term, on the other hand, is often assumed as lo- cal,mn=mn, and represented by purely phenomeno- logical Gilbert damping constant only. It describes the angular-momentum transfer between the spins and a usu- ally unspecied heat bath.On the atomistic level, the Gilbert damping must be seen as originating from microscopic couplings of the spins to the conduction-electron system (as well as to lattice degrees of freedom which, however, will not be considered here). There are numerous studies where the damping constant, or tensor, has been computed nu- merically from a more fundamental model including elec- tron degrees of freedom explicitly13{15or even from rst principles.16{21All these studies rely on two, partially related, assumptions: (i) The spin-electron coupling J is weak and can be treated perturbatively to lowest or- der, i.e., the Kubo formula or linear-response theory is employed. (ii) The classical spin dynamics is slow as compared to the electron dynamics. These assumptions appear as well justied but they are also necessary to achieve a simple eective spin-only theory by eliminat- ing the fast electron degrees of freedom. The purpose of the present paper is to explore the physics beyond the two assumptions (i) and (ii). Us- ing a computationally ecient formulation in terms of the electronic one-particle reduced density matrix, we have set up a scheme by which the dynamics of classi- cal spins coupled to a system of conduction electrons can be treated numerically exactly. The theory applies to ar- bitrary coupling strengths and does not assume a separa- tion of electron and spin time scales. Our approach is a quantum-classical hybrid theory22which may be charac- terized as Ehrenfest dynamics, similar to exact numerical treatments of the dynamics of nuclei, treated as classical objects, coupled to a quantum system of electrons (see, e.g., Ref. 23 for an overview). Some other instructive ex- amples of quantum-classical hybrid dynamics have been discussed recently.24,25 The obvious numerical advantage of an eective spin- only theory, as given by LLG equations of the form (1), is that in solving the equations of motion there is only the time scale of the spins that must be taken care of. AsarXiv:1507.08227v2 [cond-mat.mes-hall] 28 Nov 20152 compared to our hybrid theory, much larger time steps and much longer propagation times can be achieved. Op- posed to ab-initio approaches16,17,26we therefore con- sider a simple one-dimensional non-interacting tight- binding model for the conduction-electron degrees of free- dom, i.e., electrons are hopping between the nearest- neighboring sites of a lattice. Within this model ap- proach, systems consisting of about 1000 sites can be treated easily, and we can access suciently long time scales to study the spin relaxation. An equilibrium state with a half-lled conduction band is assumed as the ini- tial state. The subsequent dynamics is initiated by a sudden switch of a magnetic eld coupled to the classical spin. The present study is performed for a single spin, i.e., we consider a classical-spin Kondo-impurity model with antiferromagnetic local exchange coupling J, while the theory itself is general and can be applied to more than a single or even to a large number of spins as well. As compared to the conventional (quantum-spin) Kondo model,27,28the model considered here does not account for the Kondo eect and therefore applies to sit- uations where this is absent or less important, such as for systems with large spin quantum numbers S, strongly anisotropic systems or, as considered here, systems in a strong magnetic eld. To estimate the quality of the classical-spin approximation a priori is dicult.29{31For one-dimensional systems, however, a quantitative study is possible by comparing with full quantum calculations and will be discussed elsewhere.32 There are dierent questions to be addressed: For dimensional reasons, one should expect that linear- response theory, even for weak J, must break down at long times. It will therefore be interesting to compare the exact spin dynamics with the predictions of the LLG equation for dierent J. Furthermore, the spin dynamics in the long-time limit can be expected to be sensitively dependent on the low-energy electronic structure. We will show that this has important consequences for the computation of the damping constant and thatis even ill-dened in some cases. An advantage of a full theory of spin and electron dynamics is that a precise microscopic picture of the electron dynamics is available and can be used to discuss the precession and relaxation dynamics of the spin from another, namely from the elec- tronic perspective. This information is in principle exper- imentally accessible to spin-resolved scanning-tunnelling microscope techniques33{36and important for an atom- istic understanding of nano-spintronics devices.37,38We are particularly interested in the physics of the system in the strong- Jregime or for a strong eld Bwhere the time scales of the spin and the electron dynamics become comparable. This has not yet been explored but could become relevant to understand real-time dynamics in re- alizations of strong- JKondo-lattice models by means of ultracold fermionic Yb quantum gases trapped in optical lattices.39,40 The paper is organized as follows: We rst introduce the model and the equations of motion for the exactquantum-classical hybrid dynamics in Sec. II and discuss some computational details in Sec. III. Sec. IV provides a comprehensive discussion of the relaxation of the clas- sical spin after a sudden switch of a magnetic eld. The reversal time as a function of the interaction and the eld strength is analyzed in detail. We then set the focus on the conduction-electron system which induces the relax- ation of the classical spin by dissipation of energy. In Sec. V, the linear-response approach to integrate out the elec- tron degrees of freedom is carefully examined, including a discussion of the additional approximations that are nec- essary to re-derive the LLG equation and the damping term in particular. Sec. VI summarizes the results and the main conclusions. II. MODEL AND THEORY We consider a classical spin SwithjSj= 1=2, which is coupled via a local exchange interaction of strength Jto the local quantum spin si0at the sitei0of a system of N itinerant and non-interacting conduction electrons. The conduction electrons hop with amplitude T(T > 0) between non-degenerate orbitals on nearest-neighboring sites of aD-dimensional lattice, see Fig. 1. Lis the number of lattice sites, and n=N=L is the average conduction-electron density. The dynamics of this quantum-classical hybrid system22is determined by the Hamiltonian H=TX hiji;cy icj+Jsi0SBS: (2) Here,ciannihilates an electron at site i= 1;:::;L with spin projection =";#, andsi=1 2P 0cy i0ci0is the local conduction-electron spin at i, wheredenotes the vector of Pauli matrices. The sum runs over the dierent ordered pairs hijiof nearest neighbors. Bis an external magnetic eld which couples to the classical spin. To be denite, an antiferromagnetic exchange coupling J > 0 is assumed. If Swas a quantum spin with S= 1=2, Eq. (2) would represent the single-impurity Kondo model.27,28However, in the case of a classical spin considered here, there is no Kondo eect. The semiclas- sical single-impurity Kondo model thus applies to sys- tems where a local spin is coupled to electronic degrees of freedom but where the Kondo eect absent or sup- pressed. This comprises the case of large spin quantum numbersS, or the case of temperatures well above the Kondo scale, or systems with a ferromagnetic Kondo cou- plingJ <0 where, for a classical spin, we expect a qual- itatively similar dynamics as for J >0. We assume that initially, at time t= 0, the clas- sical spinS(t= 0) has a certain direction and that the conduction-electron system is in the corresponding ground state, i.e., the conduction electrons occupy the lowestNone-particle eigenstates of the non-interacting3 Hamiltonian Eq. (2) for the given S=S(t= 0) up to the chemical potential . A non-trivial time evolution is initiated if the initial direction of the classical spin and the direction of the eld Bare non-collinear. To determine the real-time dynamics of the electronic subsystem, it is convenient to introduce the reduced one- particle density matrix. Its elements are dened as ex- pectation values, ii0;0(t)hcy i00ciit; (3) in the system's state at time t. Att= 0 we have(0) = (T(0)). The elements of (0) are given by i;i00(0) =X kUi;k("k)Uy k;i00; (4) where  is the step function and where Uis the uni- tary matrix diagonalizing the hopping matrix T(0), i.e., UyT(0)U="with the diagonal matrix "given by the eigenvalues of T(0). The hopping matrix at time tis can be read o from Eq. (2). It comprises the physical hopping and the contribution resulting from the coupling term. Its elements are given by Ti;i00(t) =Thii0i0+ii0i0i0J 2(S(t))0:(5) Herehii0i= 1 ifi;i0are nearest neighbors and zero else. There is a closed system of equations of motion for the classical spin vector S(t) and for the one-particle density matrix(t). The time evolution of the classical spin is determined via ( d=dt)S(t) =fS;Hclass:gby the classical Hamilton function Hclass:=hHi. This equation of mo- tion is the only known way to consistently describe the dynamics of quantum-classical hybrids (see Refs. 22,41,42 and references therein for a general discussion). The Poisson bracket between arbitrary functions AandBof the spin components is given by,43,44 fA;Bg=X ;; " @A @S@B @SS ; (6) where the sums run over x;y;z and where " is the fully antisymmetric "-tensor. With this we nd d dtS(t) =Jhsi0itS(t)BS(t): (7) This is the Landau-Lifschitz equation where the expec- tation value of the conduction-electron spin at i0is given by hsi0it=1 2X 0i0;i00(t)0; (8) and where Jhsi0itacts as an eective time-dependent internal eld in addition to the external eld B. S(t)JTi0BFIG. 1: (Color online) Classical spin S(t) coupled via an an- tiferromagnetic local exchange interaction of strength Jto a system of conduction electrons hopping with nearest-neighbor hopping amplitude Tover the sites of a one-dimensional lat- tice with open boundaries. The spin couples to the central sitei0of the system and is subjected to a local magnetic eld of strength B. The equation of motion for hsiitreads as d dthsiit=ii0JS(t)hsiit +Tn:n:X j1 2iX 0(hcy i0cj0itc.c.);(9) where the sum runs over the nearest neighbors of i. The second term on the right-hand side describes the coupling of the local conduction-electron spin to its environment and the dissipation of spin and energy into the bulk of the system (see below). Apparently, the system of equa- tions of motion can only be closed by considering the complete one-particle density matrix Eq. (3). It obeys a von Neumann equation of motion, id dt(t) = [T(t);(t)] (10) as is easily derived, e.g., from the Heisenberg equation of motion for the annihilators and creators. As is obvious from the equations of motion, the real- time dynamics of the quantum-classical Kondo-impurity model on a lattice with a nite but large number of sites L can be treated numerically exactly (see also below). Nev- ertheless, the model comprises highly non-trivial physics as the electron dynamics becomes eectively correlated due to the interaction with the classical spin. In addi- tion, the eective electron-electron interaction mediated by the classical spin is retarded: electrons scattered from the spin at time twill experience the eects of the spin torque exerted by electrons that have been scattered from the spin at earlier times t0<t. III. COMPUTATIONAL DETAILS Eqs. (5), (7), (8) and (10) represent a coupled non- linear system of rst-order ordinary dierential equations which can be solved numerically. By blocking up the von Neumann equation (10), the dierential equations are written in a standard form _y=f(y(t);t), wherey(t) is a high-dimensional vector, such that an explicit Runge- Kutta method can be applied. A high-order propagation4 technique is used45which provides the numerically exact solution up to 6-th order in the time step
t. For a typ- ical system consisting of about L= 103sites this implies that106coupled equations are solved. We consider a one-dimensional system with open boundaries consisting of L= 1001 sites and a local per- turbation at the central site i0of the system, see Fig. 1. For a half-lled tight-binding conduction band the Fermi velocityvF= 2Troughly determines the maximum speed of the excitations and denes a \light cone".46,47This means that nite-size eects due to scattering at the sys- tem boundaries become relevant after a propagation time tmax500 (in units of 1 =T). A time step
t= 0:1 is usually sucient for reliable numerical results up to tmax, i.e., about 5000 time steps are performed. The compu- tational cost is moderate, and calculations can be per- formed in a few hours on a standard desktop computer. Assuming, for example, that B= (0;0;B), the Hamil- tonian is invariant under rotations around the zaxis. It is then easily veried that not only the length of the spin jSj= 1=2 is conserved but also the total number of con- duction electrons, Ntot=X ihcy icii; (11) thez-component of the total spin, Stot;z=Sz+X ihsizi; (12) as well as the total energy, Etot=hHi= tr ((t)T(t))BS(t): (13) Despite the fact that the model does not include a di- rect (e.g., Coulomb) interaction among the conduction electrons, the average occupation numbers of the basis of one-particle states in which the hopping matrix T(t) is diagonal at time tarenotconserved. This is due to the eective retarded interaction mediated by the classical spin. Hence, the system is not integrable, unlike a free fermion gas. The conservation of the above-mentioned global observables serves as a sensitive check for the ac- curacy of the numerical procedure. IV. COUPLED SPIN AND ELECTRON DYNAMICS A. Spin relaxation Fig. 2 shows the real-time dynamics of the classical spin forJ= 1. Energy and time units are xed by the nearest-neighbor hopping T= 1 throughout the paper. Initially, for t= 0, the spin is oriented (almost) antipar- allel to the external local eld B= (0;0;B) withB= 1, i.e., initially Sx(0) =1 2sin#,Sy(0) = 0,Sz(0) =1 2cos# -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 xyz 10−1100101102 timet-0.50-0.250.000.250.50/vectorSSx SzFIG. 2: (Color online) Real-time dynamics of the classical spin. Upper panel: Bloch sphere representation. Lower panel: xandzcomponent of S(t) (jSj= 1=2). Calculations for exchange coupling J= 1 and eld strength B= 1 and for a system of L= 1001 sites. ( L= 1001 is kept xed for the rest of the paper). Energy and time units are xed by the nearest-neighbor hopping T= 1. where a non-zero but small polar angle #==50 is nec- essary to slightly break the symmetry of the initial state and to start the dynamics. For the same setup, the Landau-Lifschitz-Gilbert equa- tion would essentially predict two eects: rst, a preces- sion of the classical spin around the eld direction with Larmor frequency !L=Bz, and second, a relaxation of the spin to the equilibrium state with Sparallel to B=Bezfort!1 . Both eects are also found in the full dynamics of the quantum-classical hybrid model. The frequency of the oscillation of Sx(t) that is seen in Fig. 2 is!L, andSzis reversed after a few hundred time units. The precessional motion is easily explained by the torque on the spin exerted by the eld according to Eq. (7). The explanation of the damping eect is more in- volved: Even for the high eld strength considered here, the spin dynamics is slow as compared to the characteristic electronic time scale such that it could be reasonable to assume the electronic system being in its instantaneous5 102101100101102timet3.083.103.123.14~S(t)h~si0it(t)~B(t)J=46810J=20xy FIG. 3: (Color online) Time dependence of the angle (t) enclosed by S(t) andhsi0itforB= 0:1 and dierent Jas indicated. ground state at any instant of time and corresponding to the conguration of the classical spin. This, however, would imply that the expectation value hsi0itof the local conduction-electron moment at i0is always strictly par- allel toS(t) and, hence, there would not be any torque mediated by the exchange coupling JonS(t). In fact, the direction of hsi0itis always somewhat be- hind the \adiabatic direction", i.e., behind S(t): This is shown in Fig. 3 for a eld of strength B= 0:1 where the spin dynamics is by a factor 10 slower, compared to Fig. 2, and for dierent stronger exchange couplings J. Even in this case the process is by no means adiabatic, and the angle (t) betweenS(t) andhsi0itis close to but clearly smaller than =at any instant of time. This non-adiabaticity results from the fact that the motion of the classical spin aects the conduction electrons in a retarded way, i.e., it takes a nite time until the local conduction-electron spin hsi0itati0reacts to the motion of the classical spin. This retardation eect results in a torque Jhsi0it S(t)6= 0 exerted on the classical S(t) in the +zdirection which adds to the torque due to Band which drives the spin to its new equilibrium direction. Hence, retardation is the physical origin of the Gilbert spin damping. With increasing time, the deviation of (t) from the instantaneous equilibrium value =increases in mag- nitude, i.e., the direction of hsi0itis more and more be- hind the adiabatic direction, and the torque increases. Its magnitude is at a maximum at the same time when the oscillatingx;ycomponents of S(t) are at a maximum (see Fig. 2). The z-component of the torque does not vanish before the spin has reached its new equilibrium positionS(t)/ez. Fig. 3 shows results for dierent J. Generally, non- adiabatic eects show up if the typical time scale of the dynamics is faster than the relaxation time, i.e., the time necessary to transport the excitation away from the lo- cationi0where it is created initially. Roughly this time scale is set by the inverse hopping 1 =T. One therefore expects that, for xed T, a stronger Jimplies a stronger 0 20 40 60 80 100 exchange coupling J050100150200250300350400reversal time τ100∗τ1/τ2 τ1 τ2FIG. 4: (Color online) Time for a spin reversal Sz=1 2cos( #) =1 2cos(#)!Sz=1 2cos(#) for#=#1==50 (reversal time1) and for#=#2==25 (reversal time 2). Calcula- tions as a function of Jfor xedB= 0:1. retardation of the conduction-electron dynamics. The re- sults for dierent Jshown in Fig. 3 in fact show that the maximum deviation of (t) from the adiabatic direction =increases with increasing J(for very strong Jthe dynamics becomes much more complicated, see below). This results in a stronger torque on S(t) inzdirection and thus in a stronger damping. The picture is also qual- itatively consistent with the LLG equation as the Gilbert damping constant increases with J. The spin (almost) reverses its direction after a nite reversal time which is shown in Fig. 4 as a function ofJ. Calculations have been performed for an initial direction of the classical spin with Sz(0) =(1=2) cos# with two dierent polar angles #1;2. If#is suciently small, the results for dierent #are expected to dier by aJandBindependent constant factor only. As Fig. 4 demonstrates, the ratio 1=2is in fact nearly constant. For weakJand up to coupling strengths of about J.30, we nd that the reversal time decreases with increasing J. With increasing J, the retardation eect increases, as discussed above, and the stronger damping results in a shorter reversal time. 0 2 4 6 8 10 1/B0100200300400reversal time τ J= 2 J= 4 FIG. 5: (Color online) Reversal time 1as function of Bfor xedJ= 2 andJ= 4 as indicated.6 The prediction of the LLG equation for the reversal time of a single spin can be derived analytically48and is given by /1 +2 1 Bln1=2Sz(0) 1=2 +Sz(0): (14) However, down to the smallest Jfor whichcan be cal- culated reliably, our results for the full spin dynamics do not scale as /1=J2as one would expect for weak J assuming that /J2(see discussion in Sec. V B). TheBdependence of the reversal time is shown in Fig. 5. With increasing eld strength, the classical spin S(t) precesses with a higher Larmor frequency !LBaround thezaxis. Hence, non-adiabatic eects increase. The stronger the eld, the more delayed is the precessional motion of the local conduction-electrons spin hsi0it. This results in a stronger torque in + zdirection exerted on the classical spin. Therefore, the relaxation is faster and the reversal time smaller. For weak and intermediate eld strengths, is roughly proportional to 1 =B. This is consistent with the prediction of the LLG equation, see Eq. (14). In the limit of very strong elds one would expect an increase of the reversal time with increasing Bsince the eld term will eventually dominate the dynamics, i.e., only the precessional motion survives which implies a di- verging reversal time. In fact, for a eld strength exceed- ing a critical strength Bc, which depends on J, there is no full relaxation any longer, and =1. This strong- Bregime cannot be captured by the LLG equation and deserves further studies. The strong- Jregime is interesting as well. For cou- pling strengths exceeding J30 the reversal time in- creases withJ(see Fig. 4). Eventually, the reversal time must even diverge. This is obvious as the dynamics is described by a simple two-spin model in the limit J=1 which cannot show spin relaxation. The corresponding equations of motion are obtained by from Eqs. (7) and (9) by setting t= 0 andB= 0: d dtS(t) =Jhsi0itS(t); d dthsi0it=JS(t)hsi0it (15) Note that we have jhsi0itj= 1=2 forJ!1 . The equa- tions are easily solved by exploiting the conservation of the total spin Stot=S(t) +hsi0it. Both spins precess with constant frequency !0=JStotaroundStot. Their components parallel to Stotare equal, and their compo- nents perpendicular to Stotare anti-parallel and of equal length. However, the two-spin dynamics of the J=1limit is not stable against small perturbations. Fig. 6 shows the classical spin dynamics of the full model (with T= 1 andB= 0:1) for a very strong but nite coupling J= 100. Here, the motion of the classical spin gets very complicated as compared with the highly regular -0.50.00.5 /vectorSx y z -2.00.02.0 J/angbracketleft/vector si0/angbracketright×/vectorS -0.50.00.5 −/vectorB×/vectorS 10−1100101102 timet3.103.153.20 γ=/negationslash(/vectorS/vector s0)FIG. 6: (Color online) Real-time dynamics in the strong- J regime:J= 100 and B= 0:1.First panel: Classical spin S(t).Second panel: Torque on S(t) due to the exchange interaction. Third panel: Torque on S(t) due to the eld term. Fourth panel: Angle enclosed by S(t) andhsi0it behavior in the weak- Jregime (cf. Fig. 2). In particular, thez-component of S(t) is oscillating on nearly the same scale as the xandycomponents. This characteristic time scale
t4 of the oscillation corresponds to a frequency!1:5 which diers by more than an order of magnitude from both, the Larmor frequency B= 0:1 and from the exchange-coupling strength J= 100. Note that the oscillation of Sz(t) is actually the reason for the ambiguity in the determination of the precise reversal time and gives rise to the error bars in Fig. 4 for strong J. We attribute the complexity of the dynamics to the fact that the torque due to the eld term and the torque due to the exchange coupling are of comparable magni- tudes, see second and third panel in Fig. 6. It is interest- ing that even for strong J, where one would expect S(t) andhsi0itto form a tightly bound local spin-zero state, there is actually a small deviation from perfect antiparal- lel alignment, i.e., (t)6=, as can be seen in the fourth panel of Fig. 6. This results in a nite z-component of the torque onS(t) which leads to a very fast reversal with Sz(t)+0:5 at timet2:1. Contrary to the weak- coupling limit, however, the z-component of the torque changes sign at this point and drives the spin back to the7 z-direction. At t3:2, however, the z-component of S(t) once more reverses its direction. Here, the torque due to the exchange coupling vanishes completely as S(t) andhsi0itare perfectly antiparallel (see the rst zero of (t) in the fourth panel). The motion continues due to the non-zero eld-induced torque. This pattern repeats several times. S(t) mainly oscillates within a plane in- cluding and slowly rotating around the z-axis. Eventually, there is a perfect relaxation of the classical spin for large tbut in a very dierent way as compared to the weak-coupling limit. While the deviation from = is small at any instant of time as for weak J, the most apparent dierence is perhaps that the new ground state is approached with an oscillating behavior of (t) around =, i.e.,hsi0itmay run behind or ahead ofS(t) as well. Let us summarize at this point the main dierences between the quantum-classical hybrid and the eective LLG dynamics of the classical spin: For weak JandB, the qualitative behavior, precessional motion and relax- ation, is the same in both approaches. Quantitatively, however, the LLG equation is inconsistent with the ob- servedJdependence of the reversal time when assuming /J2. TheBdependence of 1 =is linear as expected from the LLG approach. For strong J, the spin dynamics qualitatively diers from LLG dynamics and gets more complicated with a new time scale emerging. Absence of complete relaxation, as observed in the strong- Blimit, is also not accessible to an eective spin-only theory. B. Energy dissipation To complete the picture of the relaxation dynamics of the classical spin, the discussion should also comprise the dynamics of the electronic degrees of freedom. The spin relaxation must be accompanied by a dissipation of energy and spin into the bulk of the electronic system since the total energy and the total spin are conserved quantities, see Eqs. (12) and (13), while conservation of the total particle number, Eq. (11), is trivially ensured by the particle-hole symmetric setup considered here where the average conduction-electron number at every site is time-independent:P hniit= 1. The total energy is given by Etot=hHi, see Eq. (13), and is a sum over dierent contributions, Etot=EB(t)+ Ehop(t)+Eint(t), namely the interaction energy with the eld EB(t) =BS(t); (16) the kinetic (hopping) energy of the conduction-electron system Ehop(t) =TX hijiX hcy icjit; (17) and the exchange-interaction energy Eint(t) =Jhsi0itS(t): (18) -0.50.00.5 EB -1.2724-1.2718-1.2712energyEhop/L Etot/L -0.68-0.67-0.66 Eint 100101102 timet−1.30−1.25−1.20−1.15 ei0±1 ei0±3 ei0±2ei0±50ei0±100FIG. 7: (Color online) Dierent contributions to the total energy as functions of time for J= 5 andB= 1. Top panel : EB, energy of the classical spin in the external eld. Second panel :Ehop=L, kinetic (hopping) energy of the conduction- electron system per site, and Etot=L, total energy per site. Third panel :Eint, exchange-interaction energy. The fourth panel shows the time-dependence of the total-energy density at dierent distances from the site i0. The time dependence of those contributions is shown in Fig. 7 forJ= 5 andB= 1. The top panel of Fig. 7 shows that the system re- leases the interaction energy j2BSjof the classical spin in the external eld by aligning the spin to the eld di- rection. In the long-time limit, this energy is stored in the conduction-electron system: The average kinetic en- ergy per site ( L= 1001) increases by the same amount as shown in the second panel (note the dierent scales). The exchange-interaction energy changes with time but is the same for t= 0 andt!1 (see third panel). The total energy is constant (second panel). The relaxation of the classical spin in the external eld Bimplies an energy ow away from the site i0into the bulk of the conduction-electron system such that locally , in the vicinity of i0the system is in its new ground state. To discuss this energy ow, it is convenient to consider the total energy as a lattice sum Etot=P iei(t) over the8 i0−500 i0 i0+ 500 sites0255075100125150175200225250time -1.3-1.2-1.1-1.0-0.9-0.8 total energy density ei FIG. 8: (Color online) Spatiotemporal evolution of the total- energy density ei(t), as dened in Eq. (19). Calculation for J= 5,B= 1. total energy \density" dened as ei(t) =Tn:n:(i)X jX hcy icjit+ii0(Jhsi0itB)S(t); (19) where the sum over jruns over the nearest neighbors of sitei. The time dependence of the energy density in the vicinity ofi0and at distances 50 and 100 is shown in the fourth panel of Fig. 7. At any site in the conduction-electron system, the energy density increases from its ground-state value, reaches a maximum and eventually relaxes to the en- ergy density of the new ground state. Since the latter is just the ground state with the reversed classical spin, the new ground-state energy density is the same as in the initial state at t= 0. As can be seen in the fourth panel of Fig. 7, there is also a slight spatial oscillation of the ground-state energy density which just re ects the Friedel oscillations around the impurity at i0. Complete relaxation means that the excitation energy is completely removed from the vicinity of i0and trans- ported into the bulk of the system. That this is in fact the case can be seen by comparing the energy density at dierent distances from i0. It is also demonstrated by Fig. 8 which visualizes the energy-current density which symmetrically points away from i0: The total energy of the excitation owing through each pair of sites i0
iis constant, i.e., the time-integrated energy ux,R dtei(t) is the same for all i. As is seen in Fig. 7 (fourth panel) and Fig. 8, there is a considerable dispersion of the excitation wave packet carrying the energy. For example, at i0+ 100 it takes more than four times longer, as compared to i0+ 1, un- til most of the excitation has passed through (note the logarithmic time scale). The broadening of the wave packet, due to dispersion, is asymmetric and bound by an upper limit for the speed of the excitation which is roughly set by the Fermi veloc- ityvF= 2T. This Lieb-Robinson bound46,47determines i0500i0i0+5 0 0sites050100150200250300350400450500time -0.06-0.030.000.030.06hszii i0500i0i0+5 0 0sites050100150200250300350400450500time -0.04-0.020.000.020.04hsxiisx szFIG. 9: (Color online) Spatiotemporal evolution of the total- spin densityhsiit(upper panel: x-component, lower panel z- component) for J= 5 andB= 0:1. See Fig. 10 for snapshots at times indicated by the arrows. the \light cone" seen in Fig. 8. C. Spin dissipation The same upper speed limit, given by the Fermi ve- locity of the conduction-electron system, is also seen in the spatiotemporal evolution of the conduction-electron spin densityhsiit. This is shown in Fig. 9 for a dierent magnetic eld strength B= 0:1 where the classical spin dynamics is slower. Apparently, the wave packet of exci- tations emitted from the impurity not only carries energy but also spin. It symmetrically propagates away from i0 and, att300, reaches the system boundary where it is re ected perfectly. Up to t= 500 there is hardly any eect visible in the local observables close to i0that is aected by the nite system size. Snapshots of the conduction-electron spin dynamics are shown in Fig. 10 for the initial state at t= 0 and for states at four later times t >0 which are also indicated by the arrows in Fig. 9. At t= 0 the conduction-electron system is in its ground state for the given initial direction of the classical spin. The latter basically points into the zdirection, apart from a small positive x-component (#==50) which is necessary to break the symmetry9 -0.0030.0000.003 -0.0030.0000.003 -0.0030.0000.003/angbracketleft/vector si/angbracketright -0.0030.0000.003sx sz i0−100i0−60i0−20-0.0030.0000.003 t= 0 t= 80t= 60 t= 100sz sx i0i0+ 100 i0+ 500 sites t= 250 FIG. 10: (Color online) Snapshots the of conduction-electron magnetic moments hsiitat dierent times tas indicated on the right and by the corresponding arrows in Fig. 9. Red lines: z-components ofhsiit. Blue lines: xcomponents. The proles are perfectly symmetric to the impurity site i=i0but displayed up to distances jii0j100 on the left-hand side and up to the system boundary, jii0j500, on the right-hand side. Parameters J= 5;B= 0:1. of the problem and to initiate the dynamics. This tiny eect will be disregarded in the following. From the perspective of the conduction-electron sys- tem, the interaction term JSsi0acts as a local external magnetic eld JSwhich locally polarizes the conduction electrons at i0. SinceJis antiferromagnetic, the local momenthsi0ipoints into the + zdirection. At half-lling, the conduction-electron system exhibits pronounced an- tiferromagnetic spin-spin correlations which give rise to an antiferromagnetic spin-density wave structure aligned to thezaxis att= 0, see rst panel of Fig. 10. The total spin Stot= 0 att= 0, i.e., the classical spinSis exactly compensated by the total conduction- electron spinhstoti=P ihsii=Sin the ground state. This can be traced back to the fact that for a D= 1- dimensional tight-binding system with an odd number of sitesL, withN=Land with a single static magnetic impurity, there is exactly one localized state per spin pro- jection, irrespective of the strength of the impurity po- tential (here given by JS= 0:5Jez). The number of " one-particle eigenstates therefore exceeds the number of #states by exactly one. Since the energy of the excitation induced by the ex- ternal eldBis completely dissipated into the bulk, the state of the conduction-electron system at large t(but shorter than t500 where nite-size eects appear) must locally, close to i0, resemble the conduction-electron ground state for the reversed spin S= +0:5ez. This implies that locally all magnetic moments hsiitmust re- verse their direction. In fact, the last panel in Fig. 10(left) shows that the new spin conguration is reached fort= 250 at sites with distance jii0j.100, see dashed line, for example. For later times the spin con- guration stays constant (until the wave packet re ected from the system boundaries reaches the vicinity of i0). The reversal is almost perfect, e.g., hsi0it=0= 0:2649! hsi0it250=0:2645. Deviations of the same order of magnitude are also found at larger distances, e.g., i=i0100. We attribute those tiny eects to a weak de- pendence of the local ground state on the non-equilibrium state far from the impurity at t= 250, see right part of the last panel in Fig. 10. The other panels in Fig. 10 demonstrate the mecha- nism of the spin reversal. At short times (see t= 60, second panel) the perturbation of the initial equilibrium conguration of the conduction-electron moments is still weak. For t= 80 andt= 100 one clearly notices the emission of the wave packet starting. Locally, the an- tiferromagnetic structure is preserved (see left part) but superimposed on this, there is an additional spatial struc- ture of much longer size developing. This nally forms the wave packet which is emitted from the central re- gion. Its spatial extension is about
300 as can be estimated for t= 250 (last panel on the right) where it covers the region 200 .i.500. The same can be read o from the upper part of Fig. 9. Assuming that the reversal of each of the conduction-electron moments takes about the same time as the reversal of the classi- cal spin,
is roughly given by the reversal time times the Fermi velocity and therefore strongly depends on J10 andB. For the present case, we have 1150=Twhich implies
1502 = 300 in rough agreement with the data. In the course of time, the long-wave length structure superimposed on the short-range antiferromagnetic tex- ture develops a node. This can be seen for t= 100 and i40 (fourth panel, see dashed line). The node marks the spatial border between the new (right of the node, closer toi0) and the original antiferromagnetic structure of the moments and moves away from i0with increasing time. At a xed position i, the reversal of the conduction- electron moment hsiittakes place in a similar way as the reversal of the classical spin (see both panels in Fig. 9 for a xed i). During the reversal time, its xandy components undergo a precessional motion while the z component changes sign. Note, however, that during the reversaljhsiijgets much larger than its value in the initial and in the nal equilibrium state. V. EFFECTIVE CLASSICAL SPIN DYNAMICS A. Perturbation theory Eqs. (7) and (9) do not form a closed set of equations of motion but must be supplemented by the full equation of motion (10) for the one-particle conduction-electron den- sity matrix. This implies that the fast electron dynamics must be taken into account explicitly even if the spin dynamics is much slower. Hence, there is a strong mo- tivation to integrate out the conduction-electron degrees of freedom altogether and to take advantage from a much larger time step within a corresponding spin-only time- propagation method. Unfortunately, a simple eective spin-only action can be obtained in the weak-coupling (small-J) limit only.13,14This weak-coupling approxima- tion is also implicit to all eective spin-only approaches that consider the eect of conduction electrons on the spin dynamics.49 In the weak- Jlimit the electron degrees of freedom can be eliminated in a straightforward way by using standard linear-response theory:50We assume that the initial state att= 0 is given by the conduction-electron system in its ground state or in thermal equilibrium and an arbitrary state of the classical spin. This may be realized formally by suddenly switching on the interaction J(t) at time t= 0, i.e.,J(t) =J(t) and by switching the local eld from some initial value Biniatt= 0 to a nal value B fort >0. The response of the conduction-electron spin ati0and timet >0 (hsi0it= 0 fort= 0) due to the time-dependent perturbation J(t)S(t) is hsi0it=JZt 0dt0(ret)(t;t0)
S(t0) (20) up to linear order in J. Here, the free ( J= 0) local retarded spin susceptibility of the conduction electrons(ret)(t;t0) is a tensor with elements (ret) (t;t0) =i(tt0)h[s i0(t);s i0(t0)]i; (21) where; =x;y;z . Using this in Eq. (7), we get an equation of motion for the classical spins only, d dtS(t) =S(t)B J2S(t)Zt 0dt0(ret)(tt0)
S(t0) (22) which is correct up to order J2. This represents an equation of motion for the classical spin only. It has a temporally non-local structure and includes an eective interaction of the classical spin at timeS(t) with the same classical spin at earlier times t0< t. In the full quantum-classical theory where the electronic degrees of freedom are taken into account ex- actly, this retarded interaction is mediated by a non- equilibrium electron dynamics starting at site i0and time t0and returning back to the same site i0at timet > t0. Here, for weak J, this is replaced by the equilibrium and homogeneous-in-time conduction-electron spin suscepti- bility (ret)(tt0). Compared with the results of the full quantum-classical theory, we expect that the pertur- bative spin-only theory breaks down after a propagation timet1=Jat the latest. Using Wick's theorem,50the spin susceptibility is eas- ily expressed in terms of the greater and the lesser equi- librium one-particle Green's functions, G> ii;0(t;t0) = ihci(t)cy i0(t0)iandG< ii;0(t;t0) =ihcy i0(t0)ci(t)i, re- spectively: (ret) 0(tt0) = (tt0)1 2 Im tr 22h G> i0i0(t;t0)0G< i0i0(t0;t)i :(23) Assuming that the conduction-electron system is charac- terized by a real, symmetric and spin-independent hop- ping matrix Tij(as given by the rst term of Eq. (2)), G>andG<are unit matrices with respect to the spin indices. They are easily expressed as explicit functions ofT(see Ref. 51, for example). With tr ( 0) = 20, we nd (ret) 0(tt0) = (tt0)0Im  eiT(tt0) 1 +e(T)! i0i0 eiT(t0t) e(T)+ 1! i0i0(24) for a conduction-electron system at inverse temperature and chemical potential . Using Eq. (24) we have computed the spin suscepti- bility for the ground state ( =1) of the conduction- electron system. This xes the kernel in the integro- dierential equation Eq. (22) which is solved numeri- cally by standard techniques.52We again consider the11 10−1100101102 timet-0.50-0.250.000.250.50/vectorSJ= 1.0Sz Sx FIG. 11: (Color online) Components Sx(t) andSz(t) of the classical spin after a sudden switch of the eld from xto zdirection at t= 0. Calculations for B= 1 andJ= 1. Solid lines: results of the linear-response dynamics, Eq. (22). Dashed lines: results of the exact quantum-classical dynamics forL= 1001. system displayed in Fig. 1 with a single classical spin coupled via Jto the central site i0of a chain consisting ofL= 1001 sites. Finite-size artifacts do not show up beforetmax= 500. Figs. 11 and 12 show the resulting linear-response dy- namics of the classical spin after preparing the initial state of the system with the classical spin pointing into +xdirection while B= (0;0;B). The external magnetic eld induces a precessional motion of the classical spin: there is a rapid oscillation of its xcomponent (and of its ycomponent, not shown) with frequency !B(blue lines). Damping is induced by dissipation of energy and spin: for large times, the zcomponent aligns to the ex- ternal eld (red lines). For weak coupling, up to J= 1 (Fig. 11), there is an al- most perfect agreement between the results of the exact quantum-classical dynamics (full lines) and the linear- response theory (dashed lines) up to the maximum prop- agation time tmax= 500. We note that, compared to the full theory, there is a tiny deviation of the linear- response result for the zcomponent of S(t) visible in Fig. 11 for times t&10. Hence, on this level of accuracy, t110 sets the time scale up to which the linear-response theory is valid. This may appear surprising as this im- pliest1J= 10 for the \small" dimensionless parameter of the perturbation theory. One has to keep in mind, however, that even if the perturbation is \strong", its eects can be rather moderate since only non-adiabatic termsS(t)S(t0) contribute in Eq. (22). ForJ= 3, see Fig. 12, damping of the classical spin sets in much earlier. Visible deviations of the linear- response theory from the full dynamics already appear on a time scale that is almost two orders of magnitude smaller as compared to the case J= 1. A simple rea- soning based on the argument that the dimensionless expansion parameter is t1Jfails as this disregards the strong enhancement of retardation eects with increas- 10−1100101 timet-0.50-0.250.000.250.50/vectorSJ= 3.0Sz SxFIG. 12: (Color online) The same as Fig. 11 but for a dierent exchange coupling constant J= 3. ingJ, which have been discussed in Sec. IV A. These eects make the perturbation much more eective, i.e., lead to a torque, which is exerted by the conduction elec- trons on the classical spin, growing stronger than linear inJ. We conclude that linear-response theory is highly at- tractive formally as it provides a tractable spin-only eec- tive theory. On the other hand, substantial discrepancies compared to the full (non-perturbative) theory show up as soon as damping eects become stronger. Note that, with increasing time, these deviations must diminish and disappear eventually since both, the full and the eec- tive theory, predict a fully relaxed spin state for t!1 { see lower panel of Fig. 12, for example. At least for simple systems with a single classical spin, as considered here, this implies that the eective theory provides qual- itatively reasonable results. B. Landau-Lifschitz-Gilbert equation To derive the LLG equation, the linear-response the- ory must be further simplied:14,53As is obvious from Eq. (22), the spin-susceptibility (ret)(tt0) can be inter- preted as an eective retarded self-interaction of the spin. We assume that the electron dynamics is much faster than the spin dynamics. On the time scale of the spin dynamics, the self-interaction then takes place almost in- stantaneously, i.e., the memory kernel (ret) 0(tt0) = 0(ret)(tt0) in Eq. (22) is peaked at t0t. We can therefore approximate S(t0)S(t) + (t0t)_S(t) under the integral in Eq. (22). This immediately yields: dS(t) dt=S(t)B+(t)S(t)dS(t) dt; (25) where (t) =J2Zt 0d(ret)() (26)12 after substituting t07!=tt0. Eq. (25) takes the form of the standard LLG equation for a single classical spin [cf. Eq. (1)] if (t) is replaced by lim t!1(t) =J2Z1 0dtt(ret)(t): (27) Note that this is a necessary step to arrive at a constant damping parameter which can again be justied by not- ing that (ret)(t) is peaked at t= 0. Before proceeding, let us stress that Eq. (27) is, or is equivalent to, the standard expression used for com- puting the Gilbert damping constant in various studies: After Fourier transformation, (ret)(!) =Z dtei!t(ret)(t); (28) one ends up with =iJ2@ @!(ret)(!) !=0=J2@ @!Im (ret)(!) !=0; (29) which has also been derived, e.g., in Refs. 14,54 in dif- ferent contexts. The frequency-dependent spin correla- tion (ret)(!) in Eq. (29) can be obtained explicitly as the Fourier transform of (ret)(t) given by Eq. (24). A straightforward calculation yields: = 2J2Z d!df(!) d!Aloc(!)Aloc(!); (30) wheref(!) = 1=(exp(!)+1) is the Fermi function, and Aloc(!) =L1X k(!+"(k)) (31) forL!1 is the local one-particle spectral function. Here we have assumed periodic boundary conditions, i.e., spatial homogeneity, such that the hopping matrix Tis diagonalized by Fourier transformation: Tij=X kUik"(k)Uy kj(32) withUik=L1=2exp(ikRi). The eigenvalues of T are given by the tight-binding dispersion "(k) of the conduction-electron Bloch band. Within our simple tight-binding model for the conduction-electron system, Eqs. (29) and (30) are equiv- alent with Kambersky's breathing Fermi-surface the- ory, related torque-correlation models and scattering theory and have frequently been used for ab initio as well as model computations of the Gilbert damping constant.15,18,20,21,55{58 Let us remark that Eq. (30) demonstrates that <0. This results from the convention for the coupling of the magnetic eld to the spin, namely H=H(B= 0)BS [see Eq. 2)], which has been adopted here. As a conse- quence, the precession of S(t) aroundBis described by aleft-hand helix, _S=S(t)B, and thus must be negative to describe damping. 10−1100101102103 timet−0.10−0.050.000.050.10t·Πret(t) Πret(t)FIG. 13: (Color online) Local retarded spin susceptibility of the conduction electrons (ret)(t) (red line) and t(ret)(t) (blue) as obtained from Eq. (33) for a one-dimensional conduction-electron system with L= 10000 sites and peri- odic boundary conditions. C. Ill-dened Gilbert damping The above discussion shows that Eq. (27) represents the fundamental denition of the Gilbert damping con- stantand that the limit t!1 is crucial to recover the LLG equation in its standard form. The existence of the long-time limit, however, decisively depends on the long- time behavior of the retarded spin-correlation function (ret)(t). Starting from Eq. (24), this is easily computed as (ret)(t) = (t)Im1 L2X k;pei"(k)t 1 +e("(k))ei"(p)t e("(p))+ 1: (33) Fig. 13 gives an example for the time-dependence of (ret)(t). The calculations have been done at half-lling, =1forL= 104sites. We note that the susceptibility is in fact peaked at t= 0. For long times, it oscillates with frequency != 4 and decays as 1 =t. This is an im- portant observation as it implies that the limit in Eq. (27) does not exist and that, therefore, the damping constant is ill-dened. To analyze the physical origin of the divergent integral, we rewrite the spin susceptibility in Eq. (33) as (ret)(t) = (t)Im[A(unocc) loc(t)A(occ) loc(t)]: (34) Its long-time behavior is governed by the long-time be- havior of the Fourier transform A(occ;unocc) loc(t) =Z d!ei!tA(occ;unocc) loc(!) (35) of the occupied, A(occ) loc(!)f(!)Aloc(!), and of the un- occupied,A(unocc) loc(!)f(!)Aloc(!), part of the spectral density, see Eq. (31). For functions with smooth !dependence, the Fourier transform generically drops to zero exponentially fast if13 t!1 . A power-law decay, however, is obtained if there are singularities of A(occ;unocc) loc(!). We can distinguish between van Hove singularities, which are, e.g., of the form/(!!0)(!!0)k(withk>1), and the step- like singularity/(!!0) (i.e.k= 0), arising in the zero-temperature limit at !0= 0 due to the Fermi func- tion. Generally, a singularity of order kgives rise to the asymptotic behavior A(occ;unocc) loc(t)/t1k, apart from a purely oscillatory factor ei!0t. For the present case, the van Hove singularities of A(occ;unocc) loc(!) at!0= 2 explain, via Eq. (34) the oscillation of (ret)(t) with fre- quency!= 2!0= 4. Generally, the location of the van Hove singularity on the frequency axis, i.e. !0, determines the oscilla- tion period while the decay of (ret)(t) is governed by the strength of the singularity. Consider, as an exam- ple, the zero-temperature case and assume that there are no van Hove singularities. The sharp Fermi edge im- pliesA(occ;unocc) loc(t)/t1, and thus (ret)(t)/t2. The Gilbert-damping constant is well dened in this case. The strength of van Hove singularities depends on the lattice dimension D.59For a one-dimensional lat- tice, we have van Hove singularities with k=1=2, and thus (ret)(t)/t1, consistent with Fig. (13). Here, the strong van Hove singularity dominates the long-time asymptotic behavior as compared to the weaker Fermi- edge singularity. For D= 3, we have k= 1=2 and (ret)(t)/t3if <1while for=1the Fermi- edge dominates and (ret)(t)/t2. TheD= 2 case is more complicated: The logarithmic van Hove singularity /lnj!jleads to (ret)(t)/t2. This, however, applies to cases o half-lling only. At half-lling the van Hove and the Fermi-edge singularity combine to a singularity /(!) lnj!jwhich gives (ret)(t)/ln2(t)=t2. For nite temperatures, we again have (ret)(t)/t2. The existence of the integral Eq. (27) depends on the t!1 behavior and either requires a decay as (ret)(t)/ t3or faster, or an asymptotic form (ret)(t)/ei!0t=t2 with an oscillating factor resulting from a non-zero po- sition!06= 0 of the van Hove singularity. For the one- dimensional case, we conclude that the LLG equation (with a time-independent damping constant) is based on an ill-dened concept. Also, the derivation of Eqs. (29) and (30) is invalid in this case as the !derivative and the tintegral do not commute. This conclusion might change for the case of interacting conduction electrons. Here one would expect a regularization of van Hove singularities due to a nite imaginary part of the conduction-electron self-energy. VI. CONCLUSIONS Hybrid systems consisting of classical spins coupled to a bath of non-interacting conduction electrons rep- resent a class of model systems with a non-trivial real- time dynamics which is numerically accessible on longtime scales. Here we have considered the simplest vari- ant of this class, the Kondo-impurity model with a clas- sical spin, and studied the relaxation dynamics of the spin in an external magnetic eld. As a fundamental model this is interesting of its own but also makes con- tact with dierent elds, e.g., atomistic spin dynamics in magnetic samples, spin relaxation in spintronics devices, femto-second dynamics of highly excited electron systems where local magnetic moments are formed due to electron correlations, and articial Kondo systems simulated with ultracold atoms in optical lattices. We have compared the coupled spin and electron dy- namics with the predictions of the widely used Landau- Lifshitz-Gilbert equation which is supposed to cover the regime of weak local exchange Jand slow spin dynamics. For the studied setup, the LLG equation predicts a rather regular time evolution characterized by spin precession, spin relaxation and eventually reversal of the spin on a time scaledepending on J(and the eld strength B). We have demonstrated that this type of dynamics can be recovered and understood on a microscopic level in the more fundamental quantum-classical Kondo model. It is traced back to a non-adiabatic dynamics of the electron degrees of freedom and the feedback of the electronic sub- system on the spin. It turns out that the spin dynamics is essentially a consequence of the retarded eect of the local exchange. Namely, the classical spin can be seen as a perturbation exciting the conduction-electron system locally. This electronic excitation propagates and feeds back to the classical spin, but at a later time, and thereby induces a spin torque. We found that this mechanism drives the relaxation of the system to its local ground state irrespective of the strength of the local exchange J. As the microscopic dy- namics is fully conserving, the energy and spin of the initial excitation which is locally stored in the vicinity of the classical spin, must be dissipated into the bulk of the system in the course of time. This dissipation could be uncovered by studying the relaxation process from the perspective of the electron degrees of freedom. Dissipa- tion of energy and spin takes place through the emission of a dispersive spin-polarized wave packet propagating through the lattice with the Fermi velocity. In this pro- cess the local conduction-electron magnetic moment at any given distance to the impurity undergoes a reversal, characterized by precession and relaxation, similar to the motion of the classical spin. The dynamics of the classical spin can be qualitatively very dierent from the predictions of the LLG equation for strongJ. In this regime we found a complex mo- tion characterized by oscillations of the angle between the classical spin S(t) and the local conduction-electron magnetic moment at the impurity site hsi0iaround the adiabatic value =which takes place on an emergent new time scale. In the weak- Jlimit, the classical spin dynamics is qual- itatively predicted correctly by the LLG equation. At least partially, however, this must be attributed to the14 fact that the LLG approach, by construction, recovers the correct nal state where the spin is parallel to the eld. In fact, quantitative deviations are found during the relaxation process. The LLG approach is based on rst-order perturbation theory in Jand on the additional assumption that the classical spin is slow. To pinpoint the source of the deviations, we have numerically solved the integro-dierential equation that is obtained in rst- order-in-Jperturbation theory and compared with the full hybrid dynamics. The deviations of the perturbative approach from the exact dynamics are found to gradually increase with the propagation time (until the proximity to the nal state enforces the correct long-time asymp- totics). This is the expected result as the dimensionless small parameter is Jt. However, with increasing Jthe time scale on which perturbation theory is reliable de- creases much stronger than 1 =Jdue to a strong enhance- ment of retardation eects which make the perturbation more eective and produce a stronger torque. Generally, the perturbation can be rather ineective in the sense that it produces a torque /S(t)S(t0) which is very weak if the process is nearly adiabatic. This ex- plains that rst-order perturbation theory and the LLG equation is applicable at all for couplings of the order of hoppingJT. For the present study this can also be seen as a fortunate circumstance since the regime of very weak couplings JTis not accessible numerically. In this case the spin-reversal time scale gets so large that the propagation of excitations in the conduction-electron subsystem would by aected by backscattering from the edges of the system which necessarily must be assumed as nite for the numerical treatment. For the one-dimensional lattice studied here, a di- rect comparison between LLG equation and the exact quantum-classical theory is not meaningful as the damp- ing constant is ill-dened in this case. We could argue that the problem results from the strength of the vanHove singularities in the conduction-electron density of states which dictates the long-time behavior of the mem- ory kernel of the integro-dierential equation which is given by the equilibrium spin susceptibility. As the type of the van Hove singularity is characteristic for all sys- tems of a given dimension, we can generally conclude that the LLG approach reduces to a purely phenomenological scheme in the one-dimensional case. However, it is an open question, which will be interesting to tackle in the future, if this conclusion is still valid for systems where the Coulomb interaction among the conduction electrons is taken into account additionally. There are more interesting lines of research which are based on the present work and could be pursued in the future. Those include systems with more than a single spin where, e.g., the eects of a time-dependent and retarded RKKY interaction can be studied additionally. We are also working on a tractable extension of the the- ory to account for longitudinal uctuations of the spins to include time-dependent Kondo screening, and the competition with RKKY coupling, on a time-dependent mean-eld level. Finally, lattice rather than impurity variants of the quantum-classical hybrid model are highly interesting to address the time-dependent phase transitions. Acknowledgments We would like to thank M. Eckstein, A. Lichtenstein, R. Rausch, E. Vedmedenko and R. Walz for instruc- tive discussions. Support of this work by the Deutsche Forschungsgemeinschaft within the SFB 668 (project B3) and within the SFB 925 (project B5) is gratefully ac- knowledged. 1L. D. Landau and E. M. Lifshitz, Phys. Z. 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2015-07-29

The real-time dynamics of a classical spin in an external magnetic field and locally exchange coupled to an extended one-dimensional system of non-interacting conduction electrons is studied numerically. Retardation effects in the coupled electron-spin dynamics are shown to be the source for the relaxation of the spin in the magnetic field. Total energy and spin is conserved in the non-adiabatic process. Approaching the new local ground state is therefore accompanied by the emission of dispersive wave packets of excitations carrying energy and spin and propagating through the lattice with Fermi velocity. While the spin dynamics in the regime of strong exchange coupling J is rather complex and governed by an emergent new time scale, the motion of the spin for weak J is regular and qualitatively well described by the Landau-Lifschitz-Gilbert (LLG) equation. Quantitatively, however, the full quantum-classical hybrid dynamics differs from the LLG approach. This is understood as a breakdown of weak-coupling perturbation theory in J in the course of time. Furthermore, it is shown that the concept of the Gilbert damping parameter is ill-defined for the case of a one-dimensional system.

Spin dynamics and relaxation in the classical-spin Kondo-impurity model beyond the Landau-Lifschitz-Gilbert equation

1507.08227v2

Fractional Driven Damped Oscillator Fernando Olivar-Romero and Oscar Rosas-Ortiz Physics Department, Cinvestav, AP 14-740, 07000 M exico City, Mexico Abstract The resonances associated with a fractional damped oscillator which is driven by an oscillatory external force are studied. It is shown that such resonances can be manipulated by tuning up either the coecient of the fractional damping or the order of the corresponding fractional derivatives. 1 Introduction The simplest oscillating system (a harmonic oscillator ) can be modeled by a mass at the end of a spring which slides back and forth without friction. The motion is characterized by the natural frequency of oscillation !0and the total stored energy E(which is a constant of motion and denes the amplitude of oscillation) [1]. Actual oscillating systems present some loss of energy due to friction forces so that the amplitude of their oscillations is a decreasing function of time. However, the oscillations can be driven to avoid their damping down by the action of a repetitive force F(t) on the system. Such a system is called driven damped oscillator [2]. An special excitation of the system arises when the frequency of the applied force matches the natural frequency of the oscillator since the spectral energy distribution takes its maximum value. The phenomenon, known as resonance , is a subject of study in classical mechanics, electromagnetism, optics, acoustics and quantum mechanics, among other physical theories [3]. The present work is addressed to the study of the driven damped oscillator in the con- text of fractional calculus [4{6]. That is, the second-order dierential equation associated with the Newtonian law of motion for a damped oscillator that is driven by an external force will be substituted by a fractional dierential equation of order 2 , with 0<1. Special emphasis will be placed on the resonance phenomenon. 2 Fractional Oscillator with fractional damping Given an oscillator of natural frequency !0, the general expression for the displacement of the mass can be expressed as the integral equation [8,9] x(t) =x0+It( _x0) +!2 0It[It(x(t))]; (1) 1arXiv:1706.08596v1 [physics.class-ph] 15 Jun 2017wherex0and _x0are constants of integration, and It(x(t)) :=Z x(t)dt (2) represents the Riemman time-integration of x(t). The fractional generalization of (1) is performed in two steps. First we replace Itwith the Riemman-Liouville fractional time- integral operator I[4{6], and!0with! 0. The latter for consistency of units. Then we have x(t) =x0+I( _x0) +!2 0I[I(x(t))];0<1: (3) Now, a fractional dierential form of (3) can be obtained by applying twice the time- fractional derivative operator of Caputo DDx(t) +!2x(t) = 0 (4) (for details about the operator Dsee, e.g., [5]). Let us introduce a `fractional damping' which is proportional to the fractional time-derivative of the position Dx(t). That is DDx(t) + 2Dx(t) +!2x(t) = 0: (5) One can show that the solution of this last equation is of the form x(t) =x0t p 2!2 0[E;1( t)E;1( +t)] +2x0+x() 0p 2!2 0[E;1( t)E;1( +t)];(6) where E;(z) =1X k=0zk (k+);Re()>0;Re()>0; z2C; (7) is the Mittag-Leer function [7], and =q 2!2 0: 3 Driven fractional oscillator with fractional damp- ing Let us add a driving force at the right hand side of Eq. (5), we have DDx(t) +Dx(t) +!2x(t) =f0cos (!t+); (8) wheref0is a real constant. Applying the Laplace transform Land solving for X(s) = L[x(t)] we arrive at the expression X(s) =f0scos!sin (s2+!2) (s2+ 2s+!2 0) +x0s21+ x0+ 2x() 0 s1 s2+ 2s+!2 0: (9) 2Makingf0= 0 we see that the second term in (9) corresponds to transient oscillations because there is no force present which can ensure their predominance; the corresponding inverse Laplace transform has been evaluated in the previous section. In turn, for the inverse Laplace transform of the rs term one has x(t) =f0 2ilim T!1Zg+iT giTestscos!sin (s2+!2) (s2+ 2s+!2 0) ds: (10) The integrand contains a branch point at s= 0 and simple poles at s=i!, and s= ( ei)1=. Following [10] we nd that x(t) is given as the sum of three contributions: x1(t),x2(t) andx3(t). The rst one results from the calculation of (10) along the Hankel- Bromwich path shown in Fig. 1 of Ref. [10], we obtain x1(t) =f0 Z1 0ert[rcos +!sin ] r2sin(2) + 2rsin() (r2+!2) [r4+ 42r2+!4+ 4r3cos() + 4r!2 0+ 2r2!2 0cos(2)]dr: (11) The latter expression vanishes as t! 1 . On the other hand, the sum of residues associated with the poles s= ( ei)1=gives x2(t) =2f0et +cos(=) 1 +(   +)" !2 +coscos t +sin(=) (2)
+ 3 +coscos (t +sin(=)) !4+ 4 ++ 2!2 2 +cos (2=)# +2f0et cos(=) 1 (  +  )" !2 coscos t sin(=) (2)
+ 3 coscos (t sin(=)) !4+ 4 + 2!2 2 cos (2=)# +2f0et +cos(=) 1 +(   +)" !3sincos t +sin(=) (1)
+!2 2 +sincos t +sin(= ) (+ 1)
!4+ 4 ++ 2!2 2 +cos (2=)# +2f0et cos(=) 1 (  +  )" !3sincos t sin(=) (1)
+!2 2 sincos t sin(= ) (+ 1)
!4+ 4 + 2!2 2 cos (2=)# ; (12) where = 1= . The term x2(t) is parameterized by the order of the Caputo operator D, the coecient of the fractional damping, and the natural frequency !0; the ap- propriate combination of these three parameters produces x2!0 ast!1 . Further details will be reported elsewhere. On the other hand, the term associated with the poles s=i!is of the form x3(t) =Acos(!t+); (13) where the amplitude and phase are respectively given by A=f0q !4+!4 0+ 2!2!2 0cos (2) + 4! !+ (!2+!2 0) cos 22
 !4+!4 0+ 2!2!2 0cos () + 4! !+ (!2+!2 0) cos 2
;(14) and = arctan" !2sin () +!2 0sin () + 2!sin 2
!2cos () +!2 0cos () + 2!cos 2
# : (15) 3As in the conventional case, the amplitude Aof the oscillations dictated by x3(t) is proportional to the amplitude f0of the driving force. At zero frequency !(i.e., for a constant driving force), the quotient  = A=f 0becomes!2 0which, in turn, reproduces the (low frequencies) Newtonian result for = 1. At very high frequencies we nd !2, so that the external driving force is dominant. On the other hand, at !=!0 withandxed, we nd that  is as larger as approaches the value = 1 and becomes smaller for 2(0;1=2). That is, given the fractional damping parameter , the fractional system behaves as an underdamped oscillator for !1, and as an overdamped one ifapproximates 1 =2 from above. Figure 1: (Color online) The quotient  = A=f 0of the amplitude of oscillation Adened in (14) and the amplitude f0of the driving force introduced in (8) with = 0,!0= 1,= 0:1!0, for= 0:99 (dotted-black), = 0:95 (solid-red), and = 0:90 (dashed-blue). The behavior of  is depicted in Fig. 1 as a function of the frequency !with,!0 andxed, and for dierent values of . For= 1 (i.e., for the Newtonian case) we nd that  reaches its maximum value when the driving force oscillates at the natural frequency!0, as expected. Such a large response of the system to the driving force is the ngerprint of a resonance. Notice however that the maximum decreases and shifts to the left asdecreases. That is, for .1 the resonance occurs at a frequency !which is lower than !0. The latter means that the resonances can be controlled by xing the fractional-damping parameter and tuning up the order of the fractional derivative D. The same holds if one xes the value of and adjust the fractional-damping parameter . 4 Conclusions Using fractional calculus one nds that the classical harmonic oscillator is aected by an `intrinsic damping' [8, 9], such a damping is also present in the quantum-fractional case [11]. The response of the classical fractional oscillator to the presence of a driving force has been already studied in e.g. [10]. In this paper we have presented some preliminary results of our study on a driven fractional oscillator which is aected by a fractional damping of the form Dx(t), withDthe Caputo time-derivative operator and 0 < 1. In 4particular, we have shown that the resonance phenomenon can be controlled by tuning up either the coecient of the fractional-damping or the order of the Caputo operator. Further results will be reported elsewhere. 5 Acknowledgments F.O.R. acknowledges the funding received through a CONACyT scholarship. References [1] French A P, Vibrations and waves , W W Norton, New York, 1971 [2] Taylor J R, Classical Mechanics University Science Books 2005 [3] Rosas-Ortiz O, Fern andez Garc a N, Cruz y Cruz S, AIP Conf. Proc. 1077 (2008) 31 [4] Miller K S and Ross B, An introduction to the fractional calculus and fractional dierential equations , John Wiley, New York, 1993 [5] Podlubny I, Fractional Dierential Equations , Academic Press, New York, 1999 [6] Hilfer R, Applications of Fractional Calculus in Physics , World Scientic, New York, 2000 [7] Erd elyi A, Higher Transcendental Functions , Vol. III, McGraw-Hill, New York, 1955 [8] Narahari A B N, Hanneken J W, Enck T and Clarke T, Physica A 297(2001) 361 [9] Olivar-Romero F, A rst approach to the fractional quantum mechanics , M.Sc. Thesis (in Spanish), Physics Department, Cinvestav, M exico City, 2014 [10] Narahari A B N, Hanneken J W and Clarke T, Physica A 309(2002) 275 [11] Olivar-Romero F and Rosas-Ortiz O, J. Phys. Conf. Ser. 698(2016) 012025 5

2017-06-15

The resonances associated with a fractional damped oscillator which is driven by an oscillatory external force are studied. It is shown that such resonances can be manipulated by tuning up either the coefficient of the fractional damping or the order of the corresponding fractional derivatives.

Fractional Driven Damped Oscillator

1706.08596v1

arXiv:1706.03942v1 [math.AP] 13 Jun 2017Uniform Energy Decay for Wave Equations with Unbounded Damping Coefficients Ryo IKEHATA∗ Department of Mathematics, Graduate School of Education Hiroshima University Higashi-Hiroshima 739-8524, Japan Hiroshi TAKEDA† Department of Intelligent Mechanical Engineering, Faculty of Engin eering Fukuoka Institute of Technology Fukuoka 811-0295, Japan October 14, 2018 Abstract Weconsiderthe Cauchyproblemforwaveequationswith unbounded dampingcoefficients inRn. For a general class of unbounded damping coefficients, we derive u niform total energy decay estimates together with a unique existence result of a weak s olution. In this case we never impose strongassumptions such as compactness ofthe sup port of the initial data. This means that we never rely on the finite propagation speed property of the solutions, and we try to deal with an essential unbounded coefficient case. One of ou r methods comes from an idea developed in [9]. 1 Introduction We consider the mixed problem for wave equations with a local ized damping in Rn(n≥1) utt(t,x)−∆u(t,x)+a(x)ut(t,x) = 0,(t,x)∈(0,∞)×Rn, (1.1) u(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn, (1.2) where (u0,u1) are initial data chosen as: u0∈H2(Rn), u1∈H1(Rn), and ut=∂u ∂t, utt=∂2u ∂t2,∆ =n/summationdisplay j=1∂2 ∂x2 j, x= (x1,···,xn). Note that solutions and/or functions considered in this pap er are all real valued except for several parts concerning the Fourier transform. ∗Corresponding author: ikehatar@hiroshima-u.ac.jp †h-takeda@fit.ac.jp Keywords and Phrases: Unbounded damping; Wave equation; Ca uchy problem; Weighted initial data; Mul- tiplier method; Fourier analysis; Total energy decay, Weak solutions. 2010 Mathematics Subject Classification. Primary 35L05; Se condary 35B35, 35B40. 1Concerning the decay or non-decay property of the total or lo cal energy to problem (1.1)- (1.2) withx-dependent variable damping coefficients, many research man uscripts are already published by Alloui-Ibrahim-Khenissi [1], Bouclet-Royer [2], Daoulatli [5], Ikehata [7], Ikehata- Todorova-Yordanov[10],Joly-Royer [11],Kawashita[13], Khader[12],Matsumura[16], Mochizuki [18], Mochizuki-Nakazawa [19], Nakao [21], Nishiyama [23] , Nishihara [22], Radu-Todorova- Yordanov [24], Sobajima-Wakasugi [26], Todorova-Yordano v [28], Uesaka [29] and Wakasugi [30], Zhang [32] and the references therein. However, we sho uld emphasize that those cases are quite restricted to the bounded damping coefficient case, i.e.,a∈L∞(Rn). This condition seems to be essential to get the uniqueexistence of mild or we ak solutions to problem (1.1)-(1.2). So, when we do not assume the boundedness of the coefficient a(x), a natural question arises whether one can construct a unique weak or mild solution u(t,x) to problem (1.1)-(1.2) together with some decay property of the total energy or not. Quite rec ently Sobajima-Wakasugi [27] have announced an interesting result from the viewpoint of t he diffusion phenomenon of the solution to the equation (1.1) together with a unique global existence result. The mixed prob- lem to the equation (1.1) in [27] is considered in the exterio r domain of a bounded obstacle, and the Dirichlet null boundary condition is treated. They t reated typically a(x) =a0|x|αwith a0>0 andα >0. But, their results require a stronger assumption such as t he compactness of the support of the initial data. This implies that they hav e to rely on the finite speed of propagation property (FSPP for short) of the solution, so th at in an essential meaning, their framework seems to be still restricted to the bounded dampin g coefficient case for each t≥0. In the treatment of the unbounded coefficient a(x), it seems important and interesting not to assume such FSPP. Additionally, they essentially used rath er stronger regularity condition on the initial data such that [ u0,u1]∈(H2∩H1 0)×H1 0. In connection with this topic, D’Abbicco [3] (and for a more general class, see D’Abbicco-Ebert [4] an d Reissig [25]) and Wirth [31] have ever studied t-dependent unbounded damping coefficient case: utt(t,x)−∆u(t,x)+b(1+t)αut(t,x) =f(u), whereα∈[0,1] andb>0. Therefore, in the x-dependent unbounded coefficient case, a unique existence of the solution itself together with some decay pr operty of the total energy are com- pletely open in the framework of non-compactly supported in itial data class. When Sobajima- Wakasugi [27] constructs a unique solution, they have used d irectly the well-known result due to Ikawa [6]. That means their result is still under the alrea dy known framework. In our result to be announced, we have to discuss how we should construct a w eak solution itself because no any well-known theories can be applied directly. This is a ma in difficulty in our result. In this connection, originally Komatsu [14] first proposed t his open question in his Master thesis in January 2016 such that for the unbounded damping co efficient case a /∈L∞(Rn), can one construct a global in time solution? Unfortunately, Komatsu [14] could not solve his problem before finishing Master course. This paper gives an a nswer to his problem. Now, let us start with introducing our new result. Before sta ting our result, we shall give the following only one assumption ( A) on the damping coefficient a(x), and the definition of the solution to be constructed. (A)a∈C(Rn), and there exists a constant V0>0 such that 0 <V0≤a(x) (∀x∈Rn). Definition. A function u: [0,∞)×Rn→Ris called as the weak solution if it satisfies /integraldisplay∞ 0/integraldisplay Rnu(t,x)(φtt(t,x)−∆φ(t,x)−a(x)φt(t,x))dxdt =/integraldisplay Rnu1(x)φ(0,x)dx−/integraldisplay Rnu0(x)φt(0,x)dx+/integraldisplay Rna(x)u0(x)φ(0,x)dx 2for anyφ∈C∞ 0([0,∞)×Rn). Our new result reads as follows. Theorem 1.1 Letn≥3and assume (A). If the initial data [u0,u1]∈(H2(Rn)∩L1(Rn))× (H1(Rn)∩L1(Rn))further satisfies a(·)u0∈L1(Rn)∩L2(Rn), then there exists a unique weak solutionu∈L∞(0,∞;H1(Rn))∩W1,∞(0,∞;L2(Rn))to problem (1.1)-(1.2)satisfying /ba∇dblu(t,·)/ba∇dbl2≤CI2 00(1+t)−1, Eu(t)≤CI2 00(1+t)−2, with some constant C >0, where I00:=/parenleftBig /ba∇dblu0/ba∇dbl2+/ba∇dbl∇u0/ba∇dbl2+/ba∇dbla(·)u0/ba∇dbl2 1+/ba∇dbla(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2+/ba∇dblu1/ba∇dbl2 1/parenrightBig1/2. Remark 1.1 If one considers the mixed problem (1.1)-(1.2) with the Diri chlet null boundary condition on the smooth exterior domain, one can treat the tw o dimensional case. In that case, we need to assume the logarithmic type weight condition such that/ba∇dbllog(B|x|)(a(·)u0+u1)/ba∇dbl< +∞on the initial data with some constant B >0. This has a close relation to the place where one obtains Lemma 2.1 below in the two dimensional exterior d omain case. For more detail, one can refer the reader to [9]. Remark 1.2 The assumption [ u0,u1]∈H2(Rn)×H1(Rn) is not essential. It will be used only to justify the integration by parts in the course of the p roof. That condition seems not to be so rare (cf. [19]). One may be able to generalize to the mo re weak case such that [u0,u1]∈H1(Rn)×L2(Rn), however, for simplicity we do not deal with such case. Example. As the typical unbounded example for a(x), we can choose a(x) := (1+ |x|2)α 2with α∈[0,∞),a(x) :=e|x|, and so on. This paper is organized as follows. In section 2 we shall prov e Theorem 1.1 by relying on a multiplier method which was introduced in [9]. In section 3 w e give several remarks and open problems. Section 4 is devoted to the appendix to check the kn own result. Notation. Throughout this paper, /ba∇dbl · /ba∇dblqstands for the usual Lq(Rn)-norm. For simplicity of notation, in particular, we use /ba∇dbl·/ba∇dblinstead of /ba∇dbl·/ba∇dbl2. Furthermore, we denote /ba∇dbl·/ba∇dblH1as the usual H1-norm. TheL2-inner product is denoted by ( f,g) :=/integraldisplay Rnf(x)g(x)dxforf,g∈L2(Rn). The total energy Eu(t) corresponding to the solution u(t,x) of (1.1) is defined by Eu(t) :=1 2(/ba∇dblut(t,·)/ba∇dbl2+/ba∇dbl∇u(t,·)/ba∇dbl2), where |∇f(x)|2:=n/summationdisplay j=1|∂f(x) ∂xj|2. The weighted L1-space and its norm /ba∇dbl·/ba∇dbl1,mcan be defined as f∈L1,m(Rn)⇔f∈L1(Rn),/ba∇dblf/ba∇dbl1,m:=/integraldisplay Rn(1+|x|)m|f(x)|dx<+∞. The subspace Xn mofL1,mis defined by Xn m:={f∈L1,m(Rn) :/integraldisplay Rnf(x)dx= 0}. 3And also, for the Hilbert space Xwe define a class of vector valued continuous functions C0([0,∞);X) as follows:f∈C0([0,∞);X)ifandonlyif f∈C([0,∞);X)andtheclosureoftheset {t∈[0,∞);/ba∇dblf(t)/ba∇dblX/ne}ationslash= 0}is compact in [0 ,∞). On the other hand, we denote the Fourier transform of f(x) byˆf(ξ) := (1 2π)n 2/integraldisplay Rne−ix·ξf(x)dxas usual with i:=√−1, and we define the usual convolution by (f∗g)(x) :=/integraldisplay Rnf(x−y)g(y)dy. 2 Proof of Theorem 1.1. Inthecourse of theproof, thenextinequality concerningth eFourier image of theRiesz potential plays an alternative rolefortheHardyinequality. Thiscom es from [8, Proposition2.1]. Itshould beemphasizedthat this inequality holdseven inthelow dime nsional caseif wecontrol theweight parameterγ. Proposition 2.1 Letn≥1andγ∈[0,1]. (1)Iff∈L2(Rn)∩L1,γ(Rn), andθ∈[0,n 2), then there exists a constant C=Cn,θ,γ>0such that /integraldisplay Rn|ˆf(ξ)|2 |ξ|2θdξ≤C/parenleftbigg /ba∇dblf/ba∇dbl2 1,γ+|/integraldisplay Rnf(x)dx|2+/ba∇dblf/ba∇dbl2/parenrightbigg . (2)Iff∈L2(Rn)∩Xn γ, andθ∈[0,γ+n 2), then it is true that /integraldisplay Rn|ˆf(ξ)|2 |ξ|2θdξ≤C(/ba∇dblf/ba∇dbl2 1,γ+/ba∇dblf/ba∇dbl2) with some constant C=Cn,θ,γ>0. Toconstructaglobalweaksolutionwefirstdefineasequenceo ftheweaksolutions {u(m)(t,x)} (m∈N) to the approximated problem below: u(m) tt(t,x)−∆u(m)(t,x)+am(x)u(m) t(t,x) = 0,(t,x)∈(0,∞)×Rn, (2.1) u(m)(0,x) =u0(x), u(m) t(0,x) =u1(x), x∈Rn, (2.2) wheream∈C(Rn) can be chosen to satisfy am(x) =/braceleftBigga(x) (|x| ≤m) V0 (|x|>m+1),(2.3) and V0≤am(x)≤a(x), am(x)→a(x)as m→ ∞(pointwise )x∈Rn(2.4) for eachx∈Rn. 4Now let us consider the problem (2.1)-(2.2) with initial dat a [u0,u1]∈H2(Rn)×H1(Rn). Then, for each m∈Nsinceam∈C(Rn)∩L∞(Rn) it is well known that the Cauchy prob- lem (2.1)-(2.2) has a unique strong solution u(m)∈C([0,∞);H2(Rn))∩C1([0,∞);H1(Rn))∩ C2([0,∞);L2(Rn)) satisfying the energy identity: Eu(m)(t)+/integraldisplayt 0/integraldisplay Rnam(x)|u(m) s(s,x)|2dxds=E(0), (2.5) where E(0) :=1 2(/ba∇dblu1/ba∇dbl2+/ba∇dbl∇u0/ba∇dbl2). To begin with we prove the following crucial estimate. The le mma below is a combination of the method introduced in [9] (= the modified Morawetz metho d [20]) and Proposition 2.1. Lemma 2.1 Letn≥3. Under the same assumptions as in Theorem 1.1, the(unique)solution u(m)(t,x)to problem (2.1)-(2.2) satisfies /ba∇dblu(m)(t,·)/ba∇dbl2+/integraldisplayt 0/integraldisplay Rnam(x)|u(m)(s,x)|2dxds ≤C/parenleftBig /ba∇dbla(·)u0/ba∇dbl2 1+/ba∇dbla(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2 1+/ba∇dblu1/ba∇dbl2/parenrightBig =:CI2 0 with a constant C >0, whereCis independent of m. Proof.The original idea comes from [9]. For the solution u(m)(t,x) to problem (2.1)-(2.2), one introduces an auxiliary function W(t,x) :=/integraldisplayt 0u(m)(s,x)ds. ThenW(t,x) satisfies Wtt−∆W+am(x)Wt=am(x)u0+u1,(t,x)∈(0,∞)×Rn, (2.6) W(0,x) = 0, Wt(0,x) =u0(x), x∈Rn. (2.7) Multiplying (2 .6) byWtand integrating over [0 ,t]×Rnwe get 1 2(/ba∇dblWt(t,·)/ba∇dbl2+/ba∇dbl∇W(t,·)/ba∇dbl2)+/integraldisplayt 0/ba∇dbl/radicalBig am(·)Ws(s,·)/ba∇dbl2ds =1 2/ba∇dblu0/ba∇dbl2+/integraldisplayt 0(am(·)u0+u1,Ws(s,·))ds. (2.8) Next one uses (1) of Proposition 2.1 with θ= 1 andγ= 0, the Plancherel theorem and the Cauchy-Schwarz inequality to obtain a series of inequaliti es below: /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt 0(am(·)u0+u1,Ws(s,·))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt 0d ds(am(·)u0+u1,W(s,·))ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Rn(am(x)u0(x)+u1(x))W(t,x)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Rn ξ(/hatwidest(amu0)(ξ)+ ˆu1(ξ))ˆW(t,ξ)dξ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle 5≤/integraldisplay Rn ξ|/hatwidest(amu0)(ξ)+ ˆu1(ξ)| |ξ|(|ξ||ˆW(t,ξ)|)dξ ≤/parenleftBigg/integraldisplay Rn ξ|/hatwidest(amu0)(ξ)+ ˆu1(ξ)|2 |ξ|2dξ/parenrightBigg1/2/parenleftBigg/integraldisplay Rn ξ|ξ|2|ˆW(t,ξ)|2dξ/parenrightBigg1/2 ≤/integraldisplay Rn ξ|/hatwidest(amu0)(ξ)+ ˆu1(ξ)|2 |ξ|2dξ+1 4/integraldisplay Rn ξ|ξ|2|ˆW(t,ξ)|2dξ ≤C/ba∇dblamu0+u1/ba∇dbl2 1+/ba∇dblamu0+u1/ba∇dbl2+C/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay Rn(am(x)u0(x)+u1(x))dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 +1 4/ba∇dbl∇W(t,·)/ba∇dbl2 ≤C/parenleftbigg /ba∇dblam(·)u0/ba∇dbl2 1+/ba∇dblu1/ba∇dbl2 1+/ba∇dblam(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2+|/integraldisplay Rn(am(x)u0(x)+u1(x))dx|2/parenrightbigg +1 4/ba∇dbl∇W(t,·)/ba∇dbl2 (2.9) with some constant C >0. Combining (2 .8) with (2.9) we can derive 1 2/ba∇dblWt(t,·)/ba∇dbl2+1 4/ba∇dbl∇W(t,·)/ba∇dbl2+/integraldisplayt 0/integraldisplay Rnam(x)|Ws(s,x)|2dxds ≤C/parenleftbigg /ba∇dblam(·)u0/ba∇dbl2 1+/ba∇dblu1/ba∇dbl2 1+/ba∇dblam(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2+|/integraldisplay Rn(am(x)u0(x)+u1(x))dx|2/parenrightbigg with some constant C >0. Sinceam(x)≤a(x), in the case when n≥3 one has 1 2/ba∇dblWt(t,·)/ba∇dbl2+1 4/ba∇dbl∇W(t,·)/ba∇dbl2+/integraldisplayt 0/integraldisplay Rnam(x)|Ws(s,x)|2dxds ≤C/parenleftbigg /ba∇dbla(·)u0/ba∇dbl2 1+/ba∇dblu1/ba∇dbl2 1+/ba∇dbla(·)u0/ba∇dbl2+/ba∇dblu1/ba∇dbl2+|/integraldisplay Rn(a(x)|u0(x)|+|u1(x)|)dx|2/parenrightbigg . We easily see that the constant Cin the above estimates is independent of m. Thus, one has the desired estimate because of the fact Wt=u(m). ✷ Lemma 2.2 Under the same assumptions as in Theorem 1.1, the(unique)solutionu(m)(t,x) to problem (2.1)-(2.2) satisfies (1+t)Eu(m)(t)≤E(0)(1+1 V0+1 2ε)+1 2(u1,u0) =:I2 1,(t≥0), /integraldisplayt 0Eu(m)(s)ds≤I2 1,(t≥0), with some small constant ε>0, whereCis independent of m. Proof.For simplicity of notation, we use w(t,x) in place of u(m)(t,x), i.e.,w(t,x) satisfies the equation below: wtt(t,x)−∆w(t,x)+am(x)wt(t,x) = 0,(t,x)∈(0,∞)×Rn, (2.10) w(0,x) =u0(x), wt(0,x) =u1(x), x∈Rn. (2.11) Note thatEw(t) =Eu(m)(t) satisfies (2.5). Then, since d dt{(1+t)Ew(t)} ≤Ew(t), 6it follows from (2.5) that (1+t)Ew(t)≤E(0)+1 2/integraldisplayt 0/ba∇dblws(s,·)/ba∇dbl2ds+1 2/integraldisplayt 0/ba∇dbl∇w(s,·)/ba∇dbl2ds ≤E(0)+1 2V0/integraldisplayt 0/integraldisplay Rnam(x)|ws(s,x)|2dxds+1 2/integraldisplayt 0/ba∇dbl∇w(s,·)/ba∇dbl2ds ≤E(0)+1 2V0E(0)+1 2/integraldisplayt 0/ba∇dbl∇w(s,·)/ba∇dbl2ds. (2.12) On the other hand, by multiplying both sides of (2.10) by w(t,x) it follows that d dt(wt(t,·),w(t,·)) +/ba∇dbl∇w(t,·)/ba∇dbl2+1 2d dt/integraldisplay Rnam(x)|w(t,x)|2dx=/ba∇dblwt(t,·)/ba∇dbl2,(2.13) so that by integrating both sides over [0 ,t] one has /integraldisplayt 0/ba∇dbl∇w(s,·)/ba∇dbl2ds+1 2/integraldisplay Rnam(x)|w(t,x)|2dx =/integraldisplayt 0/ba∇dblws(s,·)/ba∇dbl2ds−(wt(t,·),w(t,·)) +(u1,u0) ≤1 V0/integraldisplayt 0/integraldisplay Rnam(x)|ws(s,x)|2dxds+1 2ε/ba∇dblwt(t,·)/ba∇dbl2+ε 2/ba∇dblw(t,·)/ba∇dbl2+(u1,u0) ≤1 V0/integraldisplayt 0/integraldisplay Rnam(x)|ws(s,x)|2dxds+1 εEw(t)+ε 2V0/integraldisplay Rnam(x)|w(t,x)|2dx+(u1,u0) ≤1 V0E(0)+1 εE(0)+ε 2V0/integraldisplay Rnam(x)|w(t,x)|2dx+(u1,u0), where we have just used (2.5) and the Cauchy-Schwarz inequal ity with some positive parameter ε>0. This implies /integraldisplayt 0/ba∇dbl∇w(s,·)/ba∇dbl2ds+1 2(1−ε V0)/integraldisplay Rnam(x)|w(t,x)|2dx ≤E(0)(1 V0+1 ε)+(u1,u0). (2.14) Bychoosing ε>0sufficiently small, from(2.13) and(2.14) onecanget thedes iredtwo estimates. In this final check, because of (2.6) we have to make the follow ing estimate once more: /integraldisplayt 0/ba∇dblwt(s,·)/ba∇dbl2ds≤1 V0/integraldisplay Rnam(x)|ws(s,x)|2dxds≤1 V0E(0). ✷ Lemma 2.3 Under the same assumptions as in Theorem 1.1, the(unique)solutionu(m)(t,x) to problem (2.1)-(2.2) satisfies (1+t)2Eu(m)(t)≤E(0)+1 V0(E(0)+I2 1)+(u1,u0)+1 2/integraldisplay Rna(x)|u0(x)|2dx+C 2I2 0+I2 1 ε=:I2 2, (1+t)/ba∇dblu(m)(t,·)/ba∇dbl2≤2(V0−ǫ)−1{(u1,u0)+1 2/integraldisplay Rna(x)|u0(x)|2dx+C 2I2 0+1 V0(E(0)+I2 1)+I2 1 ε}=:I2 3 with some constant C >0, whereCis independent of m. 7Proof.We use the same notation as in the proof of Lemma 2.2. Then, by m ultiplying both sides of (2.11) by (1+ t)w, and integrating it over [0 ,t]×Rnone can arrive at the important identity: 1 2/ba∇dblu0/ba∇dbl2+/integraldisplayt 0(1+s)/ba∇dbl∇w(s,·)/ba∇dbl2ds+(1+t) 2/integraldisplay Rnam(x)|w(t,x)|2dx =−(1+t)(wt(t,·),w(t,·)) +(u1,u0)+1 2/ba∇dblw(t,·)/ba∇dbl2+1 2/integraldisplayt 0/integraldisplay Rnam(x)|w(s,x)|2dxds +1 2/integraldisplay Rnam(x)|u0(x)|2dx+/integraldisplayt 0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds. (2.15) Now, by using the Cauchy-Schwarz inequality and Lemma 2.2 we can first estimate −(1+t)(wt(t,·),w(t,·))≤(1+t) 2ε/ba∇dblwt(t,·)/ba∇dbl2+ε 2(1+t)/ba∇dblw(t,·)/ba∇dbl2 ≤1+t εEw(t)+ε 2V0(1+t)/integraldisplay Rnam(x)|w(t,x)|2dx ≤I2 1 ε+ε 2V0(1+t)/integraldisplay Rnam(x)|w(t,x)|2dx. (2.16) (2.16) and (2.17) imply 1 2/ba∇dblu0/ba∇dbl2+/integraldisplayt 0(1+s)/ba∇dbl∇w(s,·)/ba∇dbl2ds+(1+t) 2(1−ε V0)/integraldisplay Rnam(x)|w(s,x)|2dx ≤(u1,u0)+1 2/integraldisplay Rna(x)|u0(x)|2dx+1 2/ba∇dblw(t,·)/ba∇dbl2+1 2/integraldisplayt 0/integraldisplay Rnam(x)|w(s,x)|2dxds +I2 1 ε+/integraldisplayt 0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds. (2.17) On the other hand, it follows from (2.5) and Lemma 2.2 that /integraldisplayt 0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds≤1 V0/integraldisplayt 0(1+s)/integraldisplay Rnam(x)|ws(s,x)|2dxds =−1 V0/integraldisplayt 0(1+s)E′ w(s)ds=−1 V0(1+t)Ew(t)+1 V0E(0)+1 V0/integraldisplayt 0Ew(s)ds ≤1 V0(E(0)+I2 1). (2.18) Let us finalize the proof of Lemma 2.3. To do so, we rely on the in equality: d dt{(1+t)2Ew(t)} ≤2(1+t)Ew(t), so that (1+t)2Ew(t)≤E(0)+/integraldisplayt 0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds+/integraldisplayt 0(1+s)/ba∇dbl∇w(s,·)/ba∇dbl2ds (2.19) Because of Lemma 2.1, (2.17) with small ε>0 and (2.18) one has (1+t)2Ew(t)≤E(0)+2 V0(E(0)+I2 1)+(u1,u0)+1 2/integraldisplay Rna(x)|u0(x)|2dx+C 2I2 0+I2 1 ε=:I2 2. 8Concerning the fast L2-decay estimate, we use (2.5) and (2.17) with small ε>0 to have (1+t) 2(1−ε V0)V0/integraldisplay Rn|w(t,x)|2dx≤(1+t) 2(1−ε V0)/integraldisplay Rnam(x)|w(t,x)|2dx ≤(u1,u0)+1 2/integraldisplay Rna(x)|u0(x)|2dx+1 2/ba∇dblw(t,·)/ba∇dbl2+1 2/integraldisplayt 0/integraldisplay Rnam(x)|w(s,x)|2dxds +I2 1 ε+/integraldisplayt 0(1+s)/ba∇dblws(s,·)/ba∇dbl2ds. (2.20) The result follows from (2.20) and (2.18) and Lemma 2.1 by cho osingε>0 small enough. ✷ Proof of Theorem 1.1. From Lemma 2.3 we first notice that {u(m)}is a bounded sequence in L∞(0,∞;H1(Rn)) and inL∞(0,∞;L2(Rn)). Furthermore, {u(m) t}is also a bounded sequence inL∞(0,∞;L2(Rn)). Therefore, there exist a subsequence {u(µ)}of the original one {u(m)}, and a function u=u(t,x)∈L∞(0,∞;H1(Rn)) satisfying ut∈L∞(0,∞;L2(Rn)) such that u(µ)→u(weakly∗)in L∞(0,∞;H1(Rn)) (µ→ ∞), (2.21) u(µ) t→ut(weakly∗)in L∞(0,∞;L2(Rn)) (µ→ ∞), (2.22) u(µ)→u(weakly∗)in L∞(0,∞;L2(Rn)) (µ→ ∞). (2.23) By multiplying both sides of the approximated equation (2.1 ) withmreplaced by µthe test functionφ∈C∞ 0([0,∞)×Rn), one can get the following weak form of the problem (2.1)-(2 .2) with the help of the integration by parts: /integraldisplay∞ 0/integraldisplay Rnu(µ)(t,x)(φtt(t,x)−∆φ(t,x)−aµ(x)φt(t,x))dxdt =/integraldisplay Rnu1(x)φ(0,x)dx−/integraldisplay Rnu0(x)φt(0,x)dx+/integraldisplay Rnaµ(x)u0(x)φ(0,x)dx. (2.24) Now, it follows from (2.23) that as µ→ ∞, /integraldisplay∞ 0/integraldisplay Rnu(µ)(t,x)(φtt(t,x)−∆φ(t,x))dxdt→/integraldisplay∞ 0/integraldisplay Rnu(t,x)(φtt(t,x)−∆φ(t,x))dxdt.(2.25) Furthermore, because of the Lebesgue dominated convergenc e theorem one can get /integraldisplay Rnaµ(x)u0(x)φ(0,x)dx→/integraldisplay Rna(x)u0(x)φ(0,x)dx(µ→ ∞). (2.26) On the other hand, for each fixed φ∈C∞ 0([0,∞)×Rn), if we take µ∈Nlarge enough, then it follows from the compact support condition on φ(t,x) that /integraldisplay∞ 0/integraldisplay Rnu(µ)(t,x)aµ(x)φt(t,x)dxdt=/integraldisplay∞ 0/integraldisplay Rnu(µ)(t,x)a(x)φt(t,x)dxdt. So, sincea(·)φt(t,·)∈L1([0,∞);L2(Rn)), it follows from (2.23) that /integraldisplay∞ 0/integraldisplay Rnu(µ)(t,x)aµ(x)φt(t,x)dxdt→/integraldisplay∞ 0/integraldisplay Rnu(t,x)a(x)φt(t,x)dxdt(µ→ ∞).(2.27) Therefore, by taking µ→ ∞in (2.24), by means of (2.25)-(2.27) one can check that the li mit functionu(t,x) is the weak solution to problem (1.1)-(1.2). 9Finally, let us check decay estimates of the energy and L2-norm of solutions such that Eu(t)≤C(1+t)−2,/ba∇dblu(t,·)/ba∇dbl2≤C(1+t)−1(2.28) with some constant C >0. For this end, one first remarks that for each φ∈L1(0,∞;C∞ 0(Rn)), it follows from the integration by parts that /integraldisplay∞ 0(∂u(µ) ∂xj(t,·),φ(t,·))dt=−/integraldisplay∞ 0(u(µ)(t,·),∂φ ∂xj(t,·))dt, so that we can have limµ→∞/integraldisplay∞ 0(∂u(µ) ∂xj(t,·),φ(t,·))dt=−/integraldisplay∞ 0(u(t,·),∂φ ∂xj(t,·))dt. Sinceu∈L∞([0,∞);H1(Rn)), it follows from the integration by parts again one can get limµ→∞/integraldisplay∞ 0(∂u(µ) ∂xj(t,·),φ(t,·))dt=/integraldisplay∞ 0(∂u ∂xj(t,·),φ(t,·))dt. By density of L1(0,∞;C∞ 0(Rn)) intoL1(0,∞;L2(Rn)), for each j= 1,2,···,nit is true that ∂u(µ) ∂xj→∂u ∂xj(weakly∗)in L∞(0,∞;L2(Rn)) (µ→ ∞). (2.29) Now, let us finalize the proof of Theorem 1.1. First of all, we p repare the following basic lemma. Lemma 2.4 Assume that a sequence {vm} ⊂L∞(0,∞;L2(Rn)satisfies vm→v(weakly∗)in L∞(0,∞;L2(Rn)) (m→ ∞), for somev∈L∞(0,∞;L2(Rn)), and /ba∇dblvm(t,·)/ba∇dbl ≤C(1+t)−γ with some constants C >0, andγ >0. Then, it is also true that /ba∇dblv(t,·)/ba∇dbl ≤C(1+t)−γ. Proof.Takeψ∈C0([0,∞);L2(Rn)), and set wm(t,x) := (1+t)γvm(t,x). Then, since (1+t)γψ(t,x)∈L1(0,∞;L2(Rn)), by assumption it follows that limm→∞/integraldisplay∞ 0/integraldisplay Rnwm(t,x)ψ(t,x)dxdt= limm→∞/integraldisplay∞ 0/integraldisplay Rnvm(t,x)((1+t)γψ(t,x))dxdt =/integraldisplay∞ 0/integraldisplay Rnv(t,x)((1+t)γψ(t,x))dxdt=/integraldisplay∞ 0/integraldisplay Rn((1+t)γv(t,x))ψ(t,x)dxdt. Because of the density of C0([0,∞);L2(Rn)) intoL1(0,∞;L2(Rn)) again (cf. Miyadera [17, Theorem 15.3]), we have wm= (1+t)γvm→(1+t)γv(weakly∗)in L∞(0,∞;L2(Rn)) (m→ ∞), 10so that it follows that /ba∇dbl(1+t)γv(t,·)/ba∇dbl ≤liminfm→∞/ba∇dbl(1+t)γvm/ba∇dblL∞(0,∞;L2(Rn))≤C for eacht≥0. This implies the desired statement. ✷ Proof of Theorem 1.1 completed. It follows from Lemma 2.3 that /ba∇dbl∂u(µ)(t,·) ∂xj/ba∇dbl ≤√ 2I2(1+t)−1,/ba∇dblu(µ) t(t,·)/ba∇dbl ≤√ 2I2(1+t)−1. Thus, it follows from (2.29) and (2.22) and Lemma 2.4 that /ba∇dbl∂u(t,·) ∂xj/ba∇dbl ≤√ 2I2(1+t)−1, (2.30) /ba∇dblut(t,·)/ba∇dbl ≤√ 2I2(1+t)−1. (2.31) Furthermore, from Lemma 2.3 one also has /ba∇dblu(µ)(t,·)/ba∇dbl ≤I3(1+t)−1/2(t≥0). Therefore, (2.23) and Lemma 2.4 imply /ba∇dblu(t,·)/ba∇dbl ≤I3(1+t)−1/2,(t≥0). (2.32) (2.30)-(2.32) imply the desired estimates (2.28) of Theore m 1.1. Note that all quantities Ij (j= 0,1,2,3) can be absorbed into CI00defined in Theorem 1.1 with some constant C >0. In this connection, we should remark the following relation because of the Cauchy-Schwarz inequality: /integraldisplay Rna(x)|u0(x)|2dx≤/radicalBigg/integraldisplay Rna(x)2|u0(x)|2dx/radicalBigg/integraldisplay Rn|u0(x)|2dx. A uniqueness argument is standard, so we shall omit its detai l. ✷ 3 Remark on the low dimensional case. Let us give some remarks on the low dimensional case and sever al open problems. (I)When one gets the result for n= 2, we may use (2) of Proposition 2.1 with θ= 1 and γ∈(0,1] to get the similar estimate to (2.9), which is an essential part of proof. But, in this case we have to assume a stronger assumption such that /integraldisplay R2(a(x)u0(x)+u1(x))dx= 0. (3.1) Unfortunately, this assumption (3.1) is not hereditary to t he approximate solution, i.e., /integraldisplay R2(am(x)u0(x)+u1(x))dx= 0 is not necessarily true. So, it is completely open to get the s imilar result in the low dimensional case (i.e.,n= 1,2). 11(II)One can not treat the associated nonlinear equation suc h that utt(t,x)−∆u(t,x)+a(x)ut(t,x) =f(u(t,x)). (3.2) This is because one encounters lack of some compactness argu ment when we want to get the limit such that (see Lions [15, (1.42)]) f(u(m))→f(u)weakly∗in L∞([0,∞);X) (asm→ ∞), whereXis a Banach space. In the unbounded coefficient case there are still many obstacl es to be overcome. In this sense, the dampedwave equation with unboundedcoefficient and non-c ompactly supportedinitial data seem to be quite difficult to be treated. (III)We have another idea to deal with the two dimensional ca se, that is, the idea to rely on the well-known inequality:/integraldisplay R2|ˆf(ξ)|2 |ξ|2dξ≤C/ba∇dblf/ba∇dbl2 H1, (3.3) in place of (2) of Proposition 2.1, where H1(R2) is the so called Hardy space. Unfortunately, in this case we also encounter the same problem as the heredity o f the property a(·)u0∈ H1(R2) toam(·)u0∈ H1(R2). 4 Appendix. In this section, let us check the density of L1(0,∞;C∞ 0(Rn)) intoL1(0,∞;L2(Rn)). Takef∈L1(0,∞;L2(Rn)) and for each L>0 set fL(t,x) =/braceleftBiggf(t,x) (|x| ≤L) 0 ( |x|>L). Next set h(t) = e−1 1−t2(|t|<1) 0 ( |t| ≥1), andρ(x) :=h(|x|2)/parenleftbigg/integraldisplay Rnh(|x|2)dx/parenrightbigg−1 , and forε>0 define ρε(x) :=1 εnρ(x ε). Under these preparations, we define an approximating sequen ce of the original function fby fε,L(t,x) := (fL(t,·)∗ρε)(x). Then it is standard to check that fε,L(t,·)∈C∞ 0(Rn) for eacht≥0 andL>0, and /ba∇dblfε,L(t,·)/ba∇dbl ≤ /ba∇dblfL(t,·)/ba∇dbl ≤ /ba∇dblf(t,·)/ba∇dbl, so thatfε,L∈L1(0,∞;L2(Rn)) for each L >0 andε >0. Furthermore, for each L >0 and t≥0 it is known that /ba∇dblfε,L(t,·)−fL(t,·)/ba∇dbl →0 (ε→0). 12Now, for an arbitrary fixed η>0, chooseM >0 so large such that /ba∇dblfM−f/ba∇dblX<η 2, (4.1) whereX:=L1(0,∞;L2(Rn)), and/ba∇dbl·/ba∇dblXis the standard norm of X: /ba∇dblg/ba∇dblX:=/integraldisplay∞ 0/ba∇dblg(t,·)/ba∇dbldt. Next, for such a fixed M >0, by taking ε >0 sufficiently small, if one applies the Lebesgue convergence theorem, one can get /ba∇dblfε,M−fM/ba∇dblX<η 2. (4.2) Therefore, it follows from (4.1) and (4.2) with such large M >0 and small ε>0 that /ba∇dblfε,M−f/ba∇dblX≤ /ba∇dblfε,M−fM/ba∇dblX+/ba∇dblfM−f/ba∇dblX<η 2+η 2=η, which implies the density of L1(0,∞;C∞ 0(Rn)) intoL1(0,∞;L2(Rn)). ✷ Acknowledgement. The work of the firstauthor (R. IKEHATA) was supportedin part by Grant- in-Aid for Scientific Research 15K04958 of JSPS. The work of t he second author (H. TAKEDA) was supported in part by Grant-in-Aid for Young Scientists ( B)15K17581 of JSPS. References [1] L. Aloui, S. Ibrahim and M. Khenissi, Energy decay for lin ear dissipative wave equations in exterior domains, J. Diff. Eqns 259 (2015), 2061-2079. [2] J.M. Bouclet andJ.Royer, Local energydecay forthedamp edwave equation, J.Functional Anal. 266 (2014), 4538-4615. [3] M. D’Abbicco, Small data solutions for semilinear wave e quations with effective damping, Discrete Continuous Dynamical Systems, Supplement (2013) , 183-191. [4] M. D’Abbicco and M. R. Ebert, A classification of structur al dissipations for evolution operators, Math. Methods Appl. Sci. 39 (2016), 2558-2582. [5] M. Daoulatli, Energy decay rates for solutions of the wav e equation with linear damping in exterior domain, Evolution Equations and Control Theory 5 ( 2016), 37-59. [6] M. Ikawa, Mixed problems for hyperbolic equations of sec ond order, J. Math. Soc. Japan 20 (1968), 580-608. [7] R. Ikehata, Fast decay of solutions for linear wave equat ions with dissipation localized near infinity in an exterior domain, J. Diff. Eqns 188 (2003), 390-40 5. [8] R. Ikehata, Fast energy decay for wave equations with a lo calized damping in the n-D half space, Asymptotic Anal. 103 (2017), 77-94. [9] R. Ikehata and T. Matsuyama, L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002), 33-42. 13[10] R. Ikehata, G. Todorova and B. Yordanov, Optimal decay r ate of the energy for linear wave equations with a critical potential, J. Math. Soc. Japan 65, No. 1 (2013), 183-236. [11] R. Joly and J. Royer, Energy decay and diffusion phenomeno n for the asymptotically peri- odic damped wave equation, arXiv: 1703.05112v1, 15 Mar 2017 .. [12] M. Khader, Global existence for the dissipative wave eq uations with space-time dependent potential, Nonlinear Anal. 81 (2013), 87-100. [13] M. Kawashita, Sufficient conditions for decay estimates of the local energy and a behavior of the total energy of dissipative wave equations in exterio r domains, Hokkaido Math. J. (in press). [14] T. Komatsu, Energy decay for damped wave equations with a localized variable damping coefficient, Graduate School of Education, Hiroshima Univer sity, Master Thesis, 2016, (in Japanese). [15] J. L. Lions, Quelques m´ ethodes de r´ esolution des prob lemes aux limites non lin´ eaires, Dunod, Gauthier-Villars, Paris, 1969. [16] A. Matsumura, Energy decay of solutions of dissipative wave equations, Proc. Japan Acad. 53 (1977), 232-236. [17] I. Miyadera, Functional Analysis, Rikougaku-sha, 197 2 (in Japanese). [18] K. Mochizuki, Scattering theory for wave equations wit h dissipative terms, Publ. RIMS 12 (1976), 383-390. [19] K. Mochizuki and H. Nakazawa, Energy decay of solutions to the wave equations with linear dissipation localized near infinity, Publ. Res. Inst . Math. Sci., Kyoto Univ. 37 (2001), 441-458. [20] C. Morawetz, The decay of solutions of the exterior init ial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961), 561-568. [21] M. Nakao, Energy decay for the linear and semilinear wav e equations in exterior domains with some localized dissipations, Math. Z. 238 (2001), 781- 797. [22] K. Nishihara, Decay properties for the damped wave equa tion with space dependent po- tential and absorbed semi-linear term, Comm. Partial Differe ntial Equations 35 (2010), 1402-1418. [23] H. Nishiyama, Remarks on the asymptotic behavior of the solution to damped wave equa- tions, J. Diff. Eqns 261 (2016), 3893-3940. [24] P. Radu, G. Todorova and B. Yordanov, Diffusion phenomeno n in Hilbert spaces and ap- plications, J. Diff. Eqns 250 (2011), 4200-4218. [25] M. Reissig, Rates of decay for structural damped models with coefficients strictly increasing in time, Complex analysis and dynamical systems IV. Part 2, 1 87-206, Contemp. Math., 554, Israel Math. Conf. Proc., Amer. Math. Soc., Providence , RI, 2011. [26] M. Sobajima and Y. Wakasugi, Diffusion phenomena for the w ave equation with space- dependent damping in an exterior domain, J. Diff. Eqns 261 (201 6), 5690-5718. 14[27] M. Sobajima and Y. Wakasugi, Diffusion phenomena for the w ave equation with space- dependent damping term growing at infinity, arXiv:1704.076 50v1, 25, April 2017. [28] G. Todorova and B. Yordanov, Weighted L2-estimates of dissipative wave equations with variable coefficients, J. Diff. Eqns 246 (2009), 4497-4518. [29] H. Uesaka, The total energy decay of solutions for the wa ve equation with a dissipative term, J. Math. Kyoto Univ. 20 (1980), 57-65. [30] Y. Wakasugi, On diffusion phenomena for the linear wave eq uation with space-dependent damping, J. Hyperbolic differ. Equ. 11 (2014), 795-819. [31] J. Wirth, Wave equations with time-dependent dissipat ion II. Effective dissipation, J. Diff. Eqns 232 (2007), 74-103. [32] Z. Zhang, Fast decay of solutions for wave equations wit h localized dissipation on noncom- pact Riemannian manifolds, Nonlinear Analysis: Real World Appl. 27 (2016), 246-260. 15

2017-06-13

We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence result of a weak solution. In this case we never impose strong assumptions such as compactness of the support of the initial data. This means that we never rely on the finite propagation speed property of the solution, and we try to deal with an essential unbounded coefficient case.

Uniform energy decay for wave equations with unbounded damping coefficients

1706.03942v1

arXiv:0904.1455v1 [cond-mat.mtrl-sci] 9 Apr 2009Evaluating the locality of intrinsic precession damping in transition metals Keith Gilmore1,2and Mark D. Stiles1 1Center for Nanoscale Science and Technology National Institute of Standards and Technology, Gaithersburg, MD 20899-6202 2Maryland Nanocenter, University of Maryland, College Park, MD 20742-3511 (Dated: December 4, 2018.) The Landau-Lifshitz-Gilbert damping parameter is typical ly assumed to be a local quantity, in- dependent of magnetic configuration. To test the validity of this assumption we calculate the precession damping rate of small amplitude non-uniform mod e magnons in iron, cobalt, and nickel. At scattering rates expected near and above room temperatur e, little change in the damping rate is found as the magnon wavelength is decreased from infinity to a length shorter than features probed in recent experiments. This result indicates that non-loca l effects due to the presence of weakly non-uniform modes, expected in real devices, should not app reciably affect the dynamic response of the element at typical operating temperatures. Conversely , at scattering rates expected in very pure samples around cryogenic temperatures, non-local effects r esult in an order of magnitude decrease in damping rates for magnons with wavelengths commensurate with domain wall widths. While this low temperature result is likely of little practical import ance, it provides an experimentally testable prediction of the non-local contribution of the spin-orbit torque-correlation model of precession damping. None of these results exhibit strong dependence on the magnon propagation direction. Magnetization dynamics continues to be a techno- logically important, but incompletely understood topic. Historically, field induced magnetization dynamics have been described adequately by the phenomenological Landau-Lifshitz (LL) equation [1] ˙m=−|γM|m×H+λˆm×(m×H),(1) or the mathematically equivalent Gilbert form [2, 3]. Equation 1 accounts for the near equilibrium dynamics of systems in the absence of an electrical current. γM is the gyromagnetic ratio and λis the phenomenological damping parameter, which quantifies the decay of the excited system back to equilibrium. The LL equation is a rather simple approximation to very intricate dy- namic processes. The limitations of the approximations entering into the LL equation are likely to be tested by the next generation of magnetodynamic devices. While manygeneralizationsforthe LLequationarepossible, we focus on investigatingthe importance of non-local contri- butions to damping. It is generally assumed in both ana- lyzing experimental results and in performing micromag- netic simulations that damping is a local phenomenon. While no clearexperimental evidence exists to contradict this assumption, the possibility that the damping is non- local – that it depends, for example, on the local gradient ofthemagnetization–wouldhaveparticularimplications for experiments that quantify spin-current polarization [4], for storage [5] and logic [6] devices based on using this spin-current to move domain-walls, quantifying vor- tex [7] and mode [8] dynamics in patterned samples, and the behavior of nano-contact oscillators [9, 10]. While several viable mechanisms have been proposed to explain the damping process in different systems [11, 12, 13, 14, 15, 16, 17], werestrictthescopeofthispa- per to investigating the degree to which the assumptionof local damping is violated for small amplitude dynam- ics within pure bulk transition metal systems where the dominant source of damping is the intrinsic spin-orbit in- teraction. For such systems, Kambersk´ y’s [14] spin-orbit torque-correlationmodel, which predicts a decay rate for the uniform precession mode of λ0=π¯hγ2 M µ0/summationdisplay nm/integraldisplay dk/vextendsingle/vextendsingleΓ− nm(k)/vextendsingle/vextendsingle2Wnm(k),(2) hasrecentlybeen demonstratedtoaccountforthe major- ity of damping [18, 19]. The matrix elements |Γ− nm(k)|2 represent a scattering event in which a quantum of the uniformmodedecaysintoasinglequasi-particleelectron- hole excitation. This annihilation of a magnon raises the angularmomentum of the system, orienting the magneti- zation closer to equilibrium. The excited electron, which has wavevector kand band index m, and the hole, with wavevector kand band index n, carry off the energy and angular momentum of the magnon. This electron-hole pair is rapidly quenched through lattice scattering. The weighting function Wnm(k) measures the rate at which the scattering event occurs. The very short lifetime of the electron-hole pair quasiparticle (on the order of fs at room temperature) introduces significant energy broad- ening (several hundred meV). The weighting function, which is a generalization of the delta function appearing in a simple Fermi’s golden rule expression, quantifies the energy overlap of the broadened electron and hole states with each other and with the Fermi level. Equation 2, which has been discussed extensively [14, 18, 19, 20], considers only local contributions to the damping rate. Non-local contributions to damping may be studied through the decay of non-uniform spin-waves. Although recent efforts have approached the problem of2 non-localcontributionsto the dissipationofnon-collinear excited states [21, 22] the simple step of generalizing Kambersk´ y’s theory to non-uniform mode magnons has not yet been taken. We fill this obvious gap, obtaining a damping rate of λq=π¯hγ2 M µ0/summationdisplay nm/integraldisplay dk/vextendsingle/vextendsingleΓ− nm(k,k+q)/vextendsingle/vextendsingle2Wnm(k,k+q) (3) for a magnon with wavevector q. We test the impor- tance of non-local effects by quantifying this expression for varying degrees of magnetic non-collinearity (magnon wavevector magnitude). The numerical evaluation of Eq. 3 for the damping rate of finite wavelength magnons in transition metal systems, presented in Fig. 1, and the ensuing physical discussion form the primary contribu- tion of this paper. We find that the damping rate ex- pected inverypuresamplesatlowtemperatureisrapidly reduced as the magnon wavevector |q|grows, but the damping rate anticipated outside of this ideal limit is barely affected. We provide a simple band structure ar- gument to explain these observations. The results are relevant to systems for which the non-collinear excita- tion may be expanded in long wavelength spin-waves, provided the amplitude of these waves is small enough to neglect magnon-magnon scattering. Calculations for the single-mode damping constant (Eq. 3) as a function of electron scattering rate are pre- sented in Fig. 1 for iron, cobalt, and nickel. The Gilbert damping parameter α=λ/γMis also given. Damp- ing rates are given for magnons with wavevectors along the bulk equilibrium directions, which are /angbracketleft100/angbracketrightfor Fe, /angbracketleft0001/angbracketrightfor Co, and /angbracketleft111/angbracketrightfor Ni. Qualitatively and quan- titatively similar results were obtained for other magnon wavevector directions for each metal. The magnons re- portedoninFig.1constitutesmalldeviationsofthemag- netization transverse to the equilibrium direction with wavevectormagnitudes between zero and 1 % of the Bril- louin zone edge. This wavevector range corresponds to magnon half-wavelengths between infinity and 100 lat- tice spacings, which is 28.7 nm for Fe, 40.7 nm for Co, and 35.2 nm for Ni. This range includes the wavelengths reported by Vlaminck and Bailleul in their recent mea- surement of spin polarization [4]. Results for the three metals are qualitatively similar. Themoststrikingtrendisadramatic,orderofmagnitude decreaseofthe damping rate at the lowestscatteringrate tested as the wavevector magnitude increases from zero to 1 % of the Brillouin zone edge. This observation holds in each metal for every magnon propagation direction investigated. For the higher scattering rates expected in devices at room temperature there is almost no change in the damping rate as the magnon wavevector increases from zero to 1 % of the Brillouin zone edge in any of the directions investigated for any of the metals. To understand the different dependences of the damp- ing rate on the magnon wavevector at low versus high scattering rates we first note that the damping rate10121013101410151091010 0.010.1λ (s-1) γ (s-1) α 1081090.01λ (s-1) α1090.01 1.0 0.1 0.01 λ (s-1) αhγ (eV) 10-310-3 10-30.0 0.001 0.003 0.005 0.01 0.0 0.001 0.003 0.005 0.01 0.0 0.001 0.003 0.005 0.01Fe Co Niq q q FIG. 1: Damping rates versus scattering rate. The preces- sion damping rates for magnons in iron, cobalt, and nickel are plotted versus electron scattering rate for several mag non wavevectors. A dramatic reduction in damping rate is ob- served at the lowest scattering rates. The Landau-Lifshitz λ (Gilbert α) damping parameter is given on the left (right) axes. Electron scattering rate is given in eV on the top axis. Magnon wavevector magnitudes are given in units of the Bril- louin zone edge and directions are as indicated in the text. (Eqs. 2 & 3) is a convolution of two factors: the torque matrix elements and the weighting function. The ma- trix elements do not change significantly as the magnon wavevector increases, however, the weighting function can change substantially. The weighting function Wnm(k,k+q)≈An,k(ǫF)Am,k+q(ǫF) (4)3 789101112 789101112 Energy (eV) H F P FIG. 2: Partial band structure of bcc iron. The horizontal black line indicates the Fermi level and the shaded region represents the degree of spectral broadening. The solid dot is a hypothetical initial electron state while the open circle is a potential final scattering state. (Initial and final state wa ve- vector separations are exaggerated for clarity of illustra tion.) The intraband magnon decay rate diminishes as the energy separation of the states exceeds the spectral broadening. contains a product of the initial and final state electron spectral functions An,k(ǫ) =1 π¯hγ (ǫ−ǫn,k)2+(¯hγ)2, (5) whichareLorentziansinenergyspace. Thespectralfunc- tion for state |n,k/angbracketright, which has nominal band energy ǫn,k, is evaluated within a verynarrowrangeofthe Fermi level ǫF. The width of the spectral function ¯ hγis given by the electron scattering rate γ= 1/2τwhereτis the orbital lifetime. (The lifetimes of all orbital states are taken to be equal for these calculations and no specific scattering mechanism is implied.) The weighting func- tion restricts the electron-hole pair generated during the magnon decay to states close in energy to each other and near the Fermi level. For high scattering rates, the elec- tron spectral functions are significantly broadened and the weighting function incorporates states within an ap- preciablerange(severalhundredmeV) ofthe Fermi level. For low scattering rates, the spectral functions are quite narrow (only a few meV) and both the electron and hole state must be very close to the Fermi level. The second consideration useful for understanding the results of Fig. 1 is that the sum in Eqs. 2 & 3 can be divided into intraband ( n=m) and interband ( n/negationslash=m) terms. For the uniform mode, these two contributions correspond to different physical processes with the intra- band contributiondominatingatlowscatteringratesand the interband terms dominating at high scattering rates [14, 18, 19, 20].101210131014101510-3 10-3 10-4 1071081090.01 0. 11 0.01 0.0 0.001 0.003 0.005 0.01λintra,inte r (s-1) γ (s-1)q in ( π/a)interband intraband αintra,inter hγ (eV)Increasing q FIG. 3: Intraband and interband damping contributions in iron. Theintrabandandinterbandcontributionstothedamp - ing rate of magnons in the /angbracketleft100/angbracketrightdirection in iron are plot- ted versus scattering rate for several magnitudes of magnon wavevector. Magnitudes are given in units of the Brillouin zone edge. For intraband scattering, the electron and hole occupy the sameband and must haveessentiallythe sameenergy (within ¯hγ). The energy difference between the electron and hole states may be approximated as ǫn,k+q−ǫn,k≈ q·∂ǫn,k/∂k. The generation of intraband electron-hole pairs responsible for intraband damping gets suppressed asq·∂ǫn,k/∂kbecomes largecomparedto ¯ hγ. Unless the bands are very flat at the Fermi level there will be few lo- cations on the Fermi surface that maintain the condition q·∂ǫn,k/∂k<¯hγfor low scattering rates as the magnon wavevectorgrows. (See Fig. 2). Indeed, at low scattering rates when ¯ hγis only a few meV, Fig. 3 shows that the intraband contribution to damping decreases markedly with only modest increase of the magnon wavevector. Since the intraband contribution dominates the inter- band term in this limit the total damping rate also de- creases sharply as the magnon wavevector is increased for low scattering rates. For higher scattering rates, the electronspectralfunctionsaresufficientlybroadenedthat the overlap of intraband states does not decrease appre- ciably as the states are separated by finite wavevector (q·∂ǫn,k/∂k<¯hγgenerally holds over the Fermi sur- face). Therefore, the intraband contribution is largelyin- dependentofmagnonwavevectorathighscatteringrates. The interband contribution to damping involves scat- tering between states in different bands, separated by the magnon wavevector q. Isolating the interband damping contribution reveals that these contributions are insensi- tive to the magnon wavevector at higher scattering rates where they form the dominant contribution to damp- ing (see Fig. 3). To understand these observations we again compare the spectral broadening ¯ hγto the quasi- particle energy difference ∆m,k+q n,k=ǫm,k+q−ǫn,k. The quasiparticle energy difference may be approximated as4 ∆m,k n,k+q·∂∆m,k n,k/∂k. The interband energy spacings are effectively modulated by the product of the magnon wavevector and the slopes of the bands. At high scatter- ing rates, when the spectral broadening exceeds the ver- tical band spacings, this energy modulation is unimpor- tant and the damping rate is independent of the magnon wavevector. At low scattering rates, when the spec- tral broadening is less than many of the band spacings, this modulation can alter the interband energy spacings enough to allow or forbid generation of these electron- hole pairs. For Fe, Co, and Ni, this produces a modest increase in the interband damping rate at low scattering rates as the magnon wavevector increases. However, this effect is unimportant to the total damping rate, which remains dominated by the intraband terms at low scat- tering rates. Lastly, we describe the numerical methods employed in this study. Converged ground state electron densities were first obtained via the linear-augmented-plane-wave method. The Perdew-Wang functional for exchange- correlation within the local spin density approximation was implemented. Many details of the ground state den- sity convergence process are given in [23]. Densities were then expanded into Kohn-Sham orbitals using a scalar- relativistic spin-orbit interaction with the magnetiza- tion aligned along the experimentally determined mag- netocrystalline anisotropy easy axis. The Kohn-Sham energies were artificially broadened through the ad hoc introduction of an electron lifetime. Matrix elements of the torque operator Γ−= [σ−,Hso] were evaluated sim- ilarly to the spin-orbit matrix elements [24]. ( σ−is the spin lowering operator and Hsois the spin-orbit Hamil- tonian.) The product of the matrix elements and the weightingfunction wereintegratedover k-spaceusingthe special points method with a fictitious smearing of the Fermi surface for numerical stability. Convergence wasobtained by sampling the full Brillouin zone with 1603 k-points for Fe and Ni, and 1602x 91 points for Co. In summary, we have investigated the importance of non-local damping effects by calculating the intrinsic spin-orbit contribution to precession damping in bulk transition metal ferromagnets for small amplitude spin- waveswith finite wavelengths. Results ofthe calculations do not contradict the common-practice assumption that damping is a local phenomenon. For transition metals, at scattering rates corresponding to room temperature, we find that the single-mode damping rate is essentially independent of magnon wavevector for wavevectors be- tween zero and 1 % of the Brillouin zone edge. It is not until low temperatures in the most pure samples that non-local effects become significant. At these scatter- ing rates, damping rates decrease by as much as an or- der of magnitude as the magnon wavevector is increased. The insensitivity of damping rate to magnon wavevector at high scattering rates versus the strong sensitivity at low scattering rates can be explained in terms of band structure effects. Due to electron spectral broadening at high scattering rates the energy conservation constraint during magnon decay is effectively relaxed, making the damping rate independent of magnon wavevector. The minimal spectral broadening at low scattering rates – seenonlyinverypureandcoldsamples–greatlyrestricts the possible intraband scattering processes, lowering the damping rate. The prediction of reduced damping at low scattering rates and non-zero magnon wavevectors is of little practical importance, but could provide an accessi- ble test of the torque-correlationmodel. Specifically, this might be testable in ferromagnetic semiconductors such as (Ga,Mn)As forwhich manyspin-waveresonanceshave been experimentally observed at low temperatures [25]. This work has been supported in part through NIST- CNST / UMD-Nanocenter cooperative agreement. [1] L.LandauandE. Lifshitz, Phys.Z.Sowjet. 8, 153 (1935). [2] T. L. Gilbert, Armour research foundation project No. A059, supplementary report, unpublished (1956). [3] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). [4] V. Vlaminck and M. Bailleul, Science 322, 410 (2008). [5] S.S.P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). [6] M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S.S.P. Parkin, Science 320, 209 (2008). [7] B.E. Argyle, E. Terrenzio, and J.C. Slonczewski, Phys. Rev. Lett. 53, 190 (1984). [8] R.D. McMichael, C.A. Ross, and V.P. Chuang, J. Appl. Phys. 103, 07C505 (2008). [9] S. Kaka, M.R. Pufall, W.H. Rippard, T.J. Silva, S.E. Russek, and J.A. Katine, Nature 437, 389 (2005). [10] F.B. Mancoff, N.D. Rizzo, B.N. Engel, and S. Tehrani, Nature437, 393 (2005). [11] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Stat. Sol. 23, 501 (1967). [12] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).[13] V. Korenman and R.E. Prange, Phys. Rev. B 6, 2769 (1972). [14] V. Kambersk´ y, Czech. J. Phys. B 26, 1366 (1976). [15] J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J.K. Furdyna , W.A. Atkinson, and A.H. MacDonald, Phys. Rev. B 69, 085209 (2004). [16] Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [17] M. Zwierzycki, Y. Tserkovnyak, P.J. Kelly, A. Brataas, and G.E.W. Bauer, Phys. Rev. B 71, 064420 (2005). [18] K. Gilmore, Y.U. Idzerda, and M.D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). [19] V. Kambersk´ y, Phys. Rev. B 76, 134416 (2007). [20] K.Gilmore, Y.U.Idzerda, andM.D.Stiles, J.Appl.Phys . 103, 07D303 (2008). [21] M. F¨ ahnle and D. Steiauf, Phys. Rev. B 73, 184427 (2006). [22] J. Foros, A.Brataas, Y.Tserkovnyak,andG.E.W. Bauer, Phys. Rev. B 78, 140402(R) (2008). [23] L.F. Mattheiss and D.R. Hamann, Phys. Rev. B 33, 8235 (1986). [24] M.D. Stiles, S.V. Halilov, R.A. Hyman, and A. Zangwill, Phys. Rev. B 64, 104430 (2001). [25] S.T.B. Goennenwein, T. Graf, T. Wassner, M.S. Brandt,M. Stutzmann, A. Koeder, S. Frank, W. Schoch, and A. Waag, Journal of Superconductivity 16, 75 (2003).

2009-04-09

The Landau-Lifshitz-Gilbert damping parameter is typically assumed to be a local quantity, independent of magnetic configuration. To test the validity of this assumption we calculate the precession damping rate of small amplitude non-uniform mode magnons in iron, cobalt, and nickel. At scattering rates expected near and above room temperature, little change in the damping rate is found as the magnon wavelength is decreased from infinity to a length shorter than features probed in recent experiments. This result indicates that non-local effects due to the presence of weakly non-uniform modes, expected in real devices, should not appreciably affect the dynamic response of the element at typical operating temperatures. Conversely, at scattering rates expected in very pure samples around cryogenic temperatures, non-local effects result in an order of magnitude decrease in damping rates for magnons with wavelengths commensurate with domain wall widths. While this low temperature result is likely of little practical importance, it provides an experimentally testable prediction of the non-local contribution of the spin-orbit torque-correlation model of precession damping. None of these results exhibit strong dependence on the magnon propagation direction.

Evaluating the locality of intrinsic precession damping in transition metals

0904.1455v1

arXiv:1406.2491v2 [cond-mat.mtrl-sci] 13 Aug 2014Influence ofTa insertions onthe magneticproperties ofMgO /CoFeB/MgOfilms probed by ferromagnetic resonance MariaPatriciaRouelliSabino,Sze TerLim,andMichaelTran DataStorage Institute,Agency for Science, Technology and Research, 5Engineering Drive 1,117608 Singapore (Dated: April 3,2018) Abstract We show by vector network analyzer ferromagnetic resonance measurements that low Gilbert damping α, downto0.006,canbeachievedinperpendicularlymagnetize dMgO/CoFeB/MgOthinfilmswithultrathininsertionsofTa intheCoFeBlayer. AlthoughincreasingthenumberofTainse rtionsallowsthickerCoFeBlayerstoremainperpendicular , the effective areal magnetic anisotropy does not improve withmore insertions, whichcome withan increase in α. Perpendicularmagnetic anisotropy (PMA) is the key to furth er downscaling of spin transfer torque magnetoresistive randommemorydevices,asitallowstwokeyrequirementstob esatisfied: lowcriticalcurrent Ic0andhighthermalstability ∆, the latter of which is proportional to the energy barrier Ebbetween the two stable magnetic states. The spin torque switching efficiency, defined as Eb/Ic0, is commonly used as a metric to account for both requirement s. For a Stoner- Wohlfarthmodel,itisgivenby[1]( /planckover2pi1/4e)·(η/α),whereαistheGilbertdampingparameter,and ηisthespinpolarization factor, which is related to the tunnel magnetoresistance ra tio (TMR) byη=[TMR(TMR+2)]1/2/[2(TMR+1)]. It thus becomes evident that for high switching e fficiency, one has to decrease αwhile keeping TMR high. Magnetic tunnel junctions(MTJs)basedonCoFeB /MgOsystemsarewell knownto providehighTMR[2] andhaverec entlybeenshown to possess PMA, which is attributed to the CoFeB /MgO interface.[3] A Ta layer is usually placed adjacent to th e CoFeB to induce the proper crystallization necessary for PMA and h igh TMR [4]. In Ta /CoFeB/MgO systems, however, spin pumping to the Ta increases α. [5] Moreover, the CoFeB layer also needs to be ultrathin (ty pically less than 1.5nm) in ordertoexhibitPMA.[3]Toimprovethethermalstabilityas devicesarescaleddowntosmallerdiameters,increasingth e effectivearealanisotropyenergydensity Kef ftisdesired. One approach to address these issues is the use of double-MgO structures, i.e., those in which both the barrier layer and cappinglayer straddlingthe free layerare made of MgO. I mproved Ic0and/or∆have beenreportedin devicesusing doubleMgOfreelayers. [6,7,8,9]Theimprovementintherma lstabilityisattributedtotheadditionalCoFeB /MgOinter- face,whereaslower Ic0isassociatedwithlow α. Indeed,αdownto0.005hasbeenmeasuredinin-planeMgO /FeB/MgO films,[10] which agrees with device measurements.[11] The s tacks investigated in these damping studies, however, did not have the Ta layer used in practical free layers with perpe ndicular anisotropy.[9, 12] In addition, although the inte r- facial 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 perpendicularanisotropywasratherlow ( Kef ft≈0.04mJ/m2) relativetothat ofa Ta /CoFeB/MgOstack[3]. In thiswork, weexploretheinfluenceofTainsertionswithintheCoFeBlay erofMgO/CoFeB/MgOfilmsbymagnetometryandvector network analyzer ferromagnetic resonance (VNA-FMR) measu rements. The insertion of extremely thin Ta layers (0.3 nm) inside the CoFeB layer aids crystallization, allowing a larger total CoFeB thickness to remain perpendicular, [13] with aneffectivearealanistropycomparableto thatofTa /CoFeB/MgO. Two sample series were deposited by magnetron sputtering on SiO2substrates with seed layers of Ta 5 /TaN 20/Ta 5 in an ultrahigh vacuum environment (all thicknesses in nm). The stack configurations of the two sample series are: (1) MgO 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 - 1.5/Ta0.3/CoFeB1.0/MgO3(“double-insertion”),wheretheCoFeBcompositionis Co40Fe40B20(at%). TheTainsertion layer thickness is in the regime allowing strong ferromagne tic coupling between the CoFeB layers. [16] Two other sample serieswere grownasreferences: (a)MgO3 /CoFeB1.0-2.5/MgO3(“zero-insertion”),and(b)seed /CoFeB1.0- 2.5/MgO3(“single-MgO”).Forallthedouble-MgOsamples,anult rathinCoFeBlayerbelowthebottomMgOlayerwas alsodepositedforgoodMgOgrowth. Weconfirmedfromseparat emeasurementsthatthislayerdoesnotcontributetothe magneticsignal. All sampleswerecappedwith15nmofTa forp rotectionandwereannealedpost-growthat 300◦Cfor1 h in vacuum. Although3 nm MgO is too thick for practical use in MTJs, it was chosen to ensure continuityof the MgO layers and lessen the influence of the layersbeyondit.[10, 1 4, 15] (Measurementsof similar sampleswith 1 nm of MgO onbothsidesofthe magneticlayeryieldedthesametrends.) Magnetization measurements were performed using an altern ating gradient magnetometer (AGM). The PMA im- proveswith doublingof the CoFeB /MgO interface and with increasingnumber of Ta insertions, n, as shown in Fig. 1(a) for samples with a similar total nominal CoFeB thickness tnom≈2.5 nm. We also confirmed that we cannot obtain a 1perpendicular easy axis in double-MgO structures without T a insertions.[17] The double-insertion sample, on the othe r hand,exhibitslargeout-of-planeremanenceas shownin the inset of Fig. 1(a). A coercivefield less than0.01T (inset) is typicalofCoFeBfilmswithPMA[18,16]. /s49 /s50 /s51 /s52/s49/s50/s51/s52/s53 /s45/s48/s46/s48/s49 /s48/s46/s48/s49 /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 /s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48 /s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49 /s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s77/s47/s65/s32/s40/s65 /s109/s50 /s47/s109/s50 /s41/s32/s120/s32/s49/s48/s45/s54 /s116 /s110/s111/s109/s32/s40/s110/s109/s41/s48/s40/s98/s41 /s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s97/s46/s117/s46/s41 /s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s40/s84/s41/s116 /s110/s111 /s109/s32 /s32/s50/s46/s53/s110/s109/s40/s97/s41 Figure 1: (Color online) (a) Out-of-planeAGM loops for samp les with single-MgO (red squares), zero-insertion(purple invertedtriangles),single-insertion(bluecircles),an ddouble-insertion(greentriangles),withtotalnominalC oFeBthick- nesstnom≈2.5nm. Insetshowsalow-fieldout-of-planeloopforthesame doubleinsertionsample. (b)Magneticmoment perunitarea( M/A)asafunctionofthetotalnominalCoFeBthicknessforallsa mpleseries. Linearfitsareshownassolid lines.MSandtMDLcanbeextractedfromtheslopeand xintercept,respectively,andaresummarizedinTable 1. It is known that Ta can create a magnetically dead layer (MDL) when it is near a magnetic layer.[19] We plot the magneticmomentperarea against tnom[Fig.1(b)]to obtainthethicknessoftheMDL foreachseries fromthexintercept ofalinearfit. TheresultsaresummarizedinTable1,alongwi ththeMSvaluesobtainedfromtheslope. WefindanMDL thickness 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 insertionthicknessintherespectiveseries,andareconsi stentwiththepictureofCoFeBintermixingwithTatoproduc ea magneticallydeadvolume. The tMDLvalueforthesingle-MgOsamples(0.26 ±0.08nm)agreeswithvaluesfoundinthe literature.[14, 20, 21] On the other hand, no dead layer was f ound for the zero-insertionsamples, which is similar to the resultsinRef. [19]. Table1: SummaryofMagneticProperties Series tMDL(nm) MS(MA/m) Ki(mJ/m2) Kv(MJ/m3) single MgO 0 .26±0.08 1.51±0.08 1.61±0.07−0.29±0.08 double MgO, n=0 0.04±0.06 1.29±0.05 0.91±0.09 0.37±0.05 double MgO, n=1 0.24±0.09 1.12±0.05 2.18±0.08−0.34±0.04 double MgO, n=2 0.7±0.1 1.10±0.04 2.4±0.1−0.25±0.03 VNA-FMR was used to measure the e ffective anisotropy field and damping parameter of the samples . In the VNA- FMR setup, the samples were placed face down on a coplanar wav eguide and situated in a dc magnetic field of up to 1.2 T applied perpendicular to the film plane. The transmission scattering parameter S21was measured at a specific frequency while the dc field was swept. For each sweep, the rea l and imaginary parts of the resonance response were fitted simultaneouslyusingthecomplexsusceptibilityequ ation χ(H)=Mef f(H−Mef f+i∆H 2) (H−Mef f)2−/parenleftBig2π/planckover2pi1f gµB/parenrightBig2+i∆H(H−Mef f)(1) wherefis the frequencyofthe ac field, Mef f=MS−H⊥ K,∆His the fullwidth at half-maximum, H⊥ Kis the anisotropy field perpendicular to the plane, gis the spectroscopic splitting factor, µBis the Bohr magneton, and /planckover2pi1is the reduced Planck’s constant. Nonmagnetic contributions to the S21parameter and a linear time-dependentdrift of the instrume nts were taken into account during the fit. We note that only one re sonance peak is observed within the range studied. A representativefitofthesusceptibilitydataisshowninFig .2(a)foradouble-insertionsamplewith tnom=2.5nm. Inusing Eq.1,a valueof g=2isfirst assumedtoobtainvaluesfor Mef fand∆H, whichdoesnotaffectthe finalresult. Foreachfrequency,a resonancefield µ0Hres(f)=2π/planckover2pi1 gµBf+µ0Mef f (2) according to Kittel’s equation is calculated and plotted ag ainst the frequency, as shown in Fig. 2(c). A linear fit, now withgandMef fas fitting parameters, is then performed. The e ffective anisotropyenergydensity Kef fcan be calculated 2from the effective anisotropy field HKef f(=−Mef f) asKef f=HKef fMS/2, noting that a positive anisotropy constant correspondsto aperpendiculareasyaxis. Toobtainα,we performa linearfit ofthe measuredFMRlinewidthasafunc tionofthefrequencyto µ0∆H(f)=4π/planckover2pi1α gµBf+µ0∆H0 (3) where∆H0is the inhomogeneous linewidth broadening, and the value of gused is the fitted value from Eq. 2. We note that two-magnon scattering contributions to the linew idth are eliminated owing to the perpendicular measurement configuration[22]. Such a fit is shown in Fig. 2(c). Only data p oints taken well beyond the saturation field for each samplewereusedinthefit, andasymptoticanalysisasdescri bedinRef. [23]fortheaccessiblefrequencyrangewasalso performed. We define an effective thickness tef f=tnom−tMDLand show the calculated Kef ftef f(to which Ebis proportional) for both sample series in Fig. 3(a). The x-axis error bars originate from the fitting error in obtainin gtMDL. We find that fortef f>2 nm,double-insertionsampleshavehigher Kef ftef fthan single-insertionsamplesforthe same tef f. However, the maximum Kef ftef fachieved for both the single- and double-insertion series d oes not significantly exceed Kef ftef f measuredin ourthinnestsingle-MgOsample( tef f=1.0nm),similar tothatobservedin MTJmeasurements.[7] Tounderstandthisfurther,we considerthedi fferentcontributionsto Kef ftef f,whichisgivenby Kef ftef f=Ki+(Kv−µ0M2 S 2)tef f (4) whereKiis the total interfacial anisotropy constant, including al l CoFeB/MgO interfaces; Kvis the volume anisotropy constant; and the demagnetizing energy is given by the M2 Sterm. We assume that any interfacial anisotropy from the Ta/CoFeB interface is negligible.[24] Kiis commonly derived from the yintercept of a linear Kef ftef fversustef ffit, whereasKvcan be calculated from the slope if MSis known. Because it is possible that for CoFeB thicknesses b elow 1.0 nm,Kiis degraded because of Ta reaching the CoFeB /MgO interface,[25] we consider only the linear region of the curve during the fit. The calculated values, given in Table 1, demonstrate that the absence of a Ta insertion leads to the lowest value of Ki(0.91±0.09mJ/m2), explaining why n=0 samples did not exhibit a perpendiculareasy axis. On the other hand, Kiforn=1 (2.18±0.08mJ/m2) andn=2 (2.4±0.1mJ/m2) are both larger than the single-MgO series (1.61±0.07mJ/m2), as would be expected from the additional PMA from the secon d CoFeB/MgO interface. However, the anisotropy per interface did not double with the additional CoFeB /MgO, which may be attributed to the di fferent degrees of crystallization for single- and double-MgO samp les. An indication of better crystallization into CoFe in th e single-MgO series is its higher MS.Kvis negative and does not vary appreciably in samples where Ta is present, in contrast to the positive value found for zero-insertion sam ples. The role of Ta with regard to Kvis not yet understood, as previouslypointed out by Sinha et al. [20], and a detailed study of the amount, proximity,and profile of Ta would be necessarytoclarifythesee ffects. Turningourattentionto α,weidentifyasingle-MgOsample( tef f≈0.8nm)andasingle-insertionsample( tef f≈1.3 nm)withacomparable Kef ftef f≈0.2mJ/m2. Weimmediatelynoticethat αforthesingle-insertionsampleisaroundtwo timeslowerthanthatforthe single-MgOsample. This dramaticdecrease in αmay be attributedto the suppressionof spin pumpingby the Mg O layers straddlingboth sides of the precessing magnet.[10, 26] Indeed, measuremen ts of zero-insertion samples show no thickness dependence [purple dashed line in Fig. 3(b)] and a low mean value of α=0.0035±0.0002 comparable to the bulk damping of Co40Fe40B20[5,27]. However,adecreasein αwithincreasing tef fcanstillbeseeninboththesingle-anddouble-insertion series. One possible reason is the alloying of CoFeB and Ta, a s Ta is known to readily intermix with CoFeB [21], and higher damping may be expected from CoFeBTa alloys [28]. The relative percentage of CoFeBTa alloy decreases with increasing CoFeB thickness, coinciding with the αdecrease. This picture is also consistent with the jump in αfrom single-to double-insertionsamples, i.e.,thereismoreCo FeBTa alloybecausetherearemoreTa insertions. [11, 10] ItmayalsobepossiblethatspinpumpingtotheTainsertionl ayeroccurs,asinthecaseofthePdinterlayerinCoFe /Pd multilayers[29]. The complexityof oursystem, however,pr eventsusfromusinga simple multilayermodel. One reason isthatthemiddleCoFeBlayer(inthedouble-insertioncase )mayhavedifferentpropertiesfromtheCoFeBlayersadjacent to MgO, because CoFeB crystallizes from the MgO interface, [ 30] with which the middle CoFeB has no contact. The degree of Ta intermixing also depends on the deposition orde r and will vary across the structure.[19] At this point, we cannot discriminate the mechanism behind the damping behav ior. It may be worthwhile to study the use of CoFeBTa alloysasinterlayerstopossiblyhavemorecontroloverthe amountanddistributionofTa inthestack.[31] Inconclusion,wehavedemonstratedPMAandlowdampingindo uble-MgOstructures. AthinTainsertionlayerwas found to significantly increase the PMA - no perpendicular ea sy axis was realized in our MgO /Co40Fe40B20/MgO films without Ta - and adding more insertionsallowed thicker CoFe B layers to remain perpendicular. However, the maximum Kef ftef findouble-MgOsamplesiscomparableonlywiththatofthesin gle-MgOsampleforthisCoFeBcomposition.[9] On the other hand, αfor double MgO films increases with the number of insertions b ut is still lower than that of single 3MgO films for the entire range. Considering both trends with n, we find that the optimal stack in the range of samples we studiedis a double-MgO, n=1 sample with tnom=1.75nm,which exhibits Kef ftef f=0.27mJ/m2at a low damping valueof0.006. Acknowledgement We expressgratitudeforsupportfromtheA*STARGraduateAc ademySINGAProgram. References [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, G.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, T.Topuria,andD.C. Worledge: Phys.Rev.B88,104426(2013) . [2] S. Yuasa: J. Phys.Soc.Jpn.77,031001(2008). [3] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno: Nat.Mater. 9,721(2010). [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. O’Sullivan,andR. P.Robertazzi: Appl.Phys.Lett.98,0225 01(2011). [5] X. Liu,W.Zhang,M.J. Carter,andG. Xiao: J. Appl.Phys.1 10,033910(2011). [6] G. Jan, Y.-J. Wang, T. Moriyama, Y.-J. Lee, M. Lin, T. Zhon g, R.-Y. Tong, T. Torng, and P.-K. Wang: Appl. Phys. Express5,093008(2013). [7] J.-H. Park, Y. Kim, W. C. Lim, J. H. Kim, S. H. Park, W. Kim, K . W. Kim, J. H. Jeong, K. S. Kim, H. Kim, Y. J. Lee, S. C. Oh, J. E. Lee, S. O. Park, S. Watts, D. Apalkov, V. Nik itin, M. Krounbi, S. Jeong, S. Choi, H. K. Kang, andC. Chung: VLSITech.Dig.57(2012). [8] H. Kubota, S. Ishibashi, T. Saruya, T. Nozaki, A. Fukushi ma, K. Yakushiji, K. Ando, Y. Suzuki, and S. Yuasa: J. Appl.Phys.111,07C723(2012). [9] H. Sato,M.Yamanouchi,S.Ikeda,S. Fukami,F. Matsukura ,andH.Ohno: Appl.Phys.Lett. 101,022414(2012). [10] M.Konoto,H.Imamura,T.Taniguchi,K.Yakushiji,H.Ku bota,A.Fukushima,K.Ando,andS.Yuasa: Appl.Phys. Express6,073002(2013). [11] S.Tsunegi,H.Kubota,S.Tamaru,K.Yakushiji,M.Konot o,A.Fukushima,T.Taniguchi,H.Arai,H.Imamura,and S. Yuasa: Appl.Phys.Express7,033004(2014). [12] K. Yakushiji,A.Fukushima,H.Kubota,M.Konoto,andS. Yuasa: Appl.Phys.Express6,113006(2013). [13] V. B. Naik,H.Meng,andR. Sbiaa: AIP Advances,2,042182 (2012). [14] D. D. Lam, F. Bonell, S. Miwa, Y. Shiota, K. Yakushiji, H. Kubota, T. Nozaki, A. Fukushima, S. Yuasa, and Y. Suzuki: J. Kor.Phys.Soc.62,1461(2013). [15] M. P.R. G.Sabino,S.T.Lim,andM.Tran: (unpublished). [16] V. Sokalski,M.T.Moneck,E.Yang,andJ.-G. Zhu: Appl.P hys.Lett 101,072411(2012). [17] H. Sato,M.Yamanouchi,S.Ikeda,S. Fukami,F. Matsukur a,andH.Ohno: IEEETrans.Mag.49,4437(2013). [18] G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagt en, and B. Koopmans: Appl. Phys. Lett. 94, 102501 (2009). [19] S. Y. Jang,C.-Y. You,S. H.Lim,andS. R.Lee.: J. Appl.Ph ys.109,013901(2011). [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. Parkin,andH.Ohno: Appl.Phys.Lett.102,242405(2013). [21] Y.-H. Wang, W.-C. Chen, S.-Y. Yang, K.-H. Shen, C. Park, M.-J. Kao, and M.-J. Tsai: J. Appl. Phys. 99, 08M307 (2006). [22] N.Mo,J.Hohlfeld,M.ulIslam,C.S.Brown,E.Girt,P.Kr ivosik,W.Tong,A.Rebei,andC.E.Patton: Appl.Phys. Lett. 92,022506(2008). 4[23] J. M.Shaw,H.T.Nembach,T.J. Silva,andC.T. Boone: J. A ppl.Phys.114,243906(2013). [24] Y. Miura,M. Tsujikawa,andM.Shirai: J. Appl.Phys.113 ,233908(2013). [25] N. Miyakawa,D.C. Worledge,andK.Kita: IEEEMagn.Lett .4,2(2013). [26] O. Mosendz,J.E. Pearson,F. Y. Fradin,S. D.Bader,andA . Hoffmann: Appl.Phys.Lett. 96,022502(2010). [27] M. Oogane,T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and T. Miyazaki: Jpn. J. Appl. Phys. 45, 3889 (2006). [28] J.O.Rantschler,R.D.McMichael,A.Castillo,A.J.Sha piro,W.F. Egelhoff,B.B.Maranville,D.Pulugurtha,A.P. Chen,andL.M.Connors: J. Appl.Phys.101,033911(2007). [29] J. M.Shaw,H.T.Nembach,andT.J.Silva: Phys.Rev.B85, 054412(2012). [30] S. Mukherjee,F. Bai, D.MacMahon,C.-L.Lee,S.Gupta,a ndS. Kurinec: J. Appl.Phys.106,033906(2009). [31] K. Tsunoda, H. Noshiro,C. Yoshida, Y. Yamazaki, A. Taka hashi, Y. Iba, A. Hatada, M. Nakabayashi,T. Takenaga, M. Aoki,andT.Sugii.IEDM,29.1.1(2012). 5/s50/s48/s48 /s50/s50/s48 /s50/s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48 /s40/s99/s41/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s70/s105/s101/s108/s100/s32/s40/s109 /s84/s41/s32 /s32 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48 /s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s40/s109 /s84/s41/s32/s68/s97/s116/s97 /s32 /s32/s70/s105/s116 /s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s71/s72/s122/s41/s32/s68/s97/s116/s97 /s32/s70/s105/s116/s83 /s50/s49/s32/s82/s101/s97/s108/s32/s40/s97/s46/s117/s46/s41 /s40/s97/s41 /s40/s98/s41/s83 /s50/s49/s32/s73/s109/s97/s103/s46/s32/s40/s97/s46/s117/s46/s41 /s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s109/s84/s41 Figure 2: (a) Real and (b) imaginary parts of the S21parameter obtained from VNA-FMR measurements for a double- insertion sample with tnom=2.5 nm at 12 GHz while a perpendiculardc magnetic field is swept. The lines are fits to an expressionusingEq.1, takingnonmagneticcontributionst oS21anda lineardriftintoaccount. (c)Field-sweptlinewidth andresonancefieldsforthesamesampleasafunctionoffrequ ency. Thelinearfitsdescribedinthetextareusedtoextract HKef f(=−Mef f)andα. /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 /s48 /s49 /s50 /s51/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s115/s105/s110/s103/s108/s101/s32/s77/s103/s79 /s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s48 /s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s49 /s32/s100/s111/s117/s98/s108/s101/s32/s77/s103/s79/s44/s32/s110/s61/s50/s75 /s101/s102/s102/s116 /s101/s102/s102/s32/s40/s109/s74/s47/s109/s50 /s41 /s40/s97/s41/s40/s49/s48/s45/s51 /s41 /s116 /s101/s102/s102/s32/s40/s110/s109/s41/s40/s98/s41 Figure3: (a) Kef ftef fand(b)αversuseffectiveCoFeBthickness tef fobtainedfromfield-sweptVNA-FMRmeasurements for all sample series. Solidlines in (a)are linear fits. Purp ledashedline in (b) correspondsto the mean αvalueaveraged overall zero-insertionsamples,whichwasfoundtobeconst antwithinerroracrosstheentirethicknessrangestudied. 6

2014-06-10

We show by vector network analyzer ferromagnetic resonance measurements that low Gilbert damping {\alpha} down to 0.006 can be achieved in perpendicularly magnetized MgO/CoFeB/MgO thin films with ultra-thin insertions of Ta in the CoFeB layer. While increasing the number of Ta insertions allows thicker CoFeB layers to remain perpendicular, the effective areal magnetic anisotropy does not improve with more insertions, and also comes with an increase in {\alpha}.

Influence of Ta insertions on the magnetic properties of MgO/CoFeB/MgO films probed by ferromagnetic resonance

1406.2491v2

Springer Nature 2021 L ATEX template Frictional weakening of a granular sheared layer due to viscous rolling revealed by Discrete Element Modeling Alexandre Sac-Morane1,2*, Manolis Veveakis1†and Hadrien Rattez2† 1*Multiphysics Geomechanics Lab, Duke University, Durham, 27708, NC, USA. 2Institute of Mechanics, Materials and Civil Engineering, UCLouvain, Louvain-la-Neuve, 1348, Belgium. *Corresponding author. E-mail: alexandre.sac-morane@uclouvain.be; Contributing authors: manolis.veveakis@duke.edu; hadrien.rattez@uclouvain.be; †These authors contributed equally to this work. Abstract Considering a 3D sheared granular layer through a discrete element modeling, it is well known the rolling resistance influences the macro friction coefficient. Even if the rolling resistance role has been deeply investigated previously because it is commonly used to represent the shape and the roughness of the grains, the rolling viscous damping coefficient is still not studied. This parameter is rarely used or only to dissipate the energy and to converge numerically. This paper revisits the physical role of those coefficients with a parametric study of the rolling friction and the rolling damping at different shear speeds and different confinement pressures. It has been observed the damping coefficient induces a frictional weakening. Hence, competition between the rolling resistance and the rolling damping occurs. Angular resistance aims to avoid grains rolling, decreasing the difference between the angular velocities of grains. Whereas, angular damping acts in the opposite, avoiding a change in the difference between the angular velocities of grains. In consequence, grains stay rolling and the sample toughness decreases. This effect must be considered to not overestimate the frictional response of a granular layer. Keywords: Discrete element method, Rolling parameter, Sheared layer friction, Granular materials 1 Introduction Accurately measuring or calculating the frictional strength of granular sheared layers is of paramount importance across all fields of granular media- related sciences, including earthquakes and fault mechanics [1, 2], landslides [3], and debris flows [4] to name but a few. It is very well accepted nowadays that the calculation of a macroscopic property like the frictional coefficient of granular media is the result of grain-to-grain interactions atthe micro-scale [5, 6]. Therefore, in order to accu- rate capture those effects and homogenize them to the friction coefficient of a layer, higher order analytical and numerical approaches need to be considered [7–9]. One of the most well accepted approaches in direct modeling of granular media is the Discrete Element Method [10], which has been designed to consider those interactions between grains [11]. The method started with a simple linear contact law[12, 13], but since then contact laws have been modified by [14]: (i) considering the grain crushing 1arXiv:2310.15945v1 [physics.geo-ph] 24 Oct 2023Springer Nature 2021 L ATEX template 2 Rolling relaxation controls friction weakening [15, 16], (ii) investigating the effect of the pres- sure solution [17–19], (iii) exploring the effect of the healing [20, 21], (iv) appreciating the influence of the cohesion in the matter [22–24] or (v) the cohesion induced by the pore fluid [25, 26] among others. Also, it allows some to focus on the tem- perature influence, identifying the pressurization of the pore fluid [27, 28] and grain melting [29, 30] as the main phenomena driving the evolution of the frictional strength of a fault zone during large crustal events. The present work is motivated by previous experiments made on antigorite [31], and the main goal is to assess the influence on global behav- ior of contact laws and parameters values. Even though different relevant outputs for dense gran- ular flows are reviewed by the French research group Groupement de Recherche Milieux Divis´ es (GDR MiDi) [32], we are focused in this paper on the macro friction coefficient at steady state. As such, we introduce and explore the influence of the rolling resistance between grains to the macroscopic strength of a granular sheared layer. Experimental results [33, 34], and numerical ones [35–37] have highlighted that grain rolling has a real impact on the sample behavior with many rolling models being formulated since [38, 39]. In the literature the elastic-plastic spring dash- pot model is identified as the benchmark for this response [40, 41] and has been extended to con- clude that: (i) rolling helps the formation of shear bands and decrease the sample strength [42–45]; (ii) the stress-dilatancy curves are modified [46– 49] when accounting for the rolling resistance coming from intragranular friction [50, 51] and roughness [52–54]. However, the computational cost of these approaches is not negligible, and especially if grains clusters [55], superquadric particles [56] or even polyhedral shapes [57–59] are assumed to approximate the shape, those simulations become quickly computationally costly. Because of this fact, geometric laws for the rolling friction have been developed [60, 61] allowing for simulations to keep using round particles with a rolling resistance stemming from an equivalent shape. However, the introduced angular damping influence is not well constrained, hence being neglected in most of the DEM simulations only used for stability reasons [38, 40] rather than for physical robustness [62] In this work we revisit the physical role of the rollingresistance in a granular sheared layer and perform a parametric study over the rolling friction and the rolling viscous damping coefficients to understand better their influence on the macroscopic friction coefficient of a granular sheared layer. 2 Theory and formulation The Discrete Element Model (DEM) is an approach developed by Cundall & Strack [11] to simulate granular materials at the particles level. The foundation of this method is to consider inside the material the individual particles and their interactions explicitly. Newton’s laws (linear and angular momentum) are used to compute the motion of the grains, as follows: m∂vi ∂t=m×gi+X fi (1) I×ρ∂ωi ∂t=X ϵijkfjRk+X Mi (2) where mis the particle mass, vthe particle velocity, gthe gravity acceleration, fthe contact forces, Ithe moment of inertia, ρthe particle den- sity, ωthe angular velocity, ϵijkthe Levi-Civita symbol, Rthe radius, Mthe contact moments. Considering two particles with radii R1and R2, the interaction between particles is computed only if the distance δbetween grains satisfies the following inequality: δ < R1+R2(3) Once contact is detected between grains 1 and 2, interactions (force and moment) are computed from relative motions ∆ uand ∆ ωas: ∆ui=u1 i−u2 i+ϵijk R1 jθ1 k−R2 jθ2 k
(4) ∆ωi=ω1 i−ω2 i (5) where uis the particle displacement, θthe angular displacement and ωthe angular velocity of the grain. The contact models between cohesionless par- ticles obey the Hertz contact theory [63]. Normal, tangential and angular models are shown at figure 1 and described in the following.Springer Nature 2021 L ATEX template Rolling relaxation controls friction weakening 3 Fig. 1 The contact between two particles obeys to normal, tangential and rolling elastic-plastic spring-dashpot laws. As the contact can happen between particles with different properties, some equivalent param- eters need to be defined. The equivalent radius R∗ and equivalent mass m∗are defined at equations 6 and 7 with an harmonic mean. 1 R∗=1 R1+1 R2(6) 1 m∗=1 m1+1 m2(7) The equivalent Young modulus Y∗and shear modulus G∗are defined at equations 8 and 9 with an harmonic mean adjusted by Poisson’s ratio. 1 Y∗=1−ν12 Y1+1−ν22 Y2 ⇐⇒Y∗=Y 2(1−ν2)(8) 1 G∗=2(2−ν1)(1 + ν1) Y1+2(2−ν2)(1 + ν2) Y2 ⇐⇒G∗=Y 4(2−ν)(1 + ν) (9) The equivalent moment of inertia I∗is defined at equation 10 with an harmonic mean of the dif- ferent moments of inertia displaced at the contact point. 1 I∗=1 I1+m1R12+1 I2+m2R22 (10) Normal model The normal force is formulated as: fn=kn∆n−γnvn (11) The reaction is divided into a spring part and a damping part, with the normal stiffness knformulated as: kn=4 3Y∗p R∗∆n (12) Following the Hertz contact theory, this parameter depends mainly on the normal overlap ∆ n. Thus, the normal force is not linear with respect to the overlap. The normal stiffness depends also on the equivalent Young modulus Y∗and the equivalent radius R∗defined before. The normal damping γn is null in this paper because the restitution coef- ficient eis taken at the value 1. This choice have been made to focus on the influence of the rolling damping. Tangential model The tangential force is formulated to verify the Coulomb friction law defined on the friction coef- ficient between particle µpand the normal force fn: ft=kt∆t−γtvt≤µpfn (13) The reaction is divided into a spring part and a damping part. The tangential stiffness ktis formulated as: kt= 8G∗p R∗∆n (14) Following the Hertz contact theory, this parameter depends mainly on the normal overlap ∆ n. Thus, the tangential force is not linear with respect to the overlap. The tangential stiffness depends also on the equivalent shear modulus G∗and the equiv- alent radius R∗defined before. The tangential damping γtis also null in this paper because the restitution coefficient eis taken at the value 1. Angular model A lot of angular models could be applied but an elastic-plastic spring-dashpot model is used because it is the most accurate choice. Hence, the model allows energy dissipation during relative rotation and provides packing support for static system, two main functions to verify in a par- ticulate system [38]. The reaction moment Mis formulated as: M=Mk+Md(15)Springer Nature 2021 L ATEX template 4 Rolling relaxation controls friction weakening This reaction is divided into a spring part Mk and a damping part Mddefined at equations 16 and 19: Mk t+∆t=Mk t−kr∆θ≤Mm(16) The incremental angle ∆ θis obtained by a time integration of the angular velocity ∆ ω× dt. The angular stiffness kris formulated at equation 17 by considering a continuously dis- tributed system of normal and tangential spring at the interface [38], [62]. kr= 2,25knµ2 rR∗2(17) The rolling friction coefficient µris introduced. This variable is a dimensionless parameter defined as [38]: µr=tan(β) (18) The angle βrepresents the maximum angle of a slope on which the rolling resistance moment counterbalances the moment due to gravity on the grain, see figure 2. The influence of µris investigated in this paper. Fig. 2 Definition of the rolling resistance coefficient µr= tan(β). Md t+∆t= −Cr∆ωifMk t+∆t< Mm 0 if Mk t+∆t=Mm (19) It appears the damping part Mdis defined with a rolling viscous damping parameter Cr formulated at equation 20. Cr= 2ηrp Irkr (20) The rolling viscous damping coefficient ηr is introduced. This variable is a dimensionless parameter and its influence is investigated in this paper. Moreover, it appears the rolling viscous damping parameter depends on the rolling stiff- ness krand so on the rolling friction coefficient µr.As described by equations 16 and 19, the spring and damping parts are restricted by a plas- tic behavior, the rolling of particles. The rolling starts when the spring part reaches the plastic limit Mmdefined at equation 21. Mm=µrR∗fn (21) This limit depends on the equivalent radius, the rolling friction coefficient and the normal force. Once rolling occurs, the reaction from the angular spring takes the value of Mmand the damping element is deleted. 3 Numerical model The simulation setup is illustrated at figure 3. The box is a 0 ,004m×0,006m×0,0024mregion. Faces x and z are under periodic conditions. The size of domain has been chosen to respect a suf- ficient number of grain over the different axis. On the axis x, the shearing direction, there are lx/d50= 4/0.26 = 15 particles. On the axis z, the minor direction, there are lz/d50= 2.4/0.26 = 10 particles. On the axis y, the size allows the parti- cles generation shown at figure 4. Sizes have been minimized to reduce the number of grains and so the computational cost, main trouble with DEM simulations. The gravity is not considered because its effect stays negligible under the vertical pres- sure applied. Fig. 3 The simulation box with triangle plates and peri- odic faces. The simulation, made by the open-source soft- ware LIGGGHTS [64], is in several steps illus- trated at figure 4: 1. The box, bottom and top triangle plates are created. The triangle pattern represents the roughness of the plates with a geometry simi- lar to experimental tests [65, 66]. The specific size of the triangle is defined to be 1 .5 times the largest particle diameter.Springer Nature 2021 L ATEX template Rolling relaxation controls friction weakening 5 2. 2500 particles are generated following the dis- tribution presented in table 1 equivalent to the one used in [21]. This number of particles allows to get 17 particles on the height where the residual shear band sizes for different distribu- tions was assumed to be between 9 ×d50and 16×d50[67]. 3. Top plate applies vertical stress of 10 MPa by moving following the y axis. This plate is free to move vertically to verify this confin- ing and allow volume change. The value of the vertical stress has been chosen from previous experiments and numerical simulations [68–70]. 4. The sample is sheared by moving the bot- tom plate at the speed of 100 µm/s until 100% strain. This step is then repeated at the speed of 300 and 1000 µm/s. Those veloci- ties have been chosen from previous numerical simulations [70] and in-situ estimations [71] to minimize the computational cost but still to represent real sheared layers. Fig. 4 The simulation is in multiple steps : creation of the box and particles, application of the normal force and shearing. Radius Percentage Number of particles R1 = 0 ,2mm 14% 2500 R2 = 0 ,15mm 29% R3 = 0 ,1mm 57% Table 1 Distribution used described by discrete radius, percentage of the mass and total number of grains. Then, the influence of the vertical pressure is investigated. The same set-up is used except the vertical pressure ( P= 1MPa and = 100 MPa ). The rolling friction coefficient is constant µr= 0,5 and the rolling viscous damping coefficient changes ηr= 0,25, 0,5 or 0 ,75. The different parameters needed for the DEM simulation are presented in table 2 and the value has been chosen from previous articles to represent rock material [19, 31, 44, 70]. We can notice the time step dtmust verify the Rayleigh conditionVariable Short NameValue Simulation variables Time step dt 1,5e−6s Height of the sampleh 0,005 m Shear rate γ′2−6−20% Contact stiffness numberκ 400 Inertial number I 10−6−10−5 Mechanical variables Density ρ 2000000 kg/m3 Youngs modulus Y 70GPa Poissons ratio ν 0,3 Restitution coef- ficiente 1 Rolling friction coefficientµr 0−0,25−0,5−0,75−1 Rolling vis- cous damping coefficientηr 0−0,25−0,5−0,75 Friction coefficientµp 0,5 Table 2 DEM parameters used during simulations. [63, 72, 73] defined as: dtR=π×r×p ρ/G 0,1631×ν+ 0,8766(22) With every computing test, the main prob- lem is the running time. The time step dtmust be selected considering the number of particles, the computing power, the stability of the sim- ulation and the time scale of the test. In our case, we are looking for a 102seconds term. If we include the default value into equation 22 the time step is around 10−8second and the running time skyrockets. To answer this we can easily change the density ρand the shear modulus G. We will see those parameters are included in two dimen- sionless numbers defined at the equation 23: the contact stiffness number κ[74–76] and the inertial number I[32, 75]. κ= Y P(1−ν2)2/3 The contact stiffness number I=γ′d50p ρ/P The inertial number (23) Where γ′=vshear/his the shear rate, his the height of the sample during the shear, d50the mean diameter, Pthe pressure applied. It appears the constitutive law is sensitive to κ because grains are not rigid enough ( κ≤104) [74].Springer Nature 2021 L ATEX template 6 Rolling relaxation controls friction weakening By this fact, it becomes not possible to change the Young modulus Y(and so the shear modulus G). The inertial number Irepresents the behavior of the grains, which can be associated with solids, liquids or gases [77]. This dimensionless parame- ter does not affect the constitutive law if the flow regime is at critical state ( I≤10−3) [32, 75]. In conclusion, the density ρcan be modified, if we stay under the condition I≤10−3, to increase the time step and solve our computing problem. 4 Results and discussion Rolling Parameters Influence A parametric study has been done on the rolling friction coefficient µrand the rolling vis- cous damping coefficient ηr. As figure 5 shows, the macro friction coefficient is plotted following the shear strain. This coefficient µis computed by considering µ=Fy/Fx, where Fy(resp. Fx) is the component following the y-axis (resp. x-axis) of the force applied on the top plate. Because of the granular aspect, there is a lot of oscillation. To reduce this noise, at least 3 simulations are run by a set of parameters ( µr,ηr) and a mean curve is computed. Moreover, only the steady-state is considered and an average value is estimated. Fig. 5 Example of µ-γcurve (dotted lines mark the dif- ferent velocity steps 100 - 300 - 1000 µm/s). The comparison of the macro friction coeffi- cient with different parameters set is highlighted at figure 6. It appears there is an increase of the sheared layer friction coefficient with the rolling resistance µruntil a critical point depending on the rolling damping ηr. This reduction of the stiffness with the rolling damping is not easy to understand at the first point. The larger is the damping, the stiffer should be the system. Twomain questions should be answered: why does the friction coefficient decrease with the rolling resis- tance if there is damping? Why is the reduction larger with the damping value? vshear = 100 µm/s vshear = 300 µm/s vshear = 1000 µm/s Fig. 6 Evolution of the macro friction coefficient with the rolling friction coefficient µrand the rolling viscous damping coefficient ηrat different shear speeds with P= 10MPa . Figure 7 helps to understand the behavior. It shows the rotation of particle (in red) during four different cases. We can notice that the fewer rota- tions there are, the stiffer the system will be. It appears the number of rolling particle increases with the rolling resistance, see the three first plots of figure 7. The decrease of the friction coefficient is explained by particles rolling. Moreover, it is shown at the second and last plot of figure 7 that damping increases the number of rolling particles and so the friction coefficient is reduced.Springer Nature 2021 L ATEX template Rolling relaxation controls friction weakening 7 µr= 0,25−ηr= 0,5 µr= 0,5−ηr= 0,5 µr= 1−ηr= 0,5 µr= 0,5−ηr= 0 Fig. 7 Slide of sample highlighting grain rotation (rolling is in red) for several cases. A focus on the model equations must be done at relation 24 to understand better those observa- tions (the input rolling parameters are emphasized in red). First, it appears the increment of the spring ∆ Mk rdepends on µ2 rwhile the plastic limit µrR∗fndepends only on µr. There is a square factor between those values. Thus, this plastic limit, and so grain rolling, is reached easier with a larger rolling resistance µrfor a same angular displacement θ. Mm=µrR∗fn ∆Mk r=−2,25knµr2R∗2∆θ Md r,t+∆t= if|Mk r,t+∆t|< µrR∗fn: −2ηrµr√2,25Irknω if|Mk r,t+∆t|=µrR∗fn: 0(24) Concerning the damping, it avoids the vari- ation of the angular position (∆ ω→0) during the elastic phase. As we have seen before, the main part of the sample is at the plastic phase and particles roll. So, it is as the damping acts in opposition of the angular spring, keeping grain into the plastic phase. We can notice that we have decided in this paper to shut down the damping moment when the angular plastic limit is reached (see equation 24). In this way, we can understand better the reduction of the friction coefficient with the rolling stiffness µrif damping is active. We can noticethere is no decrease but an increase of the friction coefficient in the case of no damping. In absence of this one, the angular spring can act normally. The larger the rolling parameter is, the stiffer the global sample is. This observation can be used if we decide to track particle size distribution and grain shape evolution. Experiments and simulations have high- lighted that particles tend to become less or more [78–80] rounded under large deformation. The work of Buscarnera and Einav, extending the Continuum Breakage Mechanics, reconciles those conflicting observations [81]. Shape descrip- tors are converging to attractors. The evolution of the aspect ratio α, related to the grain morphol- ogy, is plotted following a breakage parameter B (B= 0 means unbroken state and B= 1 complete breakage) and the stress σ. It appears that from different initial values the aspect ratio converges to the the same value, the attractor. Softening and hardening behavior can be understood thank to relations between shape and rolling friction coefficient, the evolution of the grain shape and our work. For example, if the aspect ratio decreases during the test, particles become less rounded, the rolling friction coefficient increases, the sample toughness evolves depending on the position of the critical point (softening or hardening). Particle Size Distribution Influence As equation 24 shows, the plastic moment depends on the equivalent radius R∗. Appendix A notices smallest particles tend to roll more than the others. The particle size distribution should influence the macro friction coefficient. Results from this observation are described at appendix B. This one highlights the fact that macro fric- tion coefficient evolves with mean diameter d50. There is an increase of the residual value with this parameter. Unfortunately, other simulations must be done to interpolate an accurate tendency. In facts, the number of particles (and so the height of the original sample) must be increased. Thus, particle size distribution with larger mean diame- terd50can provide a sufficient number of particle to reduce the DEM noise. Vertical Pressure InfluenceSpringer Nature 2021 L ATEX template 8 Rolling relaxation controls friction weakening Moreover, equation 24 highlights the plastic moment depends on normal forces fn(and so on vertical pressure P). Figure 8 illustrates the evolu- tion of the macro friction coefficient with different vertical pressure Pand rolling viscous damping coefficient ηr. vshear = 100 µm/s vshear = 300 µm/s vshear = 1000 µm/s Fig. 8 Evolution of the macro friction coefficient with the rolling viscous damping coefficient ηrand the vertical pres- surePat different shear speeds with µr= 0,5. It appears the critical point defined previ- ously (point before which the friction weakening appears) depends on the vertical pressure. This behaviour has already been observed [70, 82]. It appears the importance of the rolling increases with the vertical pressure. The mean angular velocity ωzon the z-axis have been computed over configurations P−ηr−vshear at table 3. It is highlighted that the vertical pressure Pfavours the grain rolling. That is why a friction weakening occurs with this parameter.vshear = 100 µm/s ηr= 0,25 0,50 0,75 1MPa 0,3 2,2 33,7 10MPa 20,3 455,6 1056,2 100MPa 2384,57467,811774 ,1 vshear = 200 µm/s ηr= 0,25 0,50 0,75 1MPa 0,6 3,6 34,1 10MPa 23,9 426,6 1034,5 100MPa 2327,37564,811685 ,2 vshear = 1000 µm/s ηr= 0,25 0,50 0,75 1MPa 1,3 4,9 29,7 10MPa 28,1 342,1 1023,3 100MPa 2070,07483,611759 ,4 Table 3 The mean angular velocity ωzon the z-axis for different configurations ηr−Pat different shear speeds. Speed Influence Figure 9 highlights the shearing speed influ- ence on the system. It is the same results as before but plotted in another way. It appears there is no speed effect visible in most simulations as the friction coefficient keeps the same value. It is not surprising that no speed effects are spot- ted because there are no other parameters except the damping parameter which depends on speed or time. A speed influence is nevertheless noticed for cases where the friction coefficient starts to decrease with rolling resistance (for example the caseµr= 0,5 and ηr= 0,5 at figure 9). As shown at figure 7 for this set, few particles (in orange or in white) are still not rolling during this critical step. The damping value is not large enough to cancel the effect of the spring and few grains are in the elastic phase. The damping creates so in this case a speed influence. If the damping value is larger, we have seen particles tend to be all in the plastic phase. If it is lower, the damping is negligible or null. In both cases, the speed effect disappears. 5 Conclusion In this paper, we have considered granular materi- als into a plane shear configuration to investigate the effect of the rolling resistance and damping on the macroscopic friction coefficient. Thank numer- ical DEM simulations, the relation between those parameters becomes clearer. It appears :Springer Nature 2021 L ATEX template Rolling relaxation controls friction weakening 9 ηr= 0 ηr= 0,25 ηr= 0,5 ηr= 0,75 Fig. 9 Evolution of the sample friction coefficient with the shear speed and the rolling friction coefficient µrat differ- ent rolling viscous damping coefficients with P= 10 MPa . 1. In the no damping case, the sample stiffness increases with the rolling resistance.2. The consideration of the rolling damping intro- duces a critical point. For a constant damping value, the sample stiffness increases the rolling parameter until this critical point is reached. Then, the stiffness starts to decrease until a residual value. Hence, the damping tend to act against the spring and grains roll. 3. No visible speed effects have been highlighted except at critical point. For the same rolling resistance value : (i) When the damping param- eter is not large enough, the angular spring is the main element and no speed dependency is spotted, (ii) when the damping parameter is too large, all grains are in the plastic phase (roll) and the residual value is reached and (iii) when the damping parameter is at critical value, there is no main element in the rolling model, speed dependency occurs. 6 Acknowledgements Computational resources have been provided by the Consortium des ´Equipements de Calcul Inten- sif (C ´ECI), funded by the Fonds de la Recherche Scientifique de Belgique (F.R.S.-FNRS) under Grant No. 2.5020.11 and by the Walloon Region. Support by the CMMI-2042325 project is also acknowledged. 7 Statements and Declarations Conflict of interest The authors declare that they have no conflict of interest. Appendix A Distribution of the rolling with the radius Figure A1 highlights the distribution of the rolling in the sample with µr= 0,5,ηr= 0,5 and vshear = 100 µm/s . It appears smallest particles are rolling more than largest particles. The obser- vation stays qualitative and should be studied quantitatively to estimate better the behaviour.Springer Nature 2021 L ATEX template 10 Rolling relaxation controls friction weakening Fig. A1 Distribution of the rolling in the sample (blue is small rolling, red is large rolling). Appendix B Influence of the Particle Size Distribution During the creation of a fault zone, the parti- cles are crushed and the particle size distribution evolves greatly. Moreover, the figure A1 from the appendix A highlights the fact that small parti- cles tend to roll more than large ones (decreasing thus the sample toughness). To investigate how this evolution can affect the frictional behavior of a fault, the Particle Size Distribution (PSD) is described by two main parameters: the mean diameter d50and the fractal dimension Ddefined at equation B1 [83–85]. N=r−D(B1) with Nis the number of particle of radius r. Simulations have been run at the shear speed of 1000 µm/s during one step. Two rolling resis- tance µrare investigated 0 ,2 (smooth particles) and 1 ,4 (rough particles) and the rolling damping ηris not considered (= 0). The different PSD used are described in table B1 and results are shown in figure B2. In the case of smooth particle, it seems curves are really similar. The only exception is the case D 1 D50 0,26 where a peak appears. More dif- ferences can be appreciated in the case of rough particles. A peak can be shown at the start of the shear movement. This peak varies with the PSD, for example, the D 2,6 D50 0,4 one is more pro- nounced than the D 1 D50 0,4 one. It has already been observed that the peak tends to increase with the fractal dimension [67]. PSD has a significant influence on the residual friction coefficient if the rolling resistance becomes large. In fact, this parameter seems to stay aroundPSD NameRadius Percentage Number of particles D 1 R1 = 0 ,25mm 27% 750 D50 0,4 R2 = 0 ,21mm 32% R3 = 0 ,16mm 41% D 2 R1 = 0 ,25mm 22% 750 D50 0,4 R2 = 0 ,21mm 31% R3 = 0 ,17mm 47% D 2,6 R1 = 0 ,25mm 20% 750 D50 0,4 R2 = 0 ,21mm 30% R3 = 0 ,17mm 50% D 1 R1 = 0 ,2mm 22% 2500 D50 0,26 R2 = 0 ,14mm 30% R3 = 0 ,09mm 48% D 2 R1 = 0 ,2mm 14% 2500 D50 0,26 R2 = 0 ,15mm 29% R3 = 0 ,1mm 57% D 2,6 R1 = 0 ,2mm 12% 2500 D50 0,26 R2 = 0 ,15mm 25% R3 = 0 ,11mm 63% Table B1 Distribution used described by discrete radius, percentage of the mass and total number of grains in the case of smooth and rough particles. µr= 0.2 µr= 1.4 Fig. B2 Evolution of the µwith the strain for several PSD. the value of 0,5 for smooth particles. Whereas, in the case of rough particles, values from tests with a mean diameter d50= 0,4mm are larger (≈0.8) than the ones obtained from tests with d50= 0,26mm(≈0.7). In conclusion, it appears the rolling resistance has an influence on the friction coefficient via the PSD (and especially via the mean diame- terd50). More tests must be done to appreciateSpringer Nature 2021 L ATEX template Rolling relaxation controls friction weakening 11 results. Hence, curves obtained from PSD with d50= 0.4mm are really noisy because there are not enough grains (the sample height have been conserved). References [1] Myers, R., Aydin, A.: The evolution of faults formed by shearing across joint zones in sand- stone. J. of Struct. Geol. 26, 947-966 (2004) https://doi.org/10.1016/j.jsg.2003.07.008 [2] Poulet, T., Veveakis, M., Herwegh, M., Bucking- ham, T., Regenauer-Lieb, K.: Modeling episodic fluid- release events in the ductile carbonates of the Glarus thrust. Geophys. Res. 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2023-10-24

Considering a 3D sheared granular layer modeled with discrete elements, it is well known the rolling resistance significantly influences the mechanical behavior. Even if the rolling resistance role has been deeply investigated as it is commonly used to represent the the roughness of the grains and the interparticle locking, the role of rolling viscous damping coefficient has been largely overlooked so far. This parameter is rarely used or only to dissipate the energy and to converge numerically. This paper revisits the physical role of those coefficients with a parametric study of the rolling friction and the rolling damping for a sheared layer at different shear speeds and different confinement pressures. It has been observed that the damping coefficient induces a frictional weakening. Hence, competition between the rolling resistance and the rolling damping occurs. Angular resistance aims to avoid grains rolling, decreasing the difference between the angular velocities of grains. Whereas, angular damping acts in the opposite, avoiding a change in the difference between the angular velocities of grains. In consequence, grains keep rolling and the sample strength decreases. This effect must be considered to not overestimate the frictional response of a granular layer.

Frictional weakening of a granular sheared layer due to viscous rolling revealed by Discrete Element Modeling

2310.15945v1

Tensor damping in metalli c magnetic multilayers Neil Smith San Jose Research Center, Hitachi Globa l Storage Technologies, San Jose, CA 95135 The mechanism of spin-pumping, described by Tserkovnyak et al. , is formally analyzed in the general case of a magnetic multilayer consisting of two or more metallic ferromagnetic (FM) films separated by normal metal (NM) layers. It is shown that the spin-pumping-induced dynamic coupling between FM layers modifies the linearized Gilbert equations in a way that replaces the scalar Gilbert damping constant with a nonlocal matrix of Cartesian dampi ng tensors. The latter are shown to be methodically calculable from a matrix algebra solution of the Valet- Fert transport equations. As an example, explicit analytical results are obtained for a 5-layer (spi n-valve) of form NM/FM/NM'/FM/NM. Comparisons with earlier well known results of Tserkovnyak et al. for the related 3-layer FM/NM/FM indicate that the latter inadvert ently hid the tensor character of the damping, and instea d singled out the diagonal element of the local damping tens or along the axis normal to the plane of the two magnetization vectors. For spin-valve devices of technological interest, the influen ce of the tensor components of the damping on thermal noise or spin -torque critical currents are st rongly weighted by the relative magnitude of the elements of the nonlocal, anisotropic stiffness-fiel d tensor-matrix, and for in-plane magnetized spin-valves are generally more sensitive to the in-plane element of the damping tensor. I. INTRODUCTION For purely scientific r easons, as well as technological applica tions such as magnetic field sensors or dc current tunable microwave oscillator s, there is significant present interest1 in the magnetization dynamics in current-perpendicular-to-plane (CPP) metallic multilayer devices comprising multiple ferromagnetic (FM) films separated by normal meta l (NM) spacer layers. The phenomenon of spin- pumping, described earlier by Tserkovnyak et al.2,3 introduces an additional source of dynamic coupling, either between the magnetization of a single FM layer and its NM elect ronic environment, or between two or more FM layers as mediated through their NM spacers. In the former case,2 the effect can resemble an enhanced magnetic damping of an individual FM layer, whic h has important practical application for substantially increasing the spin-t orque critical currents of CPP spin-valves employed as giant-magnetoresistiv e (GMR) sensors for read head applications.4 Considered in this paper is a general treatment in the case of two or more FM layers in a CPP stack, where it will be shown in Sec. II that spin-pumping modifies the linearized equations of motion in a way that replaces a scalar damping constant with a nonlocal matrix of Cartesian damping tensors.5 Analytical results for the case of a 5-layer spin-valve stack of the form NM/FM/NM'/FM/NM are discussed in de tail in Sec. III, and are in Sec. IV compared and contrasted with the early well-know n results of Tserkovnyak et al..3, as well as some very recent results of that author and colleagues.6 In the case of CPP-GMR devices of technological interest, the relativ e importance of the different elements of the damping tensor on influencing measureable thermal fluctuations or spin-t orque critical currents is shown to be strongly weighted by the anisotropic nature of the stiffness-field tensor-matrix. II. SPIN-PUMPING AN D TENSOR DAMPING As discussed by Tserkovnyak et al,2,3 the spin current pumpI flowing into the normal metal (NM) layer at an FM/NM interface (Fig. 1) due to the spin-pumping effect is described the expression ⎥⎦⎤ ⎢⎣⎡− ×π=↑↓ ↑↓ dtdgdtdgm mm IˆIm )ˆˆ( Re4pumph (1) where is a dimensionless mixing conductance, and m is the unit magnetization vector. In this paper, for any ferromagnetic (FM) layer is treated as a uniform macrospin. A restatement of (1) in terms more natural to Valet-Fert↑↓g ˆ mˆ 7form of transport equations is di scussed in Appendix A. With the notational conversion , where A is the cross sectional area of the film stack, equation (1), for the case , simplifies to pump pump) 2 / (J I A eh− → ↑↓ ↑↓> > g gIm Re interface NM/FM for " " interface, FM/NM for " "ReIm,ˆ ˆˆ) 2 / ( 22 pump + −≡ ε⎟ ⎠⎞⎜ ⎝⎛ε + ×π≅↑↓↑↓ ↑↓rr dtd dtd re h e m mm Jm (2) where is the inverse mixing conductance (with dimensions of resistance-area), and is the well known inverse conductance quantum ↑↓ ↑↓≅ r rRe22 /e h ) k 9 . 12 (Ω≅ . In the present notation, all spin current densities have the same dimensions as electron charge current density , and for conceptual simplicity are defined with a parallel (i.e., ) rather than anti-parallel alignment with magnetization . Positive J is defined as electrons flowing to the right (along in Fig. 1.). spinJeJ m Jˆ ˆspin+ = mˆ yˆ+ For a FM layer sandwiched by tw o NM layers in which the FM layer is the layer of a multilayer thj ) 0 (≥j film stack, spin-pumping contributions at the interface, i.e., either left or right thi ) (j i= ) 1 (+=j i FM-NM interfaces, (2) can be expressed as ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ε + ×−= ↑↓− + =dtd dtd r ej ij j ij i j j im m m Jˆ ˆ ˆ) 1 ( 2pump 1 ,h (3) The physical picture to now be invoked is that of small (thermal) fluctuations of m about equilibrium giving rise to the terms in (2). Since ˆ 0ˆm dt d/ˆm 1ˆ≡m , the three vector components of 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, such as found in CPP-GMR pillars sandwiched between conductive leads of much larger cross section. In the example shown, the jth layer is FM, sandwiched by NM layers, with spin pumping contributions at the ith (NM/FM) and ( 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 i=0 i=1 i-1j i+1j+1-Jpump i Jpump i+1 i=jNM NM FMmj j=N-1z y lead lead i=N N-1terms in (3) it thus is useful to work in a primed coordinate system where , through use of a Cartesian rotation matrix such that 0ˆ ˆm z=′ 3 3× )ˆ(0mℜ m mˆ ˆ′⋅ℜ= .8 To first order in linearly independent quantities , ) , (y xm m′ ′ m m m′⋅ ℜ + =~ˆ ˆ0 , where , and where ℜ ⎟⎠⎞⎜⎝⎛≡′ ′′′′ ymxmm~ denotes the matrix from the first two (i.e., x and y ) columns of 2 3× ℜ. Replacing z m′→ˆ ˆ0 , and _ˆ×′z with matrix multiplication, the linearized form of (3) becomes ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ ′′ ⋅⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ ε− ε⋅ ℜ−= ′′− + = ↑↓ dt m ddt m d r e2h y jx j ii j ij i j j i// 11 ~ ) 1 ( pump 1 ,J (4) Using the present sign convention, j i s j A t Mm S ˆ/ ) (γ = is the spin angular momentum of the FM layer with saturation magnetization-thickness product , and is the gyromagnetic ratio. Taking thj j st M) ( 0> γ sM=M as constant, it follows by angu lar momentum conservation that3 ∑+ =−× × − = ⇔γ1 ˆ ˆ ) 1 (21 ˆ ) (NMj j ij i jj i j j j s e dtd A dtd t M m J mS mh (5) is the contribution to due to the net transverse spin current entering the FM layer (Fig. 1). In (5), denotes the spin-current density in the NM layer at the FM-NM interface. Taking the cross product on both sides of (5), transforming to pr imed coordinates by matrix-multiplying by , and employing similar linearization as to obtain (4), one finds to first order that dt dj/ˆmthj NM iJthi ×mˆ Tℜ = ℜ−1 ⎥⎥ ⎦⎤ ⎢⎢ ⎣⎡ − ≡ Δ ℜ ⋅⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛−=′ ×′ ∑+ =−⋅1spin NM) 1 (~ 0 11 0 21ˆj j iij i j jj je dtd AJ JS zT h (6) where Tℜ~ is the matrix transpose of ℜ 3 2×~. By definition, 0 ˆ~ 0= ⋅ ℜj jmT. The quantities in (6) are not known a priori , but must be determined after solution of the appropriate transport equations (e.g. , Appendix B). Even in the absence of charge current flow (i.e., as considered here, the are nonzero due to the set of in (4) which appear as source terms in the boundary conditions (A 9) at each FM-NM interface. Given the linear relation of (4), one can now apply linear superposition to express spin jJΔ ) 0=eJspin jJΔpump iJ ∑ ∑+ =↑↓ ↑↓ ↑↓≡′ ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ ε− ε⋅ ℜ ⋅ = Δ1spin 1 21 1,11 ~ 1 2k k ii k kk k jk kjr r dtdC remJt h (7) in terms of the set of 3-D dimensionless Cartesian tensor jkCt . The jkCt are convenient for formal expressions such as (9), or for analytical work in algebraically simple cases, such as exampled in Sec.III. However, they are also subject to met hodical computation. For the kth magnetic layer, the columns of each are the dimensionless vectors simultaneously obtainable for all magnetic layers j from a matrix solutionrd nd st3 or , 2 , 1jkCtspin jJΔ 9 of the Valet-Fert7 transport equatio ns with nonzero dimensionless spin-pump vectors )ˆor ,ˆ,ˆ)( / ( ) 1 (pump 1 ,z y x J↑↓ ↑↓ − + =− =i kk i k k ir r . To include spin currents via (5) into the magnetization dynamics, the conventional Gilbert equations of motion for can be amended as ) (ˆtm dtd A t M dtd dtdj j sj j j j ij S m m H mm 1 ) (ˆ ˆ ) ˆ(ˆG eff γ+ × α + × γ − = (8) where is the usual (scalar) Gilbert damping paramete r. From (6) and (7), one can deduce that the rightmost term in (8) will scale linearly with G jα dt d/m′, as does the conventiona l Gilbert damping term. Combining these terms together after applying the analogous linearization procedure to (8) as was done in going from (5) to (6), one obtains ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ ε− ε⋅ ℜ ⋅ ⋅ ℜ ⋅⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ − πγ= α′α′+ δ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ αα≡ α′′⋅ α′−′⋅ − ⋅ ℜ γ =′ ×′ ↑↓∑ 11 ~ ~ 0 11 0 2 / ) 4 (00] ) ˆ(~[ ˆ 2 pumppumpeff eff GG k jk j j j sjkjk jk jj jkk kjk j j j j jj j C re h t Mdtd dtd t h tt tt TT mm H m Hm z (9) where Kronecker delta k j k jjk jk ≠=δ= = δ if 0 and , if 1 . In (9), is a 2-dimensional Cartesian "damping tensor " expressed in a coordinate system where , while is a "nonlocal tensor" spanning two such coordinate systems. This formalism follows naturally from the lineariza tion of the equations of motion for non-collinear macrospins, and is particularly useful for describing the influence of "tensor damping" on the thermal fluctuations and/or j jα′t j 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.). Due to the spin-pumping contribution pump jkα′t, the four individual (with v u jk′ ′α′ y x v u′′=′ ′ or , ) are in general nonzero with , reflecting the true tensor na ture of the damping in this circumstance, which is additionally nonl ocal between magnetic layers (i.e., ). The are somewhat arbitrary to the extent that one may replace y y jkx x jk′ ′ ′ ′α′≠ α′ 0≠ α′′ ′ ≠v u j jkv u jk′ ′α′ 2~ ~ℜ ⋅ ℜ ↔ ℜ in (9), where 2ℜ is the matrix representation of any rotation about the 2 2× z′ˆaxis. It is perhaps tempting to contemplate an "inverse linearization" of (9 ) to obtain a 3D nonlinear Gilbert equation with a fully 3D damping tensor T k jk j jkℜ ⋅ α′⋅ ℜ = αt t. However, (9) has a null zˆ′ component, and contains no information rega rding the heretofore undefined quantities or . For local, isotropic/scal ar Gilbert damping, one can independen tly argue on spatial symmetry grounds that . However, the analogous extension is not so obviously available for z u jk′ ′α′z z jk′ ′α′ G G G α = α′= α′′ ′ ′ ′u u z z pump jkα′t, given the intrinsically nonlocal, anisotropic natu re of spin-pumping. The proper general equation remains that of (8), with the rightmost term given by that in (5), or its equivalent. III. EXAMPLE: FIVE LAYER SYSTEM 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), sandwiching a central NM spacer layer (#2 ), and with outer NM cap layers (#0 and #4). For discussion purpose prototype. Although the full genera lization is straightforward, the material properties and layer s described in text, the magnetization vectors 1ˆm and 3ˆm can be considered to lie in the film plane ( zx- plane). m1 NMj=0 j=1 23 4 FM NM' FM NMm3z xθ y y1y0 y2y3y4y5thickness will be assumed symmetric about the centr al (#2) normal metal spacer layer, which will additionally be taken to have a large spin-diffusion length (with the thickness of the layer), such that the "ballistic" approximation (B3) applies. The inverse mixing conductances will also be assumed to be real. Using the outer boundary conditions described by (B5), one finds for the FM-NM interfaces at that 2 2t l> >>jtthj ↑↓ =4 1-ir , and4 1y y y= ] )) / hyp( ( [ˆ NMFM NM 1 1pump 1 3 , 1 4 , 1l t l r rrJi j i iρ + ≡ ′+ = ↑↓ ↑↓↑↓ = =Jm J (10a) FM NMFM 4 1 1 4 21] )) / hyp( ( [J l t l r r Vi ρ + ≡′= Δ −= (10b) where , , and subscript "NM" refers to either outer layer 0 or 4. In (10) and below, are used interchangeably. Inside FM la yer 3, (B1,2) have solution expressible as 4 1r r=↑↓↑↓=4 1r r j j0ˆ ˆm m↔ 3 1 1 33 3 3 3spin 33 3 3 3 4 3 3 ] ) / tanh( ) /[( ] ) / tanh( ( [() / ) sinh(( ) / ) cosh(( [ ) /( 1 ) () / ) cosh(( 2 ) / ) sinh(( 2 ) ( FM FM FMFM FM FMFM FM A 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 ′+ ρ ρ +′− =− + − ρ =− + − = ≤ ≤ Δ (11) where the expression for follows from (10b). Subscript "FM" re fers to either layers 1 or 3. The boundary conditions (A5) and (A9) applied to the FM/NM boundary at 3B 3y y= yield ) ( ˆ]ˆ ) /( )[ ( ) 2 (pump 3spin 2 2 3 3spin 2 3 2 2 2 3 21 FM J J m m J V − + ⋅ = ρ − = −↑↓ ↑↓r l A r r BΔ (12) where , . The "ballistic" values 3 2r r=↑↓↑↓=3 2r r2VΔ and are constant inside central layer 2. Using (11) to eliminate coefficient in (12), the latter may be rewritten as spin 2J 3B ⎥⎦⎤ ⎢⎣⎡ ρ ′+ρ +′+ − ≡− ⋅ ⋅ + = − ↑↓ ↑↓↑↓ FMFM )] /( ) / [tanh( 1)) / tanh( ( 21] )ˆ ˆ 2 1 [( 11 2 2 2pump 3spin 2 3 3 2 2 21 l l t rl t l rr r rqq r J J m m VTt Δ (13) where is the 3-D identity tensor, and denotes the 3-D tensor formed from the vector outer - product of with itself. 1tT 3 3ˆ ˆm m⋅ 3ˆm Working through the equivalent comput ations applied now to the NM/FM interface at 2y y= , one finds the analogous result: ] )ˆ ˆ 2 1 [(pump 2spin 2 1 1 2 2 21J J m m V − ⋅ ⋅ + = +↑↓ Tq rt Δ (14) Eliminating between (13) and (14) derives the remaining needed result for : 2VΔspin 2J 1 3 3 1 1pump 3pump 2 21 spin 2)]ˆ ˆ ˆ ˆ ( 1 [ ), (−⋅ + ⋅ + ≡ + ⋅ = ⋅T Tm m m m J J J q Q Qtt t (15) treating tensor Qt as the matrix inverse of the [ ]-bracketed te nsor in (15). Using (10a) and (15) to compute , then additional use of (4) and (6), allow computation of the 3 3× NM 4 1-=iJjkCt defined in (7): ) / 1 / 1 ( / 1 ; 2 / , /, 1 2 1 212 131 13 33 11 ↑↓ ↑↓ ↑↓ ↑↓↑↓ ↑↓ ↑↓+ = ≡′≡− = = + = = r r r r r b r r aQ b C C Q b a C Cttttttt (16) For explicit evaluation of pump jkα′t, it is convenient to assume a choice of 3 , 1~ =ℜj for which 3 1ˆ ˆy y′=′ , such that and lie in the plane. To simplify the inte rmediate algebra to obtain Q03ˆm01ˆm z x′ ′-t from (15), one can consider "in-plane" magnetizations (Fig.2), taking z mˆ ˆ03=, and in the x-z plane ( ). This allows a particularly easy determination of 01ˆm θ = ⋅cosˆ ˆ01z mjℜ~ for which : y y yˆ ˆ ˆ3 1=′=′ 0 , ;0 1 0sin 0 cos ~ 3 1 3 , 1 = θ θ = θ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ θ − θ= ℜ=j j jT (17) Using (16) and (17) with (9) allows explicit solution for the pump jkα′t: θ + +θ + −= = θ + +θ + += =⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ + δ− δ + δ πγ= α′ ↑↓ 2 231 132 22 33 112 pump sin 2 1cos ) 2 1 (, sin 2 1cos 100 ) 1 2 ( 2 / ) 4 ( q qqd d q qq qd dd b ab a re h t M jk jkjk jk j j sjkh t (18) Taking , (18) holds for arbi trary orientation of and , provided the flexibility in choosing the 03 01ˆ ˆ cos m m⋅ = θ01ˆm03ˆm 3 , 1~ =ℜj is used to maintain 3 1ˆ ˆy y′=′ . However, for multilayer film stacks with three or more magnetic layers with magnetizations that do not all lie in a singl e plane, it wi ll generally be the case that some of the off-diagonal elements of the j0ˆm pump jkα′t will be nonzero. IV. DISCUSSION It is perhaps instructive to compare and contrast the results of (9) and (18). with the prior results of Ref. 3. The latter are for a a trilaye r stack, corresponding most directly to taking ∞→ ρNM in the present model, whereby . It is also effectively equivalent to the 5-layer case with insulating outer boundaries in the limit , whereby but due to perfect cancellation by the spin current reflected from the boundaries without intervening spin- flip scattering. Either way, it corresponds to in (10) and in (16) and (18). 0NM 4 , 1pump 4 , 1= == = i iJ J 0 ) / (NM→ l t 0pump 4 , 1≠=iJ 0NM 4 , 1→=iJ 5 , 0=iy ∞ →′ ′↑↓ 1 1,r r 0→a However, a more interesting difference is that Ref. 3 treats as stationary (hence , and as undergoing a perfectly circular precession about with a possibly large cone angle 3ˆm ) 0pump 3= J 1ˆm3ˆm θ. By contrast, the present analysis treats and equally as quasi-stationary vectors which undergo small but otherwise random fluctuations about their equilibrium positions and , with . To further elucidate this distinction, one can assume the aforementioned physical model of Ref. 3, and reanalyz e that situation in terms of the present formalism. With , and by explicitly inserting the condition (e.g., from (3)) that , an explicit solution of (15) can be expressed in the form: 1ˆm3ˆm 01ˆm03ˆm θ = ⋅cos ˆ ˆ03 03m m pump 3 3 0 /J m = = dt d 0 ˆ1pump 2≡ ⋅m J ]ˆ cos ) 1 (ˆ) 1 ( ˆ cos[3pump 2 2 2 23 12 pump 2 212NMm Jm mJ J ⋅ θ − ++ − θ+ = q qq q q (19) Combining (19) with the earlier re sult from (5) and then (3) (with ) 0=ε , it is readily found that dtd q qq q re h t Mq qq q t M et M e dtd A t M dtd sss s 1 2 2 22 22 11 32 2 2pump 2 3 pump 2 1 12 1 11 1 11 1 ˆ cos ) 1 (sin ) 1 (12 / ) 8 (ˆ ˆ cos ) 1 () ˆ)( 1 (ˆ ) ( 4ˆ ) ( 21ˆ ) (ˆˆNM mm mJ mJ mJ mSmmm ⎥⎥ ⎦⎤ ⎢⎢ ⎣⎡ ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ θ − +θ +−πγ− =⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ × θ − +⋅ ++ ×γ− =×γ− = ×γ⇔ × ↑↓hhh (20) The last result in (20) uses from (3), and the fact that pump 2J θ = ×sin ˆ ˆ1 3m m , and that and are parallel vectors in the case of steady circular precession of about a fixed . It is the direct equivalent of Eq. (9) of Ref. 3 with the identification dt d/ˆ1m 1 3ˆ ˆm m×1ˆm3ˆm ) 1 /(+⇔ν q q . Although the final expression in (20) is azimuthally invariant with vector orientation of , it is most convenient to compare it with (18) at that instant where is "in-plane" as shown in Fig. 2. At that orientation,1ˆm 1ˆm dt m d dt dm dt dy y / / /ˆ1 1 1′= → m , and it is immediately confirmed from (9) and (18) (with ) that the [ ]-term in (20) is simply the tensor element 0→ay y′′α′11 of pump 11α′t. It is now seen that the analysis of Ref. 3 happens to mask the tensor nature of the spin-pump damping by its restricting attention a specific form of the mo tion of the magnetization vectors, which in this case singles out the single diagonal element of the pump 11α′t tensor along the axis perpendicular to the plane formed by vectors and . The very recent results of Ref. 6 do addr ess this deficiency of generality, and reveal the tensor nature of 1ˆm3ˆm pump 11α′t with specific results for ππ=θ and , 2 / , 0 . The present Sec. III additionally includes the nonlocal tensors pump 31pump 13α′= α′t t, as well as diagonal terms jkaδ in (18) (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 boundary condition (B4). The latter condition will likely apply in the case of the technological important example of CPP-GMR spin-valves. Speaking of such, two important practical i ssues for these devices involve thermal magnetic noise and spin-torque induced oscillat ions. As described previously8, an explicit linearization of the effH term in (9) about equilibrium state that is a minimum of the free energy 0ˆm E leads to the following matrix form of the linearized Gilber t equation including spin-pumping (with : )0=eJ mA t M pE mH H Hp p p Gp p Dt p t HdtdG D j s j jjk jk j jk kj jk j j jkkj k jk j jkj jkkj k jk j jkj j j j kk jkk kjk jk Δ≡∂∂ Δ−≡ℜ ⋅ ⋅ ℜ ≡ ′ ∂∂ − δ ⋅ ≡⎥ ⎦⎤ ⎢ ⎣⎡ γα′− α′ + δγ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛−≡′ γα′+ α′ ≡′⋅ ℜ ≡ ′= ⋅′⋅′+′⋅′+′ ∑ ∑ ) ( ,ˆ)ˆ( 1~ ~,ˆ1 ) ˆ(2 0 11 0,2) (~) ( ) ( 0 eff 0eff 0 eff 0 0 mmHmH H mh h mm t t t tt ttt ttt tt TT (21) where the are small perturbation fields. The form of ) (tjhjkDt ′ and jkG′t in (21) is chosen so that they retain the original delineation8 as symmetric and antisymmetic tensors regardless of the symmetry of . By use of a fixed "reference-moment" jkα′tmΔ in the definition of , the "stiffness-field" tensor- matrix is symmetric positive-definite, and eff jH v k u jv u jk m m E H′ ′′ ′′∂′∂ ∂ ∝ ′ / ∑ ⋅ − =δj j j j sδ t M A E m h ) ( ∑′⋅′Δ − =j j j m m h has the proper conjugate form so that (21) are now ready to directly apply fluctuation-dissipation expressions specifically suited to such linear matrix equations of motion.8 Treating the fields now as thermal fluctuat ion fields driving the ) (tjh′ ) (tjm′-fluctuations, v u jkB h hv u jkB v k u j DmT kS DmT kh hv k u j′ ′ ′ ′′ ′ ′ ′ ′ Δ= ω′⇔ τ δ′ Δ= 〉′τ′〈′ ′2) ( ) (2) 0 ( ) ( (22) are the time-correlation or cross power spectral density (PSD) F ourier transform pairs. Through their relationship described in (21), the nonlocal, tensor nature of the spin-pumping contribution pump jkα′t to jkα′t is directly translated into those of the FM FM2 2N N× system "damping tensor-matrix" v u jkD′ ′′↔′Dt , where is the number of FM layers in the multila yer film stack. The cross-PSD tensor-matrix FMN ) ( ) ( ω′↔ ω′′ ′′ ′ ′v k u jm mSm mSt for the m-fluctuations can then be expressed as′8 1)] ( [ ) () ( ) ( )] ( ) ( [ ) ( −′ ′ ′ ′ ′+′ω −′≡ ω′ω′⋅′⋅ ω′→ ω′− ω′ Δ ω= ω′ D G HS Sh h m m ttt tttt t t t im iT kB χχ χ χ χ@ @ (23) where is the complex susceptibility tensor-matrix for the ) (ω′χt} , {h m′′ system, and ) (ω′@χt its Hermitian transpose. It has been theoretically argued10 that (22), and thus the second expression in (23), remain valid when , despite spin-torque contributions to resulting in an asymmetric 0≠eJeff jH H′t (e.g., see (25)) that violates the condition of therma l equilibrium implicitly a ssumed for the fluctuation dissipation relations. Since is in general a fully nonlocal with anisotropic/tensor character, any additional tensor nature of H′t Dt will likely be altered or muted as to the influence on the detectable -fluctuations. As an example, one can again consider th e situation depicted in Fig. 2, applied to the case of a CPP-GMR spin-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 (i.e., ), the PSD will be proportional to . As was also shown previously,3 1ˆ ˆm m⋅3ˆm 0 /ˆ3→dt dm ) ( sin1 12ω′θ′ ′′x xm mS 11 it follows from (23) (and assuming azimuthal symmetry 011 11=′=′′ ′ ′′ x y y xH H ) that x x y y y y x x x x y yy y x x y y x x sB m m H H H HH H tA MT kSx x ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′′ ′ ′′′′′′ ′ ′ ′α′+′α′= ω Δ′ ′ γ = ωω Δ ω + ω − ωω α′+ ω′ ′ α′γ≅ ω′′ 11 11 11 11 11 11 02 2 2 022 112 0 11 11 11 1 ) ( ) () / ( ) (2) (1 1 (24) treating . The tensor influence of the is seen to be weighted by the relative size of the stiffness-field matrix elements . For the thin film geometries 111 11 < < < α′α′′ ′′ ′ y y x x u u′ ′α′11 v vH′ ′′11 A t< < typical of such devices, out-of-plane demagnetization fi eld contribution typically result in that are an order of magnitude larger than . Since y yH′ ′′11 x xH′ ′′11x x y y ′ ′ ′′α′≤ α′11 11 from (18), it follows that the linewidth ωΔ and the PSD in the spectral range of practical interest will both be expected to be determined primarily by . ) (01 1ω ≤ ω ′′ ′′x xm mS x x′ ′α′11 A similar circumstance also applies to the im portant problem of critical currents for spin-torque magnetization excitation in CPP-GMR spin valves with 0≠eJ . Consider the same example as above, again treating as stationary, and seeking nontri vial solutions of (21) (with of the form . Summarizing results obtainable from (5), (8), and (21) 3ˆm ) 0 ) (=′th ste t−∝′) (1m 0 detˆ ) (2 / 11 11 1111 11 111 2 10eff 1eff 1NM =⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ α′ ′−′ ′ −′′+′ α′ ′−′× + = ′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′= y y y y x yy x x x x xsJ s H s Hs H s Ht Me em J H Hh (25) where , as in (18),and where in (25) is now the solution of the transport equations with but . The cross-product form of the spin-torque contribution to explicitly yields an asymmetr ic/nonreciprocal contribution γ =′ /s sv u′ ′α′11 eJ∝NM 2J 0pump= J 0≠eJeff 1H ex y y xJ H H∝′−′′′′′ 11 11 to . The critical current density is that value of where becomes negative. Given th e basic stability criterion that , the spin-torque critical condition from (25) can be expressed as Ht ′ eJ sRe 0 det11>′Ht x y y x y y x x x x y yH H H H′′′′′′′ ′ ′ ′ ′ ′′−′=′α′+′α′11 11 11 11 11 11 (26) Like for thermal noise, the spin-torque critical point should again be determined primarily by for in-plane magnetized CPP-GMR spin-valves with typical x x′ ′α′11 x x y yH H′ ′ ′′′> >′11 11. This simply reflects the fact that the (quasi-uniform) modes of thermal fluctuati on or critical-point spin-torque oscillation tend to exhibit rather "elliptical", mostly in-plane motion when x x y yH H′ ′ ′′′> >′11 11. This is obviously different than the steady, pure circular precession described in Ref. 3, which contrastingly highlights the influence of , along with its inte resting, additional y y′ ′α′11θ-dependence. APPENDIX A The well known "circuit theory" formulation12 of the boundary conditions for the electron charge current density and the (dimensionally equiva lent) spin current density at a FM/NM interface can (taking ) be expressed as eJspin NMJ m V ˆFM FM VΔ =Δ ) ˆ )( ( ) )( (FM NM FM NM 21V G G V V G G Je Δ − ⋅ − + − + =↓ ↑ ↓ ↑m VΔ (A1) )ˆ ( Im ) ˆ ˆ( Reˆ)] ˆ )( ( ) )( [( NM NMFM NM FM NM NM 21 spin m V m V mm m V J × + × × +Δ − ⋅ + + − − = ↑↓ ↑↓↓ ↑ ↓ ↑ Δ ΔΔ G GV G G V V G G (A2) in terms of spin-indepe ndent electric potential V and accumulation VΔ (Δμe= ). Setting 0=eJ in (A1) and substituting into (A2), one obtains in the limit the result 0 Im→↑↓G )ˆ ˆ( ˆ) ˆ (2 NM FM NM NM0spinm V m m m V J × × + Δ − ⋅ +=↑↓ ↓ ↑↓ ↑ =Δ Δ G V G GG G eJ (A3) Comparing with Eq. (4) of Tserkovnyak et al.3 (with )sμΔ⇔V and remembering the present conversion of , one immediately make s the identification spin 1 spin NM NM ) 2 / (I J−− ↔ e Ah ↑↓ ↑↓= G e h A g) 2 / ( 22 (A4) relating dimensionless in (1) to , the conventional mixing conductance (per area). ↑↓g↑↓G The common approximations that inside all FM layers, and that longitudinal spin current density is conserved at FM/NM interfaces, yields the us ual interface boundary condition m J ˆspin spin FM FM J=spin spin FM NMˆJ= ⋅m J (A5) Solving for from (A2) then leads (with (A1)) to a second scalar boundary condition: m Jˆspin NM⋅ spin FM FM NM4 4J G GG GJ G GG GV Ve ↓ ↑↓ ↑ ↓ ↑↓ ↑−−+= − (A6) Equation (A6) is identical in form with the standard (collinear) Valet-Fert model,7 and immediately yields the following identifications ↓ ↑↓ ↑ ↓ ↑↓ ↑ +−= γ+= G GG G G GG Gr , 4 (A7) for the conventional Valet-Fert interface parameters . γandr The three vector terms on th e right of (A2) are mutually orthogona l. Working in a rotated (primed) coordinate system where , (A1) and (A2) can be similarly inverted to solve for the three components of the vector m z′=′ˆˆ )ˆ (FM NM m V ′Δ −′ VΔ in terms of , , and . A final transformation back to the original (umprimed) coordinate s yields the vector interface-boundary condition spin NMJ′spin FMJeJ ) / /( ) 2 / ( ) 2 /( 1ˆ Im Reˆ] ) Re [( ) ˆ ( 2spin spin spin 21 NM NM FM FM NM A g e h G rr r J r J r r Ve ↑↓ ↑↓ ↑↓↑↓ ↑↓ ↑↓ = ≡× + + γ − − = Δ − J m J m m VΔ (A8) Combined with (A4), the last relation in (A8) yields (2). Equation (A8) is a generalization of Valet- Fert to the non-collinear case. As noted by Tserkovnyak et al.,3 boundary conditions (A3) do not di rectly include spin-pumping terms, but instead involve only "backflow" terms in the NM layer. With spin-pumping physically present, arises as the response to the spin accumulation back spin NM NM J J↔ back NMJNMVΔ created by . It follows that , where is henceforth the total spin current in the NM layer. Thus, including spin-pumping in Va let-Fert transport equations is then a matter of replacing in (A8). The modified form of (A 8), for a FM/NM interface, becomes: pumpJ pump spin back NM NM J J J− =spin NMJ pump spin spin NM NM J J J− → ) (ˆ Im ) ( Reˆ] ) Re [( ) ˆ ( pump spin pump spinspin 21 NM NMFM FM NM J J m J Jm m V − × + − +γ − − = Δ − ↑↓ ↑↓↑↓ r rJ r J r r Ve Δ (A9) For an NM/FM interface, the sign is flippe d on the left sides of (A6) and (A9). APPENDIX B For 1-D transport (flow along the y-axis), the quasi-static Valet-Fert7 (drift-diffusion, quasi-static) transport equations can be written as9 ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ ∂∂+∂∂βρ==⎥ ⎦⎤ ⎢ ⎣⎡ ⎟⎟ ⎠⎞ ⎜⎜ ⎝⎛ ∂∂⋅ β +∂∂ ρ=∂∂= ∂∂ y yVy yVJy l ye Vm JVmV V ΔΔ Δ Δ 21ˆ1with along0 ˆ21 1, spin2 22 (B1) where = bulk resistivityρ13, l = spin diffusion length, and β = bulk/equilibrium spin current polarization in FM layers ( in NM layers). The solution for any one layer has the form 0≡ β m B m AB A V m V ˆ ,ˆ : layers FM for,ˆ/ / 21 B Ae e C y J Vl y l y e = =+ = ⋅ β − + ρ =−Δ Δ (B2) In the case where film thickness, one may employ an al ternative "ballistic" approximation: > > >l C V , = = = ,spinB J A V Δ (B3) It is not necessary to solve for the V and/or the C-coefficients using (A6) if only and are required. The remaining coefficients are determin ed by the interface boundary conditions (A5), (A6,7) and (A9), and external boundary conditions at the outer two surfaces of the film stack. VΔspinJ Regarding the latter, one approximation is to treat the external "leads" (with quasi-infinite cross section) as equilibrium reservoirs and set 0 ) (, 0→==N i y y VΔ at the outermost (i =0, N) lead-stack interfaces of an N -layer stack (Fig. 1). The complimentary approximation is of an insulating boundary, with . . For the case (such as in Sec. III) where the outer ( j=0, N-1) layers are NM, 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 NM NM) / hyp( ) ( 21 , 0 1 , 1 i j j N j N i l t l J V− = − =ρ ± = Δ (B4) where hyp( ) = tanh( ) or coth( ) for equipotential, or insulating boundaries, respectively. Combining (B4) with (A9), and neglecting ↑↓rIm , one finds for 0=eJ that ) / hyp( ) (ˆ)] / hyp( ) ( [ pump1 , 0 1 , 1 21 FM NMFM FM j j j ii i i ii j j N j i N i l t l rrJJ l t l r V ρ ++ =ρ + = Δ ± ↑↓↑↓− = − = Jm J (B5) ACKNOWLEDGMENT The author would like to thank Y. Tserkovnya k for bringing Ref. 6 to his attention. REFERENCES 1. D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mat. 320, 1190 (2008) and many re ferences therein. 2. Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 66, 224403 (2002). 3. Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 67, 140404 (2003). 4. S. Maat, N. Smith, M. J. Carey, and J. R. Childress, Appl. Phys. Lett., 93, 103506 (2008). 5. J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W.Bauer, Phys. Rev. B 78, 140402 (2008). This paper describes a damping mechanism distinct from Refs. 6, or this work, where nonlocal/tensor properties arise from a strong magnetization gr adient in a single FM film or wire. 6. J. Foros, A. Brataas, G. E. W.Bauer, and Y. Tserkovnyak, arXiv:con-mat/0902.3779. 7. T. Valet and A. Fert, Phys. Rev. B, 48, 7099 (1993). 8. N. Smith, J. Appl. Phys. 92, 3877 (2002); N. Smith, J. Magn. Magn. Mater. 321, 531 (2009) 9. N. Smith, J. Appl. Phys., 99, 08Q703, (2006). 10. R. Duine, A.S. Nunez, J. Si nova, A.H. MacDonald, Phys. Rev. B 75, 214420 (2007) 11. N. Smith, J. Appl. Phys. 90, 5768 (2001). 12. A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur. Phys. J. B 22, 99, (2001). 13. Some poor choice of words in the appendix of Ref. 9 c onfused the bulk resistivity, ρ, with the Valet-Fert7 parameter . *ρ

2009-04-21

The mechanism of spin-pumping, described by Tserkovnyak et al., is formally analyzed in the general case of a magnetic multilayer consisting of two or more metallic ferromagnetic (FM) films separated by normal metal (NM) layers. It is shown that the spin-pumping-induced dynamic coupling between FM layers modifies the linearized Gilbert equations in a way that replaces the scalar Gilbert damping constant with a nonlocal matrix of Cartesian damping tensors. The latter are shown to be methodically calculable from a matrix algebra solution of the Valet-Fert transport equations. As an example, explicit analytical results are obtained for a 5-layer (spin-valve) of form NM/FM/NM'/FM/NM. Comparisons with earlier well known results of Tserkovnyak et al. for the related 3-layer FM/NM/FM indicate that the latter inadvertently hid the tensor character of the damping, and instead singled out the diagonal element of the local damping tensor along the axis normal to the plane of the two magnetization vectors. For spin-valve devices of technological interest, the influence of the tensor components of the damping on thermal noise or spin-torque critical currents are strongly weighted by the relative magnitude of the elements of the nonlocal, anisotropic stiffness-field tensor-matrix, and for in-plane magnetized spin-valves are generally more sensitive to the in-plane element of the damping tensor.

Tensor damping in metallic magnetic multilayers

0904.3150v2

Emerging magnetic nutation Pascal Thibaudeau1,and Stam Nicolis2 1CEA, DAM, Le Ripault, BP 16, F-37260, Monts, FRANCE 2Institut Denis Poisson, Universit e de Tours, Universit e d'Orl eans, CNRS (UMR7013), Parc de Grandmont, F-37200, Tours, FRANCEy (Dated: August 4, 2021) Nutation has been recognized as of great signicance for spintronics; but justifying its presence has proven to be a hard problem. In this paper we show that nutation can be understood as emerging from a systematic expansion of a kernel that describes the history of the interaction of a magnetic moment with a bath of colored noise. The parameter of the expansion is the ratio of the colored noise timescale to the precession period. In the process we obtain the Gilbert damping from the same expansion. We recover the known results, when the coecients of the two terms are proportional to one another, in the white noise limit; and show how colored noise leads to situations where this simple relation breaks down, but what replaces it can be understood by the appropriate generalization of the uctuation{dissipation theorem. Numerical simulations of the stochastic equations support the analytic approach. In particular we nd that the equilibration time is about an order of magnitude longer than the timescale set by the colored noise for a wide range of values of the latter and we can identify the presence of nutation in the non-uniform way the magnetization approaches equilibrium. I. INTRODUCTION Recent progress in spintronics has led to the search for processes and materials that can realize ever shorter switching times for the magnetization{and this has opened a window to a r egime, where nutation eects of the average magnetization cannot be ignored. How to take them into account becomes, therefore, of practical interest [1]. However how to describe the emergence and the relevance of nutation from rst principles, as mag- netic moments interact with a bath, has been and re- mains a challenging problem. One reason is that it is by no means obvious how to extract its properties from the interaction with the bath. For magnetic materials a common way of describing the eects of the bath is by the so{called Gilbert damping mechanism [2]. What the two eects have in common is the vector nature of the bath; where they dier is in how this gets imprinted on the magnetization prole in each case. Providing a microscopic picture of how Gilbert damp- ing may appear has long been recognized as an out- standing question and there have been many attempts for explaining how it may occur. However whether there might be any relation with the eects of nutation has only received attention. [3], where both were assumed to be present and certain consequences for ultrafast switch- ing were set forth. In that case, though, it was assumed that the magnetic moment was not in interaction with a stochastic bath in full generality: the latter was present only indirectly, through the deterministic Gilbert term. In this paper we shall show that the nutation term and the Gilbert term can both be obtained as well{dened pascal.thibaudeau@cea.fr ystam.nicolis@lmpt.univ-tours.frcontributions from a systematic expansion of the equa- tions of motion of a magnetic moment, interacting with a vector bath, whose stochastic component is drawn from colored noise. The expansion parameter can be identied as the ratio of the timescale of the colored noise to the precession frequency of the Larmor motion, here simply reduced to a Zeeman eld only; this simplies the calcu- lations, without any loss of generality. The equation of motion of a classical or quantum mag- netic moment, in the presence of this external eld, is of rst order in the dynamical variables and describes precession [4]. This equation implies, in particular, some non-trivial conservation laws: the norm of the magnetic moment is conserved and, for a constant external eld, so is the component along it [3, 5]. When the magnetic moment interacts with a bath the conservation laws take the form of uctuation{dissipation relations, that describe the fact that the magnetic mo- ment is in equilibrium with the bath. Indeed, the pro- posal in ref. [6] describes Gilbert damping and nutation as successive, relativistic, corrections to the dynamics of a spin, in equilibrium with a quantum bath. In the present paper we wish to explore the scenario, where a magnetic moment is in equilibrium with a vector bath, described by colored noise. The correlation time of the noise sets the short{time scale, so the relativistic expansion of ref. [6] can be identied as the expansion in powers of the ratio of the correlation time to the period of precession. We nd that it is possible to recover both, Gilbert damping and nutation of the magnetic moment, as terms in such an expansion. It should be kept in mind that what is the \most ap- propriate" equation of motion (eom), that represents the motion of a collection of interacting magnetic moments, is still the subject of intense debate, that goes back a long time. Landau-Lifshitz [7] and Gilbert [2] introducedarXiv:2103.04787v3 [cond-mat.stat-mech] 3 Aug 20212 an eom that described exclusively transverse damping, whereas Bloch considered an eom that described exclu- sively longitudinal damping for the coarse{grained (spa- tial) average of the magnetization of the interacting mag- netic moments. These equations have been extensively used to interpret measurements of spin relaxation and provide a phenomenological viewpoint both for the origin of the eective eld that denes the precession axis for the average magnetization, as well as for the origin of the damping, whose eects can be reduced to a small number ofdamping constants . A lot of attempts have been made to provide a microscopic foundation for the equation of motion and, in particular, for accounting for the degrees of freedom that are behind the damping eects of the magnetization [8{11]. There have been many arguments about intrinsic and extrinsic eects, without, however, any insight into how these might be distinguished clearly in an invariant way. For example, some authors [8, 12] considered a phe- nomenological theory describing one classical spin, em- bedded in a medium, that acts as a bath. This ap- proach leads to the well-known Landau-Lifshitz equation (resp. Bloch equation) in several limiting cases, i.e. in the high temperature limit. The origin of the bath, that describes the uctuations of the average magnetization, was not spelled out, and only its role as an external ther- mostat (here called uctuostat more generally) was as- sumed. This approach highlights that the damping is then a consequence of memory eects, i.e. non-local in time. Memory eects in their own right, were investi- gated theoretically in refs [13]. Depending on the form of the memory kernel involved, it was found that these can lead to a compensation or even to an overcompensa- tion of the damping, since called \Gilbert damping". More recently and based on both a quantized spin and environment Hamiltonian, Anders et.al derive a general spin operator equation of motion that describes three- dimensional precession and damping and consistently ac- count for eects arising from memory, coloured noise and quantum statistics [14]. This reveals clearly reso- nant Lorentzian system{reservoir couplings that allow a systematic comparison of dynamics between Ohmic and non{Ohmic regimes. The quantized spin+reservoir prob- lem was also addressed before [15], rst in a attempt to justify the form taken by the classical Landau-Lifshitz- Bloch equation, without recognizing immediately the benet of keeping, as long as possible, the memory ker- nel form induced by the motion equation of the quantum spin density operator. In these approaches, it is implicitly assumed that the system does reach an equilibrium state, i.e. a state that is invariant under global time translations. How the system can, indeed, attain such a state has become a subject of considerable interest in the domain of glassy systems (for magnetic systems these are known as \spin glasses"; an example of the vast literature is refs. [16, 17]). One way can be described as due to inertia , i.e. that the medium is not innitely rigid [18]. This implies that the magneticresponse depends not only on the magnetization itself, but on its velocity, as well, and, therefore, the equation of motion is of second order in time [19, 20]. The corresponding equations of motion can be iden- tied with those of an Euler top, in a time{dependent external eld, i.e. a torque. It is the impossibility of pro- viding a local description of the dynamics in terms of one set of rst order equations that leads to non{local eects. These can be captured by the so-called \atomistic spin dynamics"{as implemented, for instance, by Bhattachar- jeeet al. [21]. A particular motivation was of capturing processes in the femtosecond regime by including the mo- ment of inertia. They derived a generalized equation of motion for the magnetization dynamics in the semiclassi- cal limit, which is non-local in both space and time. Con- sequently, they recovered a generalized Landau-Lifshitz- Gilbert equation, which includes the moment of inertia and a second derivative of the magnetization in time. Going further with this idea, Pervishko et al. [22] pro- posed an alternative derivation of the Gilbert damping in a tensor form, within a mean-eld approach. In this for- malism, the itinerant electronic subsystem is considered in the presence of a nonequilibrium, classical magneti- zation eld. When this eld is suciently smooth and slow on the scales determined by the mean free path and scattering rate of the conduction electrons, the induced nonlocal spin polarization can be approximated using a linear response Ansatz, thereby showing that the damp- ing parameter emerges due to the coupling to the itiner- ant subsystem. They derive a Kubo-St reda formula for the components of the Gilbert damping tensor and illus- trate its relevance for the two-dimensional Rashba fer- romagnet, that can be realized at the interface between nonmagnetic and ferromagnetic layers. They argue that this approach can be further applied to identify prop- erly the tensor structure of the Gilbert damping for more complicated model systems and real materials. More recently, Mondal et al. [23] identied the Gilbert damping and nutation terms as rst- and second-order relativistic eects respectively, arising from the Foldy| Wouthuysen transformation of a Dirac particle (that in- cludes spin1 2) motion under external elds, embedded in a material medium. What is particularly striking in all these approaches is that, while all end up with a description of Gilbert damp- ing and of the torque that drives nutation, they seem to allow considerable ambiguity about the relative sign between the Gilbert damping and the nutation torque contribution. While the microscopic origins of both Gilbert damping and magnetic inertia are still under debate, this uncer- tainty re ects a fundamental issue, that deserves closer scrutiny. We wish to report on our eorts to resolve this am- biguity. We shall show that the coupling of a magnetic moment to a vector bath of colored noise is sucient for describing the emergence of both Gilbert damping and nutation, along with the relative sign; in addition, it pro-3 vides a well{dened route to equilibrium. The parameter that controls the relative signicance of these eects is the ratio of the colored noise timescale to the precession period. This is where the vector nature of the bath is of relevance. The plan of our paper is as follows: In section II we describe our model for a magnetic moment in a vector bath. In section III we provide representative solutions of the equations of motion obtained by numerical inte- gration and show how Gilbert damping and nutation can be unambiguously identied. In section IV we present our conclusions and ideas for further inquiry. II. MAGNETIC MOMENT IN A BATH Consider the spatial average of the magnetization M of a block of magnetic material. The \reduced" mag- netization,mM=Ms, depends only on time and its dynamics can be described by its precession about an eective eld, which can be written as the sum of two vectors!0(t) +!(t).!0(t) is dened, in turn, as the sum of the external magnetic eld, applied on the mag- netic system, and of the magnetic eld, produced by the average magnetization of the surrounding medium, i.e. the reaction eld. !(t) is a stochastic eld, and is characterized phe- nomenologically by a single relaxation time . It de- scribes the uctuations of the magnetic response of the medium, in which the magnetic block is found. We can describe the equilibrium of the magnetic block with the medium, by the statement that h!i 0Ms1hmi= s1hmi, whereis the susceptibil- ity (not a function of time), and is the gyromagnetic ra- tio. Here the average is taken over the realizations of the surrounding medium, considered as a bath. This state- ment means that the expectation value of the uctuating eld at equilibrium is aligned with and proportional to the expectation value of the magnetization [24]. When ;which is identied as the cumulant of the spin-spin function, depends explicitly on time, a convolution be- tween the uctuating eld and the magnetization has to be used [25]. This procedure focuses on the "relevant" degrees of freedom, labelled by mand sets them apart from the "irrelevant" variables, labelled by !. We are not inter- ested, in the following, in the microscopic mechanisms that may produce the eects of these variables [26, 27], just on their collective dynamics on the \relevant" de- grees of freedom. (It has been proposed [28, 29] that the symmetry that expresses the property that the physics should not depend on how the \dynamical" from the de- grees of freedom, that can dene the \bath", are chosen, is supersymmetry.) These considerations can be expressed mathematicallyas follows: dm dt= (!0+!)m (1) d! dt=1  ! s1m
+ s (2) whereis a random eld, with ultra{local Gaussian cor- relations, that describes the bath, which will be taken as thermal, in what follows, concretely: hI(t)i= 0 (3) hI(t)J(t0)i= 2DIJ(tt0) (4) whereI;Jare the indices of the vector components. D is the amplitude of the noise and provides the denition of the temperature T, through the Boltzmann{Einstein relation,D/kBT=~, thereby expressing the uctuation- dissipation theorem, for the bath. That the tempera- ture is well{dened is ensured by the property that the noise eld(t) is drawn from a stationary stochastic pro- cess, i.e. enjoys global time translation invariance. Equa- tions (1) and (2) were rst dened in [12] and evaluated in atomistic spin simulations [30]. It should be stressed that this does not imply that the 2{point function of the magnetic moment will have a simple dependence on the temperature, due to the fact that its uctuations, gener- ically, will not be Gaussian [31]. Eq. (1) is purely transverse, therefore, the norm of mis conserved, ifm
_m= 0,(d=dt)(jjmjj2) = 0. The latter relation is, of course, true, in the absence of the bath; it does require, however, another denition in its presence, since the derivative is a singular quantity [32, 33]. Such a denition can be obtained from the so-called Schwinger{ Dyson identities [34], namely as  m
dm dt = 0: (5) The eld!is dened by the stochastic dierential equation (SDE) in eq. (2). Its solution can be shown to be an Ornstein-Uhlenbeck process [35]. Therefore, m(t) be- comes a stochastic process, as well; moreover, the noise, that enters additively in the equation for !, becomes multiplicative for m(t); which implies that its correla- tion functions acquire a non{trivial dependence on the temperature, dened through the bath. This is, often de- scribed as a \breakdown" of the uctuation{dissipation theorems [36, 37]. However, what this, simply, means is that the non-linearities induce a non{trivial, but quite transparent, dependence of the noise on the dynamics of the magnetization; the two are, just, intertwined in a way that is more subtle than hitherto acknowledged. Indeed, this can be understood in terms of the variables that can resolve the dynamics of the bath, as an expression of reparametrization invariance in the space of elds. First, suppose for simplicity that the system is not in contact with the bath; (t) is absent from eq. (2). Then, eqs.(1,2) dene the dynamics of a deterministic system4 and can be explicitly solved: First of all, equation (2) can be solved for !in terms ofm(t): != s1 Zt 1ett0 m(t0)dt0 = s1 Z1 0eu m(tu)du(6) The equation (6) can then be introduced in eq. (1) to produce an integral-dierential equation for m: dm dt= !0+ s1 Z1 0eu m(tu)du m(7) The integral highlights the dependence of the solution on the full history of the magnetization, prior to time t, as well as the putative eects of the damping induced by the memory kernel with a characteristic time , dened by eq. (2). Indeed one of the purposes of this paper is to provide an intrinsic denition of such damping eects in an invariant way. In order to nd approximate solutions, it is useful to expandmin a Taylor series about some reference time t and exchange the sum and the integral. Assuming that Fubini's theorem holds [38], we thus nd != s1 1X n=0(1)n n!dnm dtnZ1 0eu undu = s11X n=0(1)n n!dnm dtnn(1 +n) = s11X n=0()ndnm dtn:(8) Of course, it is by no means obvious either that this series converges, or that it is even legitimate to exchange sum and integral; we shall try to provide a posteriori checks that are sensitive to these issues. We shall now try to interpret the properties of the magnetization, that are sensitive to our truncating the series at a given order. When the sum stops at n= 1, eq.(1) takes the form dm dt !0+ s1m s1dm dt m =!0m+mdm dt(9) where s1can be, therefore, identied as the Gilbert damping constant, and eq. (9) is the eom writ- ten in the standard Gilbert form [2]. This expression forappears consistent with other forms reported in the literature [25, 39]. It is therefore not surprising that eventually the tensor character of both the inverse of the susceptibility and the relaxation time produces a ten- sor damping parameter , a feature already reported in ferromagnetic metals assuming a torque-torque correla- tion model [40{42] in the highly anisotropic scattering regime of magnons.Upon including the n= 2 term, the equation for the magnetization takes the form dm dt !0+ s1m s1dm dt m + s12d2m dt2m =!0m+mdm dtd2m dt2 :(10) The term proportional to m(d2m=dt2) can be in- terpreted as describing the "nutation" of the magnetiza- tion [43]. It should be noted, at this point that this is the rst term that is, manifestly, symmetric under time{ reversal. An issue of considerable interest is that of the relative sign of the coecients of the terms in the equa- tion of motion. Let us note that the sign of the inertial damping (last term) seems to be opposite to the sign of the usual damping term (second term), which is in agree- ment with the theory of dampened magnetostriction, rst introduced by Suhl [44]. This is in contrast with refer- ence [20], where the signs of the two damping terms are the same. However the microscopic description in the two cases is completely dierent. What we have thus shown is that both, the Gilbert damping and the nutation term can be deduced as the consequence of the coupling of a magnetic moment to an external eld, upon taking into account the coupling to the bath self{consistently. This constitutes the central result of the paper. In this deterministic situation and because  >0, the divergence of the volume of the phase space is negative{it shrinks, due to dissipation. At this level of truncation the existence of an equilibrium state for the magnetization that is unique and is described by a point is obvious. What is by no means obvious is what happens when the non{local eects, described by the higher order terms, are taken into account. Setting this issue aside, for the moment, let us now take into account the bath, at this approximation. When the noise eld is present, !becomes a stochastic eld, which contains an extra term. This term takes into account the noise eld in the memory kernel as follows: ! s=11X n=0()ndnm dtn+Z1 0eu (tu)du(11) Let us call (t)R1 0eu (tu)du, the extra stochastic eld. Equations (3),(4) imply that the random eld has the following properties: h I(t)i= 0 (12) h I(t) J(t0)i=D IJejtt0j  (13) which means that it describes colored noise! This im- plies, in turn, for the magnetization that its correlation5 functions are, generically, those of a centered and colored noise stochastic process, and not of a white noise process, as it is usually assumed. We shall show now show that the approximations involved in the truncation to second or- der, i.e. including the nutation term, are self{consistent by solving the equations (1) numerically. III. NUMERICAL RESULTS In order to check on how the signature of the Gilbert damping and that of nutation, produced by the uctuat- ing eld!is imprinted in the magnetization prole, we solve the coupled equations (1) numerically. Precession can be readily identied as the rotation of the magnetization around a given axis. Nutation is the additional eect produced on the mag- netization by the motion of this axis with time. Upon averaging over the realizations of the noise, if hmiis a constant vector at equilibrium, then h!ialso becomes a constant vector, proportional to hmi. That means that the magnetization spins rst around !0at short times and then settles to spinning around !0+h!i at long times. But the torque produced at that time is !0hmi, because of the proportionality between h!i andhmi. As a consequence, only during a transient time, when!strongly varies, can the motion of the average magnetization be strongly aected. All these features can be read o gure 1, that displays both the motion of the average mand!taken over more than 1000 realizations of the noise, and for dierent values of the correlation time of the noise, . -1-0.500.51<m>τ=5 01234<δω> -1-0.500.51<m> τ=0.5 01234<δω> 0.1 1 10 time t-0.500.51<m> τ=0.1 0.1 1 10 time t-3-2-101234<δω> Figure 1. (color online) Dynamics of the average magneti- zation (left panels) and uctuation eld (right panels) for a varyingparameter. Conditions are !0=<0;0;2 >, D= 50, s1=,m(0) =<1;0;0>,!(0) =<1;0;0>. Components of x,y,zare in black, red and green respectively. The norm is displayed in blue.The equations (1) are integrated globally with an ex- plicit 4thorder Runge-Kutta algorithm and a variable stepping scheme, with only a renormalization of the mag- netization at each step, in order to produce a precession and nutation motion consistent on the S2sphere. Better symplectic algorithms [45], that preserve the structure of the equations of motion, can be used, but they do not aect the conclusions drawn. What we observe here is that the average magneti- zationhmitends to align with the eective eld along thez-axis, by producing a dampened motion and a wrig- gling movement of the hmzicomponent, which is charac- teristic of a high frequency nutation eect, because of the nite values of and s1. When the suscep- tibilityis decreased, while keeping all the other pa- rameters xed, the internal precession eld, coming from the uctuations, dominates the natural precession eld !0, that increases the precession pulsation. We observe that increasing and reducing Msdoes indeed enhance the eects of the nutation term. Moreover when is large, the diusive term, that is generated by the noise, dominates the motion of the magnetization. A conse- quence is thathm(t)icannot stay constant even if, for all values of ,hm:mi= 1 by construction. When the time is long enough to capture the growing main com- ponent of the magnetization then hmialigns itself on h!i. Whenis small, in the transient regime, the uc- tuating eldh!icannot be suciently dampened and follows more closely the dynamics of the magnetization. For low values of the noise amplitude, the dynamics of the averageh!iis insensitive to the noise amplitude and its leading motion is described by h!ihmi. Whentakes values of O(1=!0), the leading motion ofhmiis given by the Gilbert equation of precession dhmi=dt(!0hdmi=dt)hmi, that produces in return a dampened motion of h!i. One conclusion of this study is identifying the appro- priate dimensionless combinations. Our results moti- vate dening the dimensionless quantities X0!0and X!. In terms of these te equations of motion 1 and 2 take the form (upon dening x=t=) dm dx= (X0+X)m dX dx=(Xm) +
(14) where
 sandhi(x)j(x0)i= 2D= ij(xx0). In the particular case where X0=0, i.e. when no external torque acts on the magnetization and with 6= 0, then hmim(0) is a constant of motion. Upon averaging over the noise realizations, the dynam- ics ofhXiis given byhXi= (X0m(0))ex+m(0). Thus the uctuating eld at equilibrium is given by hXi1=m(0) and no torque acts on the magnetiza- tion, keeping it constant over time. The gure 2 displays the dampened motion of mand Xas a function of the dimensionless time, for two con- gurations : sD= 0, i.e. without thermal noise, and6 sD= 50, for the same external eld X0=<0;0; > . The longitudinal behavior of the average magnetization is clearly visible by the decrease of the average norm khmik. Moreover because the average eom for Xis in- dependent of the noise amplitude, as depicted, this is not the case for the average magnetization, because there hXmi6=hXihmi. 0 10 20 30-1-0.500.51<m>ΩsD=0 ΩsD=50 0 10 20 30 t/τ-0.4-0.200.20.40.60.81<τδω> Figure 2. (color online) Dynamics of the average magneti- zation (up panel) and uctuation eld (down) for a varying sDparameter. Conditions are !0=<0;0; > ,= 1, m(0) =<1;0;0>,!(0) =<0;0;0>. Components of x, y,zare in black, red and green respectively. The norm is displayed in blue. IV. CONCLUSIONS AND OUTLOOK In this paper, we have shown that the mechanism of Gilbert damping of the precession, as well as the eects of nutation can be understood in terms of an eective inter- action between magnetic moments and the uctuationsof their eective elds, when the latter are described by colored noise in a systematic expansion in powers of the ratio of the correlation time of the noise to the period dened by the precession torque. We have identied a relation between the Gilbert damping parameter and the static (or spectral) inverse susceptibility of the material, with the contribution to a characteristic relaxation time, that can be assigned to magnon scattering mechanisms, in the relaxation time approximation. It is stressed that if it were possible to perform mea- surements that could resolve the contribution of the nu- tation loops, as they are superimposed on the usual pre- cession motion of the magnetic moments, it would be possible to nd which processes provide the dominant contribution leading to inertial damping, as recently been reported [1]. The relative sign between the Gilbert damping and the inertia term is negative as a consequence of the fact that these two terms represent successive contributions of the Taylor expansion. Therefore studies that assume that these terms have the same sign make additional assump- tions, that it would be very interesting to spell out. 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2021-03-08

Nutation has been recognized as of great significance for spintronics; but justifying its presence has proven to be a hard problem. In this paper we show that nutation can be understood as emerging from a systematic expansion of a kernel that describes the history of the interaction of a magnetic moment with a bath of colored noise. The parameter of the expansion is the ratio of the colored noise timescale to the precession period. In the process we obtain the Gilbert damping from the same expansion. We recover the known results, when the coefficients of the two terms are proportional to one another, in the white noise limit; and show how colored noise leads to situations where this simple relation breaks down, but what replaces it can be understood by the appropriate generalization of the fluctuation--dissipation theorem. Numerical simulations of the stochastic equations support the analytic approach. In particular we find that the equilibration time is about an order of magnitude longer than the timescale set by the colored noise for a wide range of values of the latter and we can identify the presence of nutation in the non-uniform way the magnetization approaches equilibrium.

Emerging magnetic nutation

2103.04787v3